cv.bib

@article{vonmoll2024complete,
  abstract = {In the Lady in the Lake scenario, a mobile agent, 𝐿, is pitted against an agent, 𝑀, who is constrained to move along the perimeter of a circle. 𝐿 is assumed to begin inside the circle and wishes to escape to the perimeter with some finite angular separation from 𝑀 at the perimeter. This scenario has, in the past, been formulated as a zerosum differential game wherein 𝐿 seeks to maximize terminal separation and 𝑀 seeks to minimize it. Its solution is well-known. However, there is a large portion of the state space for which the canonical solution does not yield a unique equilibrium strategy. This paper provides such a unique strategy by solving an auxiliary zero-sum differential game. In the auxiliary differential game, 𝐿 seeks to reach a point opposite of 𝑀 at a radius for which their maximum angular speeds are equal (i.e., the antipodal point). 𝐿 wishes to minimize the time to reach this point while 𝑀 wishes to maximize it. The solution of the auxiliary differential game is comprised of a Focal Line, a Universal Line, and their tributaries. The Focal Line tributaries' equilibrium strategy for 𝐿 is semi-analytic, while the Universal Line tributaries' equilibrium strategy is obtained in closed form.},
  author = {Von Moll, Alexander and Pachter, Meir},
  date = {2024-07-09},
  location = {Valladolid, Spain},
  journal = {International Symposium on Dynamic Games and Applications},
  publisher = {ISDG},
  title = {Complete Solution of the Lady in the Lake Scenario},
  url = {https://arxiv.org/abs/2401.14994},
  year = {2024},
}

@article{vonmoll2024basic,
  abstract = {This paper establishes a concise definition for an Engagement Zone which describes a region of space for which a Mobile Agent must alter its current motion strategy or otherwise risk engaging with a known Threat. In the event of an engagement, the Mobile Agent may have to actively evade the Threat in order to guarantee survival. Provided the Engagement Zone definition, several fundamental engagement models are described and their corresponding Engagement Zones are derived. In each of the engagement models, the Mobile Agent is treated as an Evader; they include pursuit-evasion against a range-limited Pursuer moving with simple motion and faster speed, a variation in which the Pursuer is slower, and a turret-evasion scenario against a range-limited, turn-rate-limited Turret. The slow Pursuer Engagement Zone is applied to a path planning scenario wherein the Mobile Agent must reach a desired target in minimum time while navigating outside the dynamically changing Engagement Zone. Numerical results demonstrate the advantage of Engagement Zone-based navigation over other path plans based on circumnavigation.},
  author = {Von Moll, Alexander and Weintraub, Isaac},
  date = {2024-06-01},
  note = {Submitted for Review},
  location = {None},
  journal = {Journal of Aerospace Information Systems},
  publisher = {AIAA},
  title = {Basic Engagement Zones},
  url = {https://arxiv.org/abs/2311.06165},
  year = {2024},
}

@article{bajaj2024shortest,
  abstract = {In this work, we formulate a novel planar motion planning problem of a Dubins-Laser system that consists of a Dubins vehicle with an attached controllable laser. The vehicle moves with unit speed and the laser, having a finite range, can rotate in a clockwise or anti- clockwise direction with a bounded angular rate. From an arbitrary initial position and orientation of the Dubins-Laser system, the objective is to steer the system so that a given static target is within the range of the laser and the laser is oriented at it in minimum time. We characterize multiple properties of the optimal path and establish that the optimal path for the Dubins-laser system is one out of the total 13 candidates. Finally, we provide insightful numerical plots to illustrate the properties characterized in this work.},
  author = {Bajaj, Shivam and Bhargav, Jha and Bopardikar, Shaunak D. and Von Moll, Alexander and Casbeer, David},
  date = {2024-06-30},
  note = {Submitted for Review},
  location = {None},
  journal = {Transactions on Automatic Control},
  publisher = {IEEE},
  title = {Shortest Trajectory of a Dubins Vehicle with a Controllable Laser},
  year = {2024},
}

@article{jha2024linear,
  abstract = {None},
  author = {Jha, Bhargav and Bopardikar, Shaunak D. and Von Moll, Alexander and Casbeer, David},
  date = {2024-06-30},
  note = {Submitted for Review},
  location = {None},
  journal = {Journal of Guidance, Dynamics, and Control},
  publisher = {AIAA},
  title = {Linear Quadratic Guidance Law for Joint Motion Planning of a Pursuer-Turret Assembly},
  year = {2024},
}

@inproceedings{maity2024optimal,
  abstract = {This paper investigates a partial-information pur- suit evasion game in which the Pursuer has a limited-range sensor to detect the Evader. Given a fixed final time, we derive the optimal evasion strategy for the Evader to maximize its distance from the pursuer at the end. Our analysis reveals that in certain parametric regimes, the optimal Evasion strategy involves a `risky' maneuver, where the Evader's trajectory comes extremely close to the pursuer's sensing boundary before moving behind the Pursuer. Additionally, we explore a special case in which the Pursuer can choose the final time. In this scenario, we determine a (Nash) equilibrium pair for both the final time and the evasion strategy.},
  author = {Maity, Dipankar and Von Moll, Alexander and Shishika, Daigo and Dorothy, Michael},
  date = {2024-07-08},
  note = {Accepted},
  location = {Toronto, Canada},
  booktitle = {American Control Conference},
  title = {Optimal Evasion From a Sensing Limited Pursuer},
  year = {2024},
}

@inproceedings{mora2024escape,
  abstract = {This paper explores a multi-agent containment problem, where a fast evader, modeled having constant speed and using constant heading, attempts to escape a circular containment region that is orbited by a slower pursuer with a nonzero capture radius. The pursuer is constrained to move along the edge of the containment region and seeks to capture the evader. This paper presents an in-depth analysis of this pursuer-evader containment scenario. First, multiple types of capture conditions for a single- pursuer case are analyzed defining the worst-case initial position for the pursuer. Second, a parametric study is performed to demonstrate the effects of speed ratio, capture radius, and initial location of the evader. Finally, a reachability analysis is performed to investigate the viable escape headings and reachable regions by the evader. This work provides a foundation for the analysis of escape under more general evader inputs as well as a multiple-pursuer version of the scenario.},
  author = {Mora, Braulio and Von Moll, Alexander and Weintraub, Isaac and Casbeer, David W. and Chakravarthy, Animesh},
  date = {2024-07-08},
  note = {None},
  location = {None},
  booktitle = {None},
  title = {Escape from an Orbiting Pursuer with a Nonzero Capture Radius},
  year = {2024},
}

@misc{weintraub2024deterministic,
  abstract = {Previous methods for avoiding dynamic engagement zones leverage the calculus of variations to obtain optimal path plans for avoiding risk and reaching a desired target location [1,2]. While this method is deterministic; it scales poorly as the number of engagement zones increases. Furthermore, it is inherently sensitive to initial guesses and finding local minimums can be problematic. This talk presents a novel approach that uses deterministic approaches to obtain a set of feasible flight plans for reaching the desired destination and reports when no such plan exists. Furthermore, it uses an efficient scheme for feeding initial guesses to a nonlinear program solver that provides flyable path plans that mitigate risk. Lastly, a tool is developed that measures the risk of the flight path for avoiding a large field of engagement zones, providing useful information to improve battle management.
The solution process begins with using the location of each weapon engagement zone and associated model to obtain a Voronoi partition of the environment. This partitioning is used to construct a feasibility graph signifying navigation through each pair of weapon engagement zones. Once constructed each edge of the graph borders two opposing weapon engagement zones for which the target vehicle aims to navigate between. If the pair of engagement zones can not be penetrated due to the distance and size of those zones; then the associated edge of the graph is deleted. This process is iteratively performed until each edge has been evaluated. Upon completion of the feasibility graph, path plans are attained using graph-based search to minimize the time of arrival. Next, the provided solution is fed to a nonlinear program solver that smooths the solution or solutions from the feasibility path according to the vehicle dynamics. The resulting solutions are analyzed through a simplified simulation to assess risk and time of arrival. 
Presented are the methods used to solve this navigational problem, results of the simulated trajectories, and future research directions.
[1] Isaac E. Weintraub, Alexander Von Moll, Christian A. Carrizales, Nicholas Hanlon and Zachariah E. Fuchs. "An Optimal Engagement Zone Avoidance Scenario in 2-D," AIAA 2022-1587. AIAA SciTech 2022 Forum. January 2022.
[2] Optimal Trajectories for Aircraft Avoidance of Multiple Weapon Engagement Zones, Patrick M. Dillon, Michael D. Zollars, Isaac E. Weintraub, and Alexander Von Moll, Journal of Aerospace Information Systems 0 0:0, 1-6.},
  address = {Laurel, MD},
  author = {Weintraub, Isaac and Wolek, Artur and Von Moll, Alexander and Casbeer, David and Manyam, Satyanarayana Gupta},
  booktitle = {AIAA Defense},
  date = {2024-04-16},
  month = {4},
  title = {Deterministic Risk-Aware Path Planning Around Multiple Threats},
  year = {2024},
}

@misc{vonmoll2024digitally,
  abstract = {This talk motivates and presents a robust and novel method for improved missile guidance. The governing guidance theories as well as the benefits and trade-offs are described in detail along with simulations.},
  address = {Laurel, MD},
  author = {Von Moll, Alexander and Weintraub, Isaac},
  booktitle = {AIAA Defense},
  date = {2024-04-16},
  month = {4},
  title = {Digitally Enhanced Aim-Point for Capture for Mobile Targets},
  year = {2024},
}

@misc{milutinovic2024stochastic,
  abstract = {Description: Recent efforts in planning paths for a vehicle around dynamic engagement zones have been based upon optimal control solved via nonlinear program comprised of a pseudospectral direct collocation of the dynamics and constraints. The engagement zone(s) rotate and scale based upon the position and aspect angle of the vehicle, respectively. Initially, the problem setup was based upon a vehicle navigating around a stationary, gimballed threat to reach a specified goal position; the entire engagement took place on a 2D plane [1]. As a follow-on effort, two dynamic engagement zones were considered along with an added vehicle turning-rate constraint [2]. In all of these works, the solution consisted of a single optimal trajectory associated with a particular initial condition. In this presentation, three extensions are considered and demonstrated: 1) stochasticity via the inclusion of a Wiener process (Brownian motion) in the dynamics, 2) computation of an optimal feedback control law, and 3) multiple engagement zones (even beyond 2). The stochastic process may represent uncertainty due to wind, other environmental factors, or position measurement errors. In the case of a single engagement zone the stochastic process may represent the vehicle's uncertainty about the location of the engagement zone. Regarding the feedback control law, this is a much more powerful result than a single path plan as it requires that the control law be computed once, a priori, and can then be used for any position in the state space, whereas the existing path plan-based solutions require re-computation for each new position in the state space. Having the feedback control law and Value function everywhere is also illustrative of the overall behavior of optimal trajectories \textendash{} e.g., some optimal trajectories fly around while others fly between neighboring engagement zones. Lastly, the methodology can be applied to larger numbers of engagement zones. When based on Value iteration [3], the Value function computation scales at least sub-exponentially with the number of engagement zones.
[1] Isaac E. Weintraub, Alexander Von Moll, Christian A. Carrizales, Nicholas Hanlon and Zachariah E. Fuchs. "An Optimal Engagement Zone Avoidance Scenario in 2-D," AIAA 2022-1587. AIAA SciTech 2022 Forum. January 2022.
[2] Optimal Trajectories for Aircraft Avoidance of Multiple Weapon Engagement Zones, Patrick M. Dillon, Michael D. Zollars, Isaac E. Weintraub, and Alexander Von Moll, Journal of Aerospace Information Systems 0 0:0, 1-6.
[3] Thrun, S., Burgard, W., and Fox, D., 2005, Probabilistic Robotics, The MIT Press, Cambridge, MA. 

Notes: Alexander Von Moll from Air Force Research Laboratory will be presenting on behalf of the author due to security restrictions in place at the AIAA Defense.},
  address = {Laurel, MD},
  author = {Milutinovi\'{c}, Dejan and Von Moll, Alexander and Weintraub, Isaac and Casbeer, David},
  booktitle = {AIAA Defense},
  date = {2024-04-16},
  month = {4},
  title = {Stochastic Risk-Aware Path Planning Around Multiple Threats},
  year = {2024},
}

@misc{vonmoll2019game,
  address = {Williamsburg, VA},
  author = {Von Moll, Alexander},
  booktitle = {Aerospace Control and Guidance Systems Committee},
  date = {2019-08-16},
  month = {8},
  title = {Game Theoretic Control Strategies in Adversarial Environments},
  year = {2019},
}

@misc{weintraub2023automatic,
  address = {Laurel, MD},
  author = {Weintraub, Isaac E. and Von Moll, Alexander L. and Hanlon, Nicholas},
  booktitle = {AIAA Defense},
  date = {2023-04-11},
  month = {4},
  title = {Automatic Engagement Zone Avoidance Theory and Practice},
  year = {2023},
}

@inproceedings{vonmoll2024pure,
  abstract = {This investigation seeks to obtain estimates on the intercept time associated with a fast Pursuer implementing Pure Pursuit against a Target whose trajectory is a circular arc. The analytical solution of the kinematic equations of motion is not feasible and thus the baseline approach for computing the intercept time is numerical simulation, i.e., numerical integration of the kinematics. This method can obtain the intercept time in (essentially) real time for a single Pursuer against a single Target. Here, the goal is to estimate the intercept time using orders of magnitude less computational time. Although not explicitly explored in this work, the purpose of such approximations is to make feasible the rapid evaluation of Pursuer-to-Target assignments in a massive many-on-many engagement scenario wherein the intercept time may be a key measure. Several approximation methods are proposed here and their merits (or lack thereof) are demonstrated through simulation. Ultimately, assuming the Target is not actually turning may be suitable when its turn radius is relatively large, while assuming intercept occurs at the closest point on the turn circle may be suitable when its turn radius is relatively small.},
  author = {Von Moll, Alexander and Casbeer, David W. and Weintraub, Isaac E. and Pachter, Meir},
  date = {2024-01-30},
  doi = {10.2514/6.2024-0956},
  note = {Accepted},
  location = {Orlando, FL},
  booktitle = {AIAA SciTech},
  title = {Pure Pursuit of a Target on a Circular Trajectory},
  url = {www.isaacew.com/publication/vonmoll2024pure/vonmoll2024pure.pdf},
  year = {2024},
}

@inproceedings{weintraub2024virtual,
  abstract = {This paper considers an M-pursuer N-evader scenario involving virtual targets. The virtual targets serve as an intermediary target for the pursuers, allowing the pursuers to delay their final assignment to the evaders. However, upon reaching the virtual target, the pursuers must decide which evader to capture. It is assumed that there are more pursuers than evaders and that the pursuers are faster than the evaders. The objective is two-part: first, assign each pursuer to virtual target and evader such that the the pursuer team's energy is minimized, and second, choose the the virtual targets' locations for this minimization problem. The approach taken is to consider the Apollonius geometry between each pursuer's virtual target location and each evader. Using the constructed Apollonius circles, the pursuer's travel distance and maneuver at a virtual target are obtained. These metrics serve as a gauge for the total energy required to capture a particular evader and are used to solve the joint virtual target selection and pursuer-evader assignment problem. This paper provides a mathematical definition of this problem, the solution approach taken, and an example.},
  author = {Weintraub, Isaac E. and Von Moll, Alexander and Casbeer, David W. and Manyam, Satyanarayana G.},
  date = {2024-01-30},
  doi = {10.2514/6.2024-0123},
  note = {Accepted},
  location = {Orlando, FL},
  title = {Virtual Target Selection for a Multiple-Pursuer Multiple-Evader Scenario},
  url = {https://arxiv.org/abs/2305.19399},
  year = {2024},
}

@inproceedings{weintraub2023range-limited,
  abstract = {In pursuit-evasion the objective of the pursuer is to capture the evader. In this work, the faster pursuer is modeled to have limited range and therefore optimal strategies for the pursuer and evader change. This paper describes the optimal strategies and nuances that appear for point-capture or when the pursuer is endowed with a non-zero capture radius.},
  author = {Weintraub, Isaac and Von Moll, Alexander and Pachter, Meir},
  booktitle = {2023 National Aerospace and Electronics Conference (NAECON)},
  date = {2023-08-28},
  doi = {10.1109/NAECON58068.2023.10365808},
  location = {Dayton, OH},
  title = {Range-Limited Pursuit-Evasion},
  year = {2023},
}

@article{bajaj2023competitive,
  abstract = {We consider perimeter defense problem in a planar conical environment with two cooperative heterogeneous defenders, i.e., a turret and a mobile vehicle, that seek to defend a concentric perimeter against mobile intruders. Arbitrary numbers of intruders are released at the circumference of the environment at arbitrary time instants and locations. Upon release, they move radially inwards with fixed speed towards the perimeter. The defenders are heterogeneous in terms of their motion and capture capabilities. Specifically, the turret has a finite engagement range and can only turn (clockwise or anti-clockwise) in the environment with fixed angular rate whereas, the vehicle has a finite capture radius and can move in any direction with unit speed. We present a competitive analysis approach to this perimeter defense problem by measuring the performance of multiple cooperative online algorithms for the defenders against arbitrary inputs, relative to an optimal offline algorithm that has information about the entire input sequence in advance. Specifically, we establish necessary conditions on the parameter space to guarantee finite competitiveness of any online algorithm. We then design and analyze four cooperative online algorithms and characterize parameter regimes in which they have finite competitive ratios. In particular, our first two algorithms are 1-competitive in specific parameter regimes, our third algorithm exhibits different competitive ratios in different regimes of problem parameters, and our fourth algorithm is 1.5-competitive in specific parameter regimes. Finally, we provide multiple numerical plots in the parameter space to reveal additional insights into the relative performance of our algorithms.},
  author = {Bajaj, Shivam and Bopardikar, Shaunak D. and Von Moll, Alexander and Torng, Eric and Casbeer, David W.},
  date = {2023-02-27},
  doi = {10.3389/fcteg.2023.1128597},
  journal = {Frontiers in Control Engineering},
  title = {Competitive Perimeter Defense with a Turret and Mobile Vehicle},
  year = {2023},
}

@article{dillon2023optimal,
  abstract = {The work herein expresses an optimal control problem for an air vehicle to reach a desired location through two dynamic keep out zones. In today's contested world, there are many instances where air vehicles are denied a portion of the airspace when trying to reach a desired target. Under these conditions the objective is to find the optimal route to the desired location while obeying path constraints. In this work, the path constraint is defined as the enemy's ground weapon engagement zone, and is established in a way that simulates an anti-aircraft defense system. The vehicle's objective is to reach the desired location in minimum time while completely avoiding the engagement zone. This optimal control problem is considered in the Cartesian plane, has a fixed final state, and free final time. The dynamics of the aircraft are a function of the aircraft's velocity and heading. The velocity of the aircraft is assumed to be constant and the vehicle's control is its heading rate. The engagement zone, which acts as a path constraint, is dynamic in that its effective range is a function of the engagement zone's instantaneous line of sight angle and the aircraft's instantaneous bearing angle. The desired final location is defined to be a point opposite the engagement zone from the aircraft's initial location. Several scenarios are conducted to show path trajectories through multiple dynamic engagement zones. Results illustrate how aircraft dynamic maneuvers can minimize total flight time to a target while maintaining a safe distance from enemy threats. Further, while investigating multiple engagement zones, it is shown that an aircraft can maintain safe separation from multiple overlapping threats by manipulating relative bearing angles and the aircraft's heading.},
  author = {Dillon, Patrick M. and Zollars, Michael D. and Weintraub, Isaac E. and Von Moll, Alexander},
  date = {2023-06-12},
  doi = {10.2514/1.I011224},
  issue = {8},
  pages = {520--525},
  journal = {Journal of Aerospace Information Systems},
  publisher = {AIAA},
  title = {Optimal Trajectories for Aircraft Avoidance of Multiple Weapon Engagement Zones},
  volume = {20},
  year = {2023},
}

@article{vonmoll2023turret,
  abstract = {In this paper, a zero-sum differential game is formulated and solved in which a mobile Evader seeks to escape from within a circle at whose origin lies a stationary, turn-constrained Turret. The scenario is a variant of the famous Lady in the Lake game in which the shore-constrained Pursuer has been replaced with the Turret. As in the former, it is assumed that the Turret's maximum angular rate is greater than the linear velocity of the Evader. Since two outcomes are possible, a Game of Kind arises - either the Evader wins by reaching the perimeter of the circle, or the Turret wins by aligning with the latter's position. A barrier surface partitions the state space into two regions corresponding to these two outcomes and a Game of Degree is solved within each region. The solutions to the Games of Degree are comprised of the Value functions (i.e., the equilibrium value of the cost/utility as a function of the state) and the saddle-point equilibrium control policies for the two players. Like the Lady in the Lake game, the equilibrium policy of the Evader is not uniquely de ned where it has angular rate advantage over the Turret. Unlike the Lady in the Lake game, the losing region for the Evader is present for all speed ratios, and there is an additional semi-permeable surface separating center- and shore-bound Evader trajectories. The solution depends heavily upon the speed ratio of the agents; in particular, there are two speed ratio regimes with distinctive solution structures.},
  address = {New York},
  author = {Von Moll, Alexander and Fuchs, Zachariah and Shishika, Daigo and Maity, Dipankar and Dorothy, Michael and Pachter, Meir},
  date = {2023-09-01},
  doi = {10.3934/jdg.2023012},
  issue = {2},
  note = {Presented at the 19th ISDG.},
  pages = {100--114},
  journal = {Journal of Dynamics and Games},
  publisher = {American Institute of Mathematical Sciences},
  title = {Turret Escape Differential Game},
  url = {https://avonmoll.github.io/files/turret-escape-differential-game.pdf},
  volume = {11},
  year = {2023},
}

@article{bajaj2024multi-vehicle,
  abstract = {We consider a perimeter defense problem in a planar conical environment in which M identical vehicles, each having a finite capture radius, seek to defend a concentric perimeter from mobile intruders. The intruders are released at the circumference of the environment at arbitrary times and in any number. Upon release, each intruder moves radially toward the perimeter with fixed speed. We provide a worst-case analysis of this problem. Specifically, we present a competitive analysis approach to this problem by measuring the performance of decentralized and cooperative online algorithms for the vehicles against arbitrary inputs, relative to an optimal offline algorithm that has information about entire intruder release sequence in advance. In particular, apart from the minimum number of vehicles required to capture all intruders, we establish a necessary condition on the parameter space to guarantee finite competitive- ness of any algorithm. We then design and analyze three decen- tralized and two cooperative online algorithms and characterize parameter regimes in which they have finite competitive ratios. Specifically, our first two decentralized algorithms are provably 1, and 2-competitive, respectively, whereas our third decentralized algorithm exhibit different competitive ratios in different regimes of problem parameters. Similarly, our first cooperative algorithm is 1.5-competitive and our second cooperative algorithm exhibits different competitive ratios in different regimes of problem parameters. We then provide multiple numerical plots in the parameter space to reveal additional insights into the relative performance of our algorithms and discuss an extension to the case of heterogeneous vehicles.},
  address = {New York},
  author = {Bajaj, Shivam and Bopardikar, Shaunak and Torng, Eric and Von Moll, Alexander and Casbeer, David W.},
  date = {2024-01-10},
  doi = {10.1109/TRO.2024.3351556},
  journal = {Transactions on Robotics and Automation},
  publisher = {IEEE},
  title = {Multi-vehicle Perimeter Defense in Conical Environments},
  year = {2024},
}

@inproceedings{bajaj2023perimeter,
  abstract = {We consider a perimeter defense problem in a planar conical environment comprising a single turret that has a finite range and non-zero service time. The turret seeks to defend a concentric perimeter against N \geq{} 2 intruders. Upon release, each intruders move radially towards the perimeter with a fixed speed. To capture an intruder, the turret's angle must be aligned with that of the intruder's angle and must spend a specified capture time at that orientation. We present an online and an offline approach to this optimal problem. Specifically, for the offline approach, we establish that for general parameter regimes, this problem is equivalent to solving a Travelling Repairperson Problem with Time Windows (TRP- TW). We then identify specific parameter regimes in which there is a polynomial time algorithm that maximizes the number of intruders captured. For the online approach we present a competitive analysis technique in which we establish a fundamental guarantee on the existence of at best (N − 1)- competitive algorithms and design two online algorithms that are provably 1 and 2-competitive in specific parameter regimes.},
  author = {Bajaj, Shivam and Bopardikar, Shaunak and Von Moll, Alexander and Torng, Eric and Casbeer, David},
  date = {2023-05-31},
  doi = {10.23919/ACC55779.2023.10155838},
  location = {San Diego, USA},
  location = {San Diego, USA},
  booktitle = {American Control Conference},
  publisher = {IEEE},
  title = {Perimeter Defense using a Turret with Finite Range and Startup Times},
  year = {2023},
}

@article{weintraub2023surveillance,
  abstract = {The maximum surveillance of a target which is holding course is considered, wherein an observer vehicle aims to maximize the time that a faster target remains within a fixed-range of the observer. This entails two coupled phases: an approach phase and observation phase. In the approach phase, the observer strives to make contact with the faster target, such that in the observation phase, the observer is able to maximize the time where the target remains within range. Using Pontryagin's Minimum Principle, the optimal control laws for the observer are found in closed-form. Example scenarios highlight various aspects of the engagement.},
  address = {New York},
  author = {Weintraub, Isaac and Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Pachter, Meir},
  date = {2023-05-31},
  doi = {10.1109/TAES.2023.3237129},
  note = {None},
  journal = {Transactions on Aerospace \textbackslash{}\& Electronic Systems},
  publisher = {IEEE},
  title = {Surveillance of a Faster Fixed-Course Target},
  url = {http://arxiv.org/abs/2209.11289},
  year = {2023},
}

@inproceedings{pourghorban2022target,
  abstract = {We consider a variant of the target defense problem where a single defender is tasked to capture a sequence of incoming intruders. The intruders' objective is to breach the target boundary without being captured by the defender. As soon as the current intruder breaches the target or gets captured by the defender, the next intruder appears at a random location on a fixed circle surrounding the target. Therefore, the defender's final location at the end of the current game becomes its initial location for the next game. Thus, the players pick strategies that are advantageous for the current as well as for the future games. Depending on the information available to the players, each game is divided into two phases: partial information and full information phase. Under some assumptions on the sensing and speed capabilities, we analyze each agent's strategy in both phases. We derive equilibrium strategies for both the players to optimize the capture percentage using the notions of engagement surface and capture circle. We quantify the percentage of capture for both finite and infinite sequences of incoming intruders.},
  author = {Pourghorban, Arman and Dorothy, Michael and Shishika, Daigo and Von Moll, Alexander and Maity, Dipankar},
  date = {2022-12-14},
  doi = {10.1109/CDC51059.2022.9992425},
  location = {Cancun, MX},
  booktitle = {Conference on Decision and Control},
  publisher = {IEEE},
  title = {Target Defense against Sequentially Arriving Attackers},
  url = {https://arxiv.org/pdf/2212.06628.pdf},
  year = {2022},
}

@article{manyam2022shortest,
  abstract = {We present a path planning problem for a pursuing Unmanned Aerial Vehicle (UAV) to intercept a target traveling on a circle. The target is cooperative, and its position, heading and speed are precisely known. The pursuing UAV has nonholonomic motion constraints, and therefore the path traveled must satisfy the minimum turn radius constraints. We consider the class of Dubins paths as candidate solutions, and analyze the characteristics of the six modes of Dubins paths where the final position is restricted to lie on the target circle with heading in the tangential direction of the circle. For each Dubins mode, we derive the feasibility limits, discontinuities and local extrema. Using this analysis the intercepting paths are found by a systematic bisection search in the feasible regions of each of the Dubins modes. We prove that the algorithm finds the optimal (shortest length) intercepting path if it is a Dubins path. If the shortest intercepting path is not a Dubins path for any given instance, the algorithm finds a tight lower bound and an upper bound to the optimal solution.},
  author = {Manyam, Satyanarayana Gupta and Casbeer, David W. and Von Moll, Alexander and Fuchs, Zachariah},
  date = {2022-06-01},
  doi = {10.2514/1.G005748},
  journal = {Journal of Guidance, Control, and Dynamics},
  publisher = {AIAA},
  title = {Shortest Dubins Paths to Intercept a Target Moving on a Circle},
  year = {2022},
}

@article{vonmoll2022pure,
  abstract = {The study of pursuit curves is valuable in the context of air-to-air combat as pure pursuit guidance (heading directly at the target) is oftentimes implemented. The problems considered in this paper concern a Pursuer, implementing pure pursuit (i.e., line of sight guidance), chasing an Evader who holds course. Previous results are applicable to the case in which capture is defined as the two agents being coincident, i.e., point capture. The focus here is on obtaining results for the more realistic case  where the pursuer is endowed with an effector whose range is finite. The scenario in which the Evader begins inside the Pursuer's effector range is also considered (i.e., escape from persistent surveillance, among other potential applications). Questions herein addressed include: does the engagement end in head-on collision or tail chase, will the Evader be captured or escape, what is the minimum distance the Pursuer will attain, for two Pursuers, is simultaneous capture/escape optimal and, if so, what is the optimal heading for the Evader (max time to capture, or min time to escape), and the feasibility for a fast Evader to escape from many Pursuers. Where possible, closed-form, analytic results are obtained, otherwise attention is given to computability with an eye towards real-time, on-board implementation.},
  author = {Von Moll, Alexander and Pachter, Meir and Fuchs, Zachariah},
  date = {2022-11-02},
  doi = {10.1007/s13235-022-00481-9},
  pages = {961\textendash{}-979},
  journal = {Dynamic Games and Applications},
  title = {Pure Pursuit with an Effector},
  url = {https://avonmoll.github.io/files/pure-pursuit-with-an-effector.pdf},
  year = {2022},
}

@thesis{vonmoll2022skirmish-level,
  abstract = {Supremacy in armed conflict comes not merely from superiority in capability or numbers but from how assets are used, down to the maneuvers of individual vehicles and munitions. This document outlines a research plan focused on skirmish-level tactics to militarily relevant scenarios. Skirmish-level refers to both the size of the adversarial engagement -- generally one vs. one, two vs. one, and/or one vs. two -- as well as the fact that the goal or objective of each team is well-established. The problem areas include pursuit-evasion and target guarding, either of which may be considered as sub-problems within military missions such as air-to-air combat, suppression/defense of ground-based assets, etc. In most cases, the tactics considered are comprised of the control policy of the agents (i.e., their spatial maneuvers), but may also include role assignment (e.g, whether to act as a decoy or striker) as well as discrete decisions (e.g., whether to engage or retreat). Skirmish-level tactics are important because they can provide insight into how to approach larger scale conflicts (many vs. many, many objectives, many decisions). Machine learning approaches such as reinforcement learning and neural networks have been demonstrated to be capable of developing controllers for large teams of agents. However, the performance of these controllers compared to the optimal (or equilibrium) policies is generally unknown. Differential Game Theory provides the means to obtain a rigorous solution to relevant scenarios in the form of saddle-point equilibrium control policies and the min/max (or max/min) cost / reward in the case of zero-sum games. When the equilibrium control policies can be obtained analytically, they are suitable for onboard / real-time implementation. Some challenges associated with the classical Differential Game Theory approach are explored herein. These challenges arise mainly due to the presence of singularities, which may appear in even the simplest differential games. The utility of skirmish-level solutions is demonstrated in (i) the multiple pursuer, single evader differential games, (ii) multi-agent turret defense scenarios, and (iii) engage or retreat scenarios. In its culmination, this work contributes differential game and optimal control solutions to novel scenarios, numerical techniques for computing singular surfaces, approximations for computationally-intensive solutions, and techniques for addressing scenarios with multiple stages or outcomes.},
  author = {Von Moll, Alexander},
  date = {2022-03-09},
  doi = {10.13140/RG.2.2.31559.78242},
  title = {Skirmish-Level Tactics via Game-Theoretic Analysis},
  school = {University of Cincinnati},
  url = {https://avonmoll.github.io/files/dissertation.pdf},
  year = {2022},
}

@article{vonmoll2022lock-evade,
  abstract = {In this paper we model and analyze a scenario in the two-dimensional plane involving a mobile Attacker and stationary Defender. There are two possibilities for termination: the Attacker can collide with the Defender (engagement) or maneuver to a safe zone away from the Defender (retreat). The Defender is equipped with a directional turret, which it can rotate with a bounded rate. If the turret is aligned with the Attacker's position, the Defender has a lock on the Attacker and may choose to fire on the Attacker. Thus, whether engaging or retreating, the Attacker has incentive to evade the turret's line of sight and thereby avoid being locked-on. In the case of retreat, if lock-on occurs the Defender cooperates with the Attacker by withholding fire to allow the Attacker to retreat. However, if the Attacker chooses to engage and lock-on occurs, the Defender will open fire on the Attacker. We model the scenario as a set of differential games with different cost functionals depending on the type of termination. The agents are assumed to have full state information. In the case that the Defender can lock onto the Attacker the pre- and post-lock portions of the game are solved individually and combined together. Otherwise, the Attacker can terminate the game prior to lock-on and the game is comprised of a single stage. The equilibrium strategies are derived in each case, and a partitioning of the state space wherein a particular goal selection is optimal is constructed.},
  author = {Von Moll, Alexander and Fuchs, Zachariah},
  date = {2022-10-01},
  note = {In preparation},
  journal = {None},
  publisher = {IEEE},
  title = {A Lock-Evade, Engage or Retreat Game},
  year = {2022},
}

@article{dorothy2021one,
  abstract = {This paper investigates obstacle-free simple motion pursuit-evasion problems where the pursuer is faster and game termination is point capture. It is well known that the interior of the Apollonius Circle (AC) is the evader's dominance region, however, it was unclear whether the evader could reach outside the initial AC without being captured. We construct a pursuit strategy that guarantees the capture of an evader within an arbitrarily close neighborhood of the initial AC. The pursuer strategy is derived by reformulating the game into a nonlinear control problem, and the guarantee holds against any admissible evader strategy. Our result implies that the evader can freely select the capture location, but only inside the initial AC. Therefore, a class of problems, including those where the payoff is determined solely based on the location of capture, are now trivial.},
  archiveprefix = {arXiv},
  author = {Dorothy, Michael and Maity, Dipankar and Shishika, Daigo and Von Moll, Alexander},
  date = {2021-11-17},
  eprint = {2111.09205},
  journal = {Automatica},
  note = {Accepted},
  primaryclass = {math.OC},
  publisher = {IFAC},
  title = {One Apollonius Circle is Enough for Many Pursuit-Evasion Games},
  url = {https://arxiv.org/abs/2111.09205},
  year = {2021},
}

@inproceedings{bajaj2022competitive,
  abstract = {This work addresses a perimeter defense problem in a planar conical environment in which a single vehicle, having a finite capture radius, aims to defend a concentric perimeter from mobile intruders. The intruders are arbitrarily released at the circumference of the environment and move radially toward the perimeter with fixed speed. We present a competitive analysis approach to this problem by measuring the performance of multiple online algorithms for the vehicle against arbitrary inputs relative to an optimal offline algorithm that has access to all future inputs. In particular, we first establish a necessary condition on the parameter space to guarantee finite competitiveness of any algorithm and then characterize parameter regime in which one must expect a competitive ratio of at least 2. We then design and analyze three online algorithms and characterize parameter regimes for which they have finite competitive ratios. Specifically, our first two algorithms are provably 1, and 2-competitive, respectively, whereas our third algorithm exhibits a finite competitive ratio that depends on the problem parameters. Finally, we provide numerous parameter space plots providing insights into the relative performance of our algorithms.},
  author = {Bajaj, Shivam and Torng, Eric and Bopardikar, Shaunak D. and Von Moll, Alexander and Weintraub, Isaac and Garcia, Eloy and Casbeer, David W.},
  date = {2022-06-08},
  doi = {10.1109/CDC51059.2022.9993007},
  booktitle = {Conference on Decision and Control},
  title = {Competitive Perimeter Defense of Conical Environments},
  url = {https://arxiv.org/pdf/2110.04667.pdf},
  year = {2022},
}

@inproceedings{weintraub2022optimal,
  abstract = {In this paper, an optimal control problem is considered where a target vehicle aims to reach a desired location in minimum time while avoiding a dynamic engagement zone. Using simple motion, four potential approaches are considered. First, the min-time strategy which ignores the engagement zone is posed and solved. Second, the min-time strategy which avoids the engagement zone entirely is considered. Third, the min-time strategy which allows for some time in the engagement zone; but, still strives to stay away from the center of the engagement zone is posed. Lastly, a fixed final-time strategy is considered, wherein the target tries to avoid the engagement zone; but, is required to arrive at the desired location at a specific time. Using a nonlinear program solver, the optimal strategies are numerically solved. From the results of the numeric solutions, the optimal strategies are discussed and comparisons are drawn.},
  author = {Weintraub, Isaac E. and Von Moll, Alexander and Carrizales, Christian and Hanlon, Nicholas and Fuchs, Zachariah},
  date = {2022-01-03},
  doi = {10.2514/6.2022-1587},
  location = {San Diego},
  booktitle = {Scitech},
  publisher = {AIAA},
  title = {An Optimal Engagement Zone Avoidance Scenario in 2-D},
  url = {https://arxiv.org/pdf/2111.05904.pdf},
  year = {2022},
}

@article{vonmoll2022turret-runner-penetrator,
  abstract = {A scenario is considered in which two cooperative Attackers aim to infiltrate a circular target guarded by a Turret.  The engagement plays out in the two dimensional plane; the holonomic Attackers have the same speed and move with simple motion and the Turret is stationary, located at the target circle's center, and has a bounded turn rate.  When the Turret's look angle is aligned with an Attacker, that Attacker is neutralized.  In this paper, we focus on a region of the state space wherein only one of the Attackers is able to reach the target circle -- and even then, only with the help of its partner Attacker.  The Runner distracts the Turret and ends up being neutralized in order that the Penetrator can be guaranteed to hit the target circle.  We formulate the Turret-Runner-Penetrator scenario as a differential game over the Value of the subsequent game of min/max terminal angle which takes place between the Turret and Penetrator once the Runner has been neutralized.  The solution to the Game of Degree, including equilibrium Turret, Runner, and Penetrator strategies, as well as the Value function are given.  The case in which the Penetrator can reach the target before the Turret can neutralize the Runner is formulated and solved.  Finally, the assumption of a priori defined roles/goals is relaxed and the minimum of the solutions to the two fixed-role games is shown to be a Global Stackelberg Equilibrium.},
  author = {Von Moll, Alexander and Shishika, Daigo and Fuchs, Zachariah and Dorothy, Michael},
  date = {2022-05-23},
  doi = {10.1109/TAES.2022.3176599},
  issue = {6},
  journal = {Transactions on Aerospace \textbackslash{}\& Electronic Systems},
  pages = {5687--5702},
  publisher = {IEEE},
  title = {The Turret-Runner-Penetrator Differential Game with Role Selection},
  url = {https://avonmoll.github.io/files/trpdg-with-role-selection.pdf},
  volume = {58},
  year = {2022},
}

@inproceedings{weintraub2021engagement,
  abstract = {This paper considers a three agent scenario consisting of a pursuer, evader, and a defender. The pursuer's objective is to capture the non-maneuvering evader in minimum time while a defender aims at keeping the pursuer inside his circular engagement zone for as long as possible; the pursuer is considered to be faster than both the evader and the defender.  Using the calculus of variations the optimal control problem for the defender which maximizes the amount of time that the pursuer is inside his area of effect is posed and solved. In the event that the evader is captured by the pursuer before the pursuer escapes the area of effect of the defender, some suboptimal strategies of the defender provide equivalent contact time. The derivation of the headings which the defender can take to maximally contact the pursuer is presented, along with examples which highlight the importance of the initial conditions of the engagement scenario.},
  author = {Weintraub, Isaac and Von Moll, Alexander and Casbeer, David and Garcia, Eloy and Pachter, Meir},
  date = {2021-08-31},
  doi = {10.1109/CCTA48906.2021.9659042},
  location = {San Diego, CA},
  booktitle = {Conference on Control Technology and Applications},
  title = {Engagement Zone Defense of a Non-Maneuvering Evader},
  year = {2021},
}

@inproceedings{fuchs2021engage,
  abstract = {A two-player, Engage or Retreat (EoR) differential game is presented in which an attacker must choose whether to capture one of N static targets or retreat across a defined retreat boundary. Throughout the engagement, a defending player is capable of activating defensive assets located at each of the target locations. These defensive assets inflict a cost on the attacker as a function of distance and allow the defender to present a deterrent in an effort to persuade the attacker to elect retreat. The solution of the game is constructed by examining two related subproblems: the Game of Engagement (GoE) and the Optimal Constrained Retreat (OCR). Each subproblem examines a different combination of attacker termination strategies, capture vs retreat, and defender strategies, resist vs cooperate. When solving for the optimal constrained retreat strategy, a value function constraint is imposed on the retreat trajectory in order to ensure that the attacker does not maneuver into an advantageous position. Several solutions are examined to illustrate the types of behaviors found in the equilibrium game solutions.},
  author = {Fuchs, Zachariah E. and Von Moll, Alexander and Casbeer, David},
  date = {2021-08-31},
  doi = {10.1109/CCTA48906.2021.9658922},
  location = {San Diego, CA},
  booktitle = {Conference on Controls Technology and Applications},
  title = {Engage or Retreat Differential Game with N-Targets and Distributed Defensive Assets},
  year = {2021},
}

@inproceedings{vonmoll2021turret,
  abstract = {In this paper we model and analyze a scenario in the two dimensional plane involving a mobile Attacker and stationary Defender. There are two possibilities for termination: the Attacker can collide with the Defender (engagement) or maneuver to a safe zone away from the Defender (retreat). The Defender is equipped with a directional turret which it can rotate with a bounded rate. If the turret is aligned with the Attacker's position, the Defender has a lock on the Attacker and may choose to fire on the Attacker. Thus, whether engaging or retreating, the Attacker has incentive to evade the turret's line of sight and thereby avoid being locked-on. In the case of retreat, if lock-on occurs the Defender cooperates with the Attacker by withholding fire to allow the Attacker to retreat. However, if the Attacker chooses to engage and lock-on occurs, the Defender will open fire on the Attacker. We model the scenario as a set of differential games with different cost functionals depending on the type of termination. The agents are assumed to have full state information. In the case that the Defender can align with the Attacker the pre- and post-lock portions of the game are solved individually and stitched together. The equilibrium strategies are derived in each case, and a partitioning of the state space wherein a particular termination condition is optimal is constructed.},
  author = {Von Moll, Alexander and Fuchs, Zachariah},
  date = {2021-05-28},
  doi = {10.23919/ACC50511.2021.9483106},
  pages = {3188-3195},
  location = {New Orleans},
  booktitle = {American Control Conference},
  publisher = {IEEE},
  title = {Turret Lock-on in an Engage or Retreat Game},
  url = {https://avonmoll.github.io/files/turret-lockon-in-an-engage-or-retreat-game.pdf},
  year = {2021},
}

@inproceedings{vonmoll2021turret-runner-penetrator,
  abstract = {A scenario is considered in which two cooperative Attackers aim to infiltrate a circular target guarded by a Turret. The engagement plays out in the two dimensional plane; the holonomic Attackers have the same speed and move with simple motion and the Turret is stationary, located at the target circle's center, and has a bounded turn rate. When the Turret's look angle is aligned with an Attacker, that Attacker is terminated. In this paper, we focus on a region of the state space wherein only one of the Attackers is able to reach the target circle -- and even then, only with the help of its partner Attacker. The Runner distracts the Turret and ends up being terminated in order that the Penetrator can be guaranteed to hit the target circle. We formulate the Turret-Runner-Penetrator scenario as a differential game over the Value of the subsequent game of \textdollar{}\textbackslash{}min\textbackslash{}max\textdollar{} terminal angle which takes place between the Turret and Penetrator once the Runner has been terminated. The solution to the Game of Degree, including equilibrium Turret, Runner, and Penetrator strategies, as well as the Value function are given. In addition, the Game of Kind solution, which is the manifold of states in which the Penetrator will be terminated exactly on the target circle, is constructed numerically.},
  author = {Von Moll, Alexander and Shishika, Daigo and Fuchs, Zachariah and Dorothy, Michael},
  date = {2021-05-26},
  doi = {10.23919/ACC50511.2021.9483094},
  location = {New Orleans},
  booktitle = {American Control Conference},
  title = {The Turret-Runner-Penetrator Differential Game},
  url = {https://avonmoll.github.io/files/turret-runner-penetrator-differential-game.pdf},
  year = {2021},
}

@inproceedings{garcia2021cooperativeblocking,
  abstract = {This paper considers a pursuit-evasion problem with two cooperative pursuers and one evader. The problem is posed as a zero-sum differential game where the evader aims at reaching a goal line which is protected by the pursuers. When reaching this goal is not possible, the evader strives to position itself as close as possible with respect to the goal line at the time of capture. The pursuers try to capture the evader as far as possible from the goal line. Leveraging differential game theory, state feedback strategies are both synthesized and verified in this paper. In addition, the Barrier surface that partitions the state space into two winning sets, one for the pursuer team and one for the evader, is obtained in analytical form. Under optimal play, the winning team is determined by evaluating the associated Barrier function.},
  author = {Garcia, Eloy and Casbeer, David W. and Von Moll, Alexander and Pachter, Meir},
  date = {2021-12-31},
  doi = {10.1109/CCTA48906.2021.9658654},
  journal = {Conference on Control Technology and Applications},
  publisher = {IEEE},
  title = {The Cooperative Blocking Differential Game},
  year = {2021},
}

@article{vonmoll2022circular,
  abstract = {In this paper, the problem of guarding a circular target wherein the Defender(s) is constrained to move along its perimeter is posed and solved using a differential game theoretic approach. Both the one-Defender and two-Defender scenarios are analyzed and solved. The mobile Attacker seeks to reach the perimeter of the circular target, whereas the Defender(s) seeks to align itself with the Attacker, thereby ending the game. In the former case, the Attacker wins, and the Attacker and Defender play a zero sum differential game where the payoff/cost is the terminal angular separation. In the latter case, the Defender(s) wins, and the Attacker and Defender play a zero sum differential game where the cost/payoff is the Attacker's terminal distance to the target. This formulation is representative of a scenario in which the Attacker inflicts damage on the target as a function of its terminal distance. The state-feedback equilibrium strategies and Value functions for the Attacker-win and Defender(s)-win scenarios are derived for both the one- and two-Defender cases, thus providing a solution to the Game of Degree. Analytic expressions for the separating surfaces between the various terminal scenarios are derived, thus providing a solution to the Game of Kind.},
  address = {New York},
  author = {Von Moll, Alexander and Pachter, Meir and Shishika, Daigo and Fuchs, Zachariah},
  date = {2022-10-05},
  doi = {10.1109/TAC.2022.3203357},
  issue = {7},
  pages = {4065 -- 4078},
  journal = {Transactions on Automatic Control},
  publisher = {IEEE},
  title = {Circular Target Defense Differential Games},
  url = {https://avonmoll.github.io/files/circular-target-defense-differential-games.pdf},
  volume = {68},
  year = {2022},
}

@article{weintraub2021maximum,
  abstract = {In this work, we consider a two-agent scenario consisting of an observer and non-maneuvering target. In the scenario, the observer is considered to be maneuverable and slower than the target. The observer is endowed with a nonzero radius of observation within which he strives at keeping the target for as long as possible. Using the calculus of variations, we pose and solve the optimal control problem, solving for the heading and flight path angle of the observer to maximize the amount of time the target vehicle is contained within his observation radius. Using the optimal observer heading and flight path angle, the exposure time is computed, based upon the initial azimuth and elevation by which the target is captured by the observer. The special cases, where the engagement may be represented in a plane, rather than 3-D, are also provided. Presented, along with examples, are the zero-time of exposure conditions, maximum exposure time conditions, and a proof that observation is persistent under the optimal observer strategy.},
  author = {Weintraub, Isaac E. and Von Moll, Alexander and Garcia, Eloy and Pachter, Meir},
  date = {2021-12-15},
  doi = {10.2514/1.G005619},
  issue = {3},
  journal = {Journal of Guidance, Control, and Navigation},
  publisher = {AIAA},
  title = {Maximum Observation of a Target by a Slower Observer in Three Dimensions},
  volume = {44},
  year = {2021},
}

@article{manyam2020curvature,
  abstract = {We present a path planning problem for a pursuing agent to intercept a target traveling on a circle. The target is cooperative, and its position, heading and speed are precisely known by the pursuing agent. The pursuing agent has nonholonomic motion constraints, and therefore the path traveled by the pursuing agent must satisfy the minimum turn radius constraints. We consider the class of Dubins paths as candidate solutions, and analyze the characteristics of the six modes of Dubins paths where the final position is restricted to lie on the target circle with heading in the tangential direction of the circle. For each Dubins mode, we derive the feasibility limits, discontinuities and local extrema. Using this analysis the intercepting paths are found by a systematic bisection search in the feasible regions of each of the Dubins modes. We prove that the algorithm outputs the optimal solution, if the optimal intercepting path is a Dubins path; otherwise, it outputs the best found intercepting path and a tight lower bound to the optimal solution.},
  author = {Manyam, Satyanarayana G. and Casbeer, David W. and Von Moll, Alexander and Fuchs, Zachariah},
  date = {2020-12-31},
  note = {Submitted for Review},
  journal = {Transactions on Automation Science and Engineering},
  publisher = {IEEE},
  title = {Curvature Constrained Paths to Intercept a Target Moving on a Circle},
  year = {2020},
}

@inproceedings{garcia2020pride,
  abstract = {A reach-avoid differential game with multiple players is considered in this paper. A group of defenders is in charge of capturing a spy before it leaves the game set. This paper addresses cooperation among the defenders which protect a circular area on interest and try to intercept an intelligent adversary/intruder. The optimal strategies for each team, the defenders team and the spy, are obtained. These strategies and the Value function are obtained in analytical and explicit form. Furthermore, it is verified that the Value function is continuously differentiable and it is the solution of the Hamilton-Jacobi-Isaacs equation. Finally, the Barrier surface that separates the winning regions of each team is explicitly obtained as well. Illustrative examples show the Barrier surface cross-sections of this high dimensional differential game.},
  author = {Garcia, Eloy and Casbeer, David W. and Von Moll, Alexander and Pachter, Meir},
  date = {2020-12-08},
  doi = {10.1109/CDC42340.2020.9304060},
  location = {Jeju Island, South Korea},
  booktitle = {Conference on Decision and Control},
  title = {Pride of Lions and Man Differential Game},
  year = {2020},
}

@inproceedings{vonmoll2020guarding,
  abstract = {In this paper, the problem of guarding a circular target wherein the Defender(s) is constrained to move along its perimeter is posed and solved using a differential game theoretic approach. Both the one-Defender and two-Defender scenarios are analyzed and solved. The mobile Attacker seeks to reach the perimeter of the circular target, whereas the Defender(s) seeks to align itself with the Attacker, thereby ending the game. In the former case, the Attacker wins, and the Attacker and Defender play a zero sum differential game where the payoff/cost is the terminal angular separation. In the latter case, the Defender(s) wins, and the Attacker and Defender play a zero sum differential game where the cost/payoff is the Attacker's terminal distance to the target. This formulation is representative of a scenario in which the Attacker inflicts damage on the target as a function of its terminal distance. The state-feedback equilibrium strategies and Value functions for the Attacker-win and Defender(s)-win scenarios are derived for both the one- and two-Defender cases, thus providing a solution to the Game of Degree. Analytic expressions for the separating surfaces between the various terminal scenarios are derived, thus providing a solution to the Game of Kind.},
  author = {Von Moll, Alexander and Pachter, Meir and Shishika, Daigo and Fuchs, Zachariah},
  date = {2020-12-11},
  doi = {10.1109/CDC42340.2020.9304018},
  pages = {1658--1665},
  location = {Jeju Island, South Korea},
  booktitle = {Conference on Decision and Control},
  publisher = {IEEE},
  title = {Guarding a Circular Target By Patrolling its Perimeter},
  url = {https://avonmoll.github.io/files/guarding-a-circular-target-by-patrolling-its-perimeter.pdf},
  year = {2020},
}

@inproceedings{vonmoll2020evolutionary,
  abstract = {In this paper a scenario is considered in which a group of predators cooperate to maximize the number of prey captures over a finite time horizon on a two-dimensional plane. The emphasis is on developing predator strategies, and thus the behavior of the prey agents is fixed to a Boids-like flocking model which incorporates avoidance of nearby predators. At each time instant, the predators have control authority over their heading angle; however, we restrict the headings to be governed by one of five different pre-specified behaviors. No communication occurs between the predator agents - each agent determines its own control without knowledge of what the other predators will implement; thus, the predator strategies are fully decentralized. The full state space of the system is collapsed to a set of features which is independent of the number of prey. An evolutionary algorithm is used to evolve an anchor point controller wherein the anchor point lies in the feature space and has a particular predator behavior associated, thus providing a candidate solution to the question of "what to do when". The two predator case is the focus in this work, although the ideas could be applied to larger groups of predators. The strategies resulting from the evolutionary algorithm favor aiming at the nearest prey mostly, and also avoiding having the predators getting too close and then pursuing the same prey. Thus useless behaviors are generally not present among the elite at the end of the evolutionary process.},
  author = {Von Moll, Alexander and Androulakakis, Pavlos and Fuchs, Zachariah and Vanderelst, Dieter},
  date = {2020-08-24},
  doi = {10.1109/CoG47356.2020.9231945},
  location = {Osaka, Japan},
  booktitle = {Conference on Games},
  publisher = {IEEE},
  title = {Evolutionary Design of Cooperative Predation Strategies},
  url = {https://avonmoll.github.io/files/evolutionary-design.pdf},
  year = {2020},
}

@article{milutinovic2021rate,
  abstract = {The scope of this work is the well-known wall pursuit game which has been used in the literature to illustrate the existence of a singular surface (dispersal line) and the associated game dilemma. We derive an analytical expression for the value function of the game and use a hold time to derive the rate of change for the loss of the time to capture along the dispersal line. Then we resolve the dilemma along the dispersal line using actions defined by the saddle point of the rate of change for the loss. Finally, we analyze the same game in a version with a non-zero hold time and show that in that case, the actions from the dispersal line have to be applied not only on the dispersal line, but in a narrow band around it. To illustrate that, we use an example to compute the band around the line.},
  author = {Milutinovi\'{c}, Dejan and Casbeer, David W. and Von Moll, Alexander and Pachter, Meir and Garcia, Eloy},
  date = {2021-12-27},
  doi = {10.1109/TAC.2021.3137786},
  issue = {1},
  journal = {Transactions on Automatic Control},
  pages = {242 -- 256},
  publisher = {IEEE},
  title = {Rate of Loss Characterization that Resolves the Dilemma of the Wall Pursuit Game Solution},
  volume = {68},
  year = {2021},
}

@inproceedings{vonmoll2014recent,
  abstract = {Thermal management has emerged as one of the primary challenges for modern military gas turbine engines. Fuel actuator pumping systems contribute to the challenge by constantly injecting waste heat into the fuel system. This is due to a fundamental mismatching between an actuation system with a low duty cycle and a pump which is always running. Since the pump is directly coupled to the gearbox, the detriment is twofold: horsepower is always being drawn to drive the pump, and heat is generated. Furthermore, the emergence of variable cycle engines pushes the actuation system in the direction of more effectors and more loads, which drives requirements for high pressure and high flows from the pumping system. Piston pumps have been seen as favorable for the actuation pump application due to their high single stage pressure rise capability. However reducing the number of pumping components alone has not solved the thermal management issue. Variable displacement pumps have been designed to operate over a range of conditions in order to deliver flow more appropriately as demanded by the control system. These pumps allow pressure and flow to be modulated independent of one another and of the pump's rotational speed. Variable displacement pumps may be considered as a component of a partial-demand actuation pumping system. Even in a partial-demand actuation pumping system, a variable displacement pump will still be coupled to the gearbox and require some minimum flow for lubrication and cooling. On-demand actuation fuel systems deliver only the amount of flow that is being demanded at a given time so that waste heat is minimized. In other words, the duty cycle of the pumping system matches that of the actuation system. The development of an on-demand actuation fuel system will require a new architecture and new components.},
  author = {Von Moll, Alexander and Semega, Ken and Behbahani, Alireza and Hoying, John},
  date = {2014-01-01},
  booktitle = {JANNAF Interagency Propulsion Committee 34th Airbreathing Propulsion},
  publisher = {JANNAF Interagency Propulsion Committee},
  title = {Recent Progress, Challenges, and Future Development Needs of Thermally/Energy Efficient Fuel Actuator Pumping Systems for Military Gas Turbine Engine Applications},
  year = {2014},
}

@inproceedings{vonmoll2013comparison,
  abstract = {A distributed engine control system (DECS) offering flexibility and scalability is envisioned for the next generation of propulsion controls. Perhaps the most touted benefit of a DECS is the potential to reduce the amount of harnessing which connects throughout the engine. Such a system is comprised of several network sections incorporating control nodes or data concentrators (DCs). These DCs contain control logic to perform control function computations and are connected to the full authority digital engine control (FADEC) via a high-speed data communication bus. A novel approach for analyzing and evaluating three topologies\textendash{}ring, star, and bus\textendash{}in the context of a relevant military engine was described. In this study, the algorithm uses a particle swarm optimization process to evolve solutions to a multi-objective optimization problem. The results of this study indicate there is potential for large wire length savings in a distributed control architecture.},
  author = {Von Moll, Alexander and Behbahani, Alireza R.},
  date = {2013-02-01},
  booktitle = {59th IIS},
  publisher = {ISA},
  title = {Comparison of Communication Architectures and Network Topologies for Distributed Propulsion Controls},
  url = {http://www.dtic.mil/dtic/tr/fulltext/u2/a586909.pdf},
  year = {2013},
}

@inproceedings{manyam2019coordinating,
  abstract = {We present a two vehicle, cooperating path planning problem in adversarial environments. The first vehicle is the lead vehicle and its path is prescribed to accomplish a certain mission,and there exists multiple attackers along this path. The second vehicle is a defender that travelsalong the lead vehicle, and it is responsible to protect the lead vehicle from the attackers. The defender launches defender missile to intercept the attackers before it reaches the lead vehicle. The defender and lead vehicle cooperate to maximize the distance between the lead vehicle and the point of interception, while the attacker tries to minimize this distance. This sub problem is modeled as Target-Attacker-Defender (TAD) game with an objective to maximize the terminal separation distance between lead vehicle and attacker. We aim to generate an optimal pathfor the defender, such that if the TAD game is played at any point along the path, the defender should be optimally positioned that maximizes the terminal separation distance. We reduce this problem to a discrete space, and methodically pose it as a shortest path problem on a graph, the solution of which gives the discrete optimal path for the defender.},
  author = {Manyam, Satyanarayana G. and Casbeer, David and Von Moll, Alexander and Garcia, Eloy},
  date = {2019-01-11},
  doi = {10.2514/6.2019-0388},
  location = {San Diego, CA},
  booktitle = {AIAA Infotech},
  publisher = {AIAA},
  title = {Coordinating Defender Path Planning for Optimal Target-Attacker-Defender Game},
  year = {2019},
}

@inproceedings{behbahani2014aircraft,
  abstract = {Modern propulsion system designers face challenges that require that aircraft and engine manufacturers improve performance as well as reduce the life-cycle cost (LCC). These improvements will require a more efficient, more reliable, and more advanced propulsion system. The concept of smart components is built around actively controlling the engine and the aircraft to operate optimally. Usage of smart components intelligently increases efficiency and system safety throughout the flight envelope, all while meeting environmental challenges. This approach requires an integration and optimization, both at the local level and the system level, to reduce cost. Interactions between the various subsystems must be understood through the use of modeling and simulation. This is accomplished by starting with individual subsystem models and combining them into a complete system model. Hierarchical, decentralized control reduces cost and risk by enabling integration and modularity. This process involves defining, developing, and validating against requirements for key integrated propulsion, power, and thermal management system capabilities.},
  author = {Behbahani, Alireza R. and Von Moll, Alex and Zeller, Robert and Ordo, James},
  date = {2014-09-01},
  doi = {10.4271/2014-01-2161},
  booktitle = {2014 SAE Aerospace Systems and Technology Conference},
  publisher = {SAE International},
  title = {Aircraft Integration Challenges and Opportunities for Distributed Intelligent Control, Power, Thermal Management, Diagnostic and Prognostic Systems},
  year = {2014},
}

@inproceedings{vonmoll2019multiple,
  abstract = {Ensuring border security against potential threats has been a long standing problem and it becomes much more challenging for land borders due to its long distances coupled with terrain variations. In this paper, we consider a scenario in which a team of UAVs (the pursuers)  attempt to prevent an intruder ground vehicle (the evader) from escaping through the border. The ground intruder's presence is made known by a laser fence positioned inside the border. Upon detection, the UAVs are activated and given the objective to cooperatively capture the intruder. We formulate the scenario as a zero-sum differential game whereupon all agents exhibit simple motion and the pursuers have a speed advantage. For cases where capture inside the border is possible, the payoff/cost for the pursuers/evader is the distance from capture to the border. The solution to the game is derived based on the geometric properties of the game and verified for the single pursuer case with a border comprised of straight-line segments. This game may contain a dispersal surface which has interesting practical consequences when the optimal strategies are applied in discrete time.},
  author = {Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Manickam, Suresh and Swar, Sufal Chandra},
  date = {2019-01-11},
  doi = {10.2514/6.2019-1162},
  location = {San Diego, CA},
  booktitle = {AIAA SciTech},
  publisher = {AIAA},
  title = {Multiple Pursuer Single Evader Border Defense Differential Game},
  url = {https://avonmoll.github.io/files/mp1e-border.pdf},
  year = {2019},
}

@inproceedings{vonmoll2014review,
  abstract = {The exhaust gas temperature (EGT) is typically defined as the gas temperature at the exit of the turbine; the sensors used to measure this parameter are considered the most vulnerable elements of the entire turbine engine instrumentation. EGT measurement is considered a key parameter for optimizing fuel economy, diagnosis, and prognosis. The reason is that turbine blade temperature is a good indicator for normal life consumption of that blade. Currently, direct sensor measurements made on turbine modules are limited due to the extremely hot environment. Exhaust gas temperature (EGT) sensors located downstream from the highest temperature sections provide a means to roughly infer the temperatures seen by the turbine blades/disks. But these sensors, which themselves are subject to frequent failures, provide a fairly inaccurate indication of the actual metal temperature profiles. This is of particular importance for high performance military turbine engines where the margin between hot-section operating conditions and material limitations is shrinking. A future state is envisioned that would use emerging pyrometric, fiber optic, and laser-based sensing to accurately assess the condition and life usage of the hot section, on a blade-by-blade basis.},
  author = {Von Moll, Alexander and Behbahani, Alireza R. and Fralick, Gustave C. and Wrbanek, John D. and Hunter, Gary W.},
  date = {2014-07-25},
  doi = {10.2514/6.2014-3977},
  booktitle = {50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference},
  publisher = {American Institute of Aeronautics and Astronautics},
  series = {AIAA Propulsion and Energy Forum},
  title = {A Review of Exhaust Gas Temperature Sensing Techniques for Modern Turbine Engine Controls},
  url = {http://arc.aiaa.org/doi/10.2514/6.2014-3977},
  year = {2014},
}

@inproceedings{black2018integrated,
  abstract = {Next-generation integrated propulsion systems require reliable, robust, low-cost and high performance sensing systems beyond what is expected to be delivered by conventional sensor networks. Emerging fiber-optic sensor networks offer distinct advantages, including multiplexability, cybersecurity, inherent EMI/RFI immunity, high speed and bandwidth, and compatibility with the avionics environment. Ongoing development of a novel reliability analysis methodology to quantify sensor network reliability in harsh environments is discussed. Failure modes, mechanisms, and effects analysis (FMMEA) was performed for jet engine environments. System design and deployment guidelines were then developed based upon experience in sensor network deployments to harsh environments. Addressing this key technology gap, an integrated fiber-optic network reliability analysis technique based upon  predictive failure models is being developed via rigorous environmental testing of critical optical and optoelectronic components comprising the sensor network. The results of this effort are expected to empower propulsion system sensing and control network engineers to enhance design and deployment for reliability.},
  author = {Black, Richard J. and Moslehi, Behzad and Price, W and Sotoudeh, Vahid and Osterman, Michael and Das, Diganta and Behbahani, Alireza and Von Moll, Alexander and Semega, Kenneth},
  date = {2018-01-07},
  doi = {10.2514/6.2018-0714},
  booktitle = {2018 AIAA Information Systems-AIAA Infotech @ Aerospace},
  publisher = {American Institute of Aeronautics and Astronautics},
  series = {AIAA SciTech Forum},
  title = {Integrated Fiber-Optic Sensor Network System Reliability Modeling and Analysis for Aerospace Applications},
  url = {http://arc.aiaa.org/doi/10.2514/6.2018-0714},
  year = {2018},
}

@inproceedings{vonmoll2018genetic,
  abstract = {We employ a genetic algorithm approach to solving the persistent visitation problem for UAVs. The objective is to minimize the maximum weighted revisit time over all the sites in a cyclicly repeating walk. In general, the optimal length of the walk is not known, so this method (like the exact methods) assume some fixed length. Exact methods for solving the problem have recently been put forth, however, in the absence of additional heuristics, the exact method scales poorly for problems with more than 10 sites or so. By using a genetic algorithm, performance and computation time can be traded off depending on the application. The main contributions are a novel chromosome encoding scheme and genetic operators for cyclic walks which may visit sites more than once. Examples show that the performance is comparable to exact methods with better scalability.},
  author = {Von Moll, Alexander and Kalyanam, Krishna and Casbeer, David and Manyam, Satyanarayana G.},
  date = {2018-10-01},
  doi = {10.1115/DSCC2018-8950},
  location = {Atlanta, GA},
  publisher = {ASME},
  title = {Genetic Algorithm Approach for UAV Persistent Visitation Problem},
  url = {https://drive.google.com/file/d/0B0yHktr7udqgZERlUHRfaXNjOWQ5ZWJwNi05bDBvOVYtbDBV/view?usp=sharing},
  year = {2018},
}

@inproceedings{moslehi2018high-bandwidth,
  abstract = {The insertion of advanced technologies into gas turbine engines is expected to reduce costs and improve performance. Engines will be getting smaller and hotter for efficiency reasons, requiring diverse types of sensors, in particular pressure and temperature sensors, with extended and enhanced performance. Emerging fiber-optic sensing approaches could provide a unique solution to surpassing the limitations and burden of conventional sensors. Two novel multiplexable fiber optic-based pressure sensor designs are discussed for high-temperature operations up to 1,800 \textdegree{}F and 3,700 \textdegree{}F. These innovative sensor designs enable more direct measurements over long durations, resulting in higher accuracy and stability. These sensors could be used in ground test cells, in-flight control, and diagnostics and prognostic health management (PHM). Highly reliable all-optical sensor networking and interface designs with a high-speed, high-bandwidth fiber-optic backbone is also presented.},
  author = {Moslehi, Behzad and Price, W and Black, Richard J. and Han, Ming and Behbahani, Alireza and Von Moll, Alexander and Semega, Kenneth},
  date = {2018-01-07},
  doi = {10.2514/6.2018-0715},
  booktitle = {2018 AIAA Information Systems-AIAA Infotech @ Aerospace},
  publisher = {American Institute of Aeronautics and Astronautics},
  series = {AIAA SciTech Forum},
  title = {High-Bandwidth Fiber-Optic Pressure Sensors for High-Temperature Aerospace Applications},
  url = {http://arc.aiaa.org/doi/10.2514/6.2018-0715},
  year = {2018},
}

@article{salmon2020single,
  abstract = {An interest in border defense, surveillance, and interdiction has recently increased for a variety of reasons related to issues of illegal immigration, terrorism, drug/human trafficking, and other potential threats. UAVs offer an attractive alternative to supporting and defending various threats at borders. This article applies a differential game to define a border defense scenario where one UAV (pursuer) seeks to capture two intruders (evaders) before they reach a designated border. The evaders can be UAVs, marine/ground vehicles, or human agents, but have a lower maximum speed than the pursuer throughout the game. Simple motion is assumed for the pursuer and evaders with complete state information shared across all agents.  The game is played within a rectangular area with a parallel top and bottom border of length L and left and right borders with a length of W, for a game aspect ratio of L/W. The value of the game is the minimum distance to the bottom border achieved by the evaders at any time before capture of both evaders. Within the region where the pursuer wins, the game of degree is explored and the optimal policy for both the evaders and pursuers is derived using geometric},
  author = {Salmon, John L. and Willey, Landon C. and Casbeer, David and Garcia, Eloy and Von Moll, Alexander},
  date = {2020-03-19},
  doi = {10.2514/1.I010766},
  journal = {Journal of Aerospace Information Systems},
  publisher = {AIAA},
  title = {Single Pursuer Multiple-Cooperative Evaders in the Border Defense Differential Game},
  year = {2020},
}

@thesis{vonmoll2015machine,
  abstract = {The intersection between machine learning and control theory is emerging rapidly; many rich applications are currently being developed. This report explores one particular area of interest: trajectory optimization. The Linear Quadratic Regulator and Linear Quadratic Gaussian algorithms are introduced as predecessors to the Iterative Linear Quadratic Gaussian (iLQG) algorithm, which is re-derived in this report and demonstrated on 3 systems of interest: cartpole, double cartpole, and quadrotor. iLQG is robust and widely applicable to a vast number of different systems. Probabilistic Differential Dynamic Programming (PDDP) is introduced as an important extension to iLQG which incorporates uncertainty into the trajectory optimization process. The conceptual advances presented in this report were centered around the use of a Gaussian Process Regression Network (GPRN) to represent the unknown system dynamics in lieu of the standard Gaussian Process representation employed by PDDP. GPRNs offer a more sophisticated noise model and a more adaptive regression framework. More speciffcally, they incorporate input-dependent noise and signal correlation between multiple outputs. This should enhance the predictive capability of the algorithm to represent the unknown system dynamics. Work remains on fully incorporating GPRN into a PDDP-like algorithm: effort continues in the areas of understanding variational Bayes and variational message passing for learning GPRN hyperparameters, and in creating a MATLAB implementation.},
  author = {Von Moll, Alexander},
  date = {2015-12-01},
  note = {Special Project AE8900},
  title = {Machine Learning Applications in Complex Control Systems},
  school = {Georgia Institute of Technology},
  url = {https://www.slideshare.net/AlexanderVonMoll/vonmollpaper},
  year = {2015},
}

@article{vonmoll2020multiple,
  abstract = {Ensuring border security against potential threats has been a long standing problem and it becomes much more challenging for international borders due to their massive physical extent coupled with varied terrain. In this paper, we consider a scenario in which a team of UAVs (the pursuers) attempt to prevent an intruder ground vehicle (the evader) from escaping through the border. The ground intruder's presence is made known by a laser fence positioned inside the border. Upon detection, the UAVs are activated and given the objective to cooperatively capture the intruder. We formulate the scenario as a zero-sum differential game whereupon all agents exhibit simple motion and the pursuers have a speed advantage. For cases where capture inside the border is possible, the payoff/cost for the pursuers/evader is the distance from capture to the border. The solution to the game is derived based on the geometric properties of the game and verified for the single pursuer case with a border comprised of straight-line segments. This game may contain a dispersal surface which has interesting practical consequences when the optimal strategies are applied in discrete time.},
  author = {Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Manickam, Suresh and Swar, Sufal Chandra},
  date = {2020-02-01},
  doi = {10.2514/1.I010740},
  journal = {Journal of Aerospace Information Systems},
  publisher = {AIAA},
  title = {Multiple Pursuer Single Evader Border Defense Differential Game},
  url = {https://avonmoll.github.io/files/mp1e-border.pdf},
  year = {2020},
}

@article{pachter2020cooperative,
  abstract = {We consider pursuit-evasion differential games in the Euclidean plane where an evader is engaged by multiple pursuers and point capture is required. The players have simple motion \`{a} la Isaacs and the pursuers are faster than the evader. We confine our attention to the case where the pursuers have the same speed, so the game's parameter is the evader/pursuers speed ratio 0 < \mathrm{\mu} < 1. State feedback capture strategies and an evader strategy which yields a lower bound on his time-to-capture are devised using a geometric method. It is shown that in group/swarm pursuit, when the players are in general position, capture is effected by one, two, or by three critical pursuers, and this irrespective of the size N (> 3) of the pursuit pack.  Group pursuit devolves into pure pursuit by one of the pursuers or into a pincer movement pursuit by two or three pursuers who isochronously capture the evader, a m\`{e}nage \`{a} trois. The critical pursuers are identified. However, these geometric method-based pursuit and evasion strategies are optimal only in a part of the state space where a strategic saddle point is obtained and the Value of the differential game is established, and as such, are suboptimal. To fully explore the differential game's high dimensional state space and get a better understanding of group pursuit, numerical experimentation is undertaken. The state space region where the geometric method-based suboptimal solution of the group pursuit differential game is the optimal solution becomes larger the smaller the speed ratio parameter is.},
  author = {Pachter, Meir and Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Milutinovi\'{c}, Dejan},
  date = {2020-02-01},
  doi = {10.2514/1.I010739},
  journal = {Journal of Aerospace Information Systems},
  title = {Cooperative Pursuit by Multiple Pursuers of a Single Evader},
  url = {https://avonmoll.github.io/files/ManyPursuers.pdf},
  year = {2020},
}

@inproceedings{kalyanam2018scalable,
  abstract = {We are interested in the persistent surveillance of an area of interest comprised of stations/ data nodes. The data collection task is undertaken by a UAV which autonomously executes the mission. Each node has a priority/ weight associated with it that characterizes the relative importance between timely collection of data from the nodes. In this context, a prudent performance metric is the maximal weighted time between consecutive visits to each node. We wish to minimize the maximum of this metric among all nodes for a given cycle length. Here, the cycle length refers to the total number of nodes (including repeats) visited per cycle. When the cycle length is greater than the number of nodes, some nodes, arguably ones with a higher priority, will be visited more often. We pose the problem as a Mixed Integer Linear Program (MILP), that computes the optimal visit sequence and number of visits to each node for a given cycle length. Not surprisingly, the exact method is not scalable with the number of nodes and the cycle length. We therefore present sub-optimal methods  by directly enforcing a pre-specified number of visits to each node. We present an iterative scheme that computes solutions for increasing cycle length with a recipe for updating the number of visits based on the solution from the previous iteration. We compare the optimal and sub-optimal iterative schemes and show that the sub-optimal scheme is an order of magnitude faster than the optimal scheme with minimal loss in performance.},
  author = {Kalyanam, Krishna and Manyam, Satyanarayana and Von Moll, Alexander and Casbeer, David and Pachter, Meir},
  date = {2018-08-01},
  doi = {10.1109/CCTA.2018.8511587},
  pages = {337-342},
  publisher = {IEEE},
  title = {Scalable and Exact MILP Methods for UAV Persistent Visitation Problem},
  url = {https://sites.google.com/site/krishnakalyanam/ccta2018.pdf},
  year = {2018},
}

@article{garcia2020multiple,
  abstract = {In this paper an N-pursuer vs. M-evader team conflict is studied. The differential game of border defense is addressed and we focus on the game of degree in the region of the state space where the pursuers are able to win. This work extends classical differential game theory to simultaneously address weapon assignments and multi-player pursuit-evasion scenarios. Saddle-point strategies that provide guaranteed performance for each team regardless of the actual strategies implemented by the opponent are devised. The players' optimal strategies require the co-design of cooperative optimal assignments and optimal guidance laws. A representative measure of performance is proposed and the Value function of the game is obtained. It is shown that the Value function is continuous, continuously differentiable, and that it satisfies the Hamilton-Jacobi-Isaacs equation - the curse of dimensionality is overcome and the optimal strategies are obtained. The cases of N=M and N>M are considered. In the latter case, cooperative guidance strategies are also developed in order for the pursuers to exploit their numerical advantage. This work provides a foundation to formally analyze complex and high-dimensional conflicts between teams of N pursuers and M evaders by means of differential game theory.},
  author = {Garcia, Eloy and Casbeer, David W and Von Moll, Alexander and Pachter, Meir},
  date = {2020-05-01},
  doi = {10.1109/TAC.2020.3003840},
  journal = {Transactions on Automatic Control},
  publisher = {IEEE},
  title = {Multiple Pursuer Multiple Evader Differential Games},
  year = {2020},
}

@inproceedings{garcia2019cooperative,
  abstract = {A pursuit-evasion problem with two pursuers and one evader is considered. The evader aims at reaching a goal line which is protected by the pursuers. When reaching this goal is not possible, the Evader strives to position itself as close as possible with respect to the goal line at the time of capture. The pursuers try to capture the evader as far as possible from the goal line. The problem is posed as a zero-sum differential game where the two pursuers cooperate against the evader. State feedback strategies are derived in this paper and the Value function is obtained. It is also shown that the Value function is continuous, continuously differentiable, and it satisfies the Hamilton-Jacobi-Isaacs equation.},
  author = {Garcia, Eloy and Casbeer, David W. and Von Moll, Alexander and Pachter, Meir},
  date = {2019-07-10},
  doi = {10.23919/ACC.2019.8814294},
  title = {Cooperative Two-Pursuer One-Evader Blocking Differential Game},
  year = {2019},
}

@inproceedings{manyam2019shortest,
  abstract = {The Dubins path problem had enormous applications in path planning for autonomous vehicles. In this paper, we consider a generalization of the Dubins path planning problem, which is to find a shortest Dubins path that starts from a given initial position and heading, and ends on a given target circle with the heading in the tangential direction. This problem has direct applications in Dubins neighborhood traveling salesman problem, obstacle avoidance Dubins path planning problem etc. We characterize the length of the four CSC paths as a function of angular position on the target circle, and derive the conditions which to find the shortest Dubins path to the target circle.},
  author = {Manyam, Satyanarayana Gupta and Casbeer, David and Von Moll, Alexander and Fuchs, Zachariah},
  date = {2019-01-07},
  doi = {10.2514/6.2019-0919},
  location = {San Diego, CA},
  booktitle = {AIAA SciTech},
  publisher = {AIAA},
  title = {Shortest Dubins Path to a Circle},
  url = {http://arxiv.org/abs/1804.07238},
  year = {2019},
}

@inproceedings{manyam2019optimal,
  abstract = {We present a trajectory planning problem for a pursuer to intercept a target traveling on a circle. The pursuer considered here have limited yaw rate, and therefore the trajectories should satisfy the kinematic constraints. We assume that the distance between initial position of the pursuer and any point on the target circle is greater than four times the minimum turn radius of the pursuer, and prove the continuity of the Dubins paths of type Circle-Straight line-Circle with respect to the position on the target circle. This is used to prove that the optimal interception paths is a Dubins path, and present an iterative scheme to find the interception point on the target circle.},
  author = {Manyam, Satyanarayana G. and Casbeer, David and Von Moll, Alexander and Fuchs, Zachariah},
  booktitle = {American Control Conference},
  date = {2019-07-10},
  doi = {10.23919/ACC.2019.8814913},
  title = {Optimal Dubins Paths to Intercept a Moving Target on a Circle},
  year = {2019},
}

@inproceedings{vonmoll2020optimal,
  abstract = {We extend the Turret Defense Differential Game by considering the possibility for the mobile Attacker to retreat to a particular region of the state space in lieu of engaging the Turret. In this case, the Turret cooperates with the Attacker by escorting the Attacker to the retreat surface, thereby minimizing its cost. Along the retreat trajectory, the cost associated with the Value of the Game of Engagement must be greater than the cost associated with retreating. We refer to this scenario as Optimal Constrained Retreat, wherein the aforementioned requirement is imposed as a path constraint. The optimality conditions are developed herein. We specify a boundary value problem with fixed initial state and compute the optimal trajectory numerically via backwards shooting. The corresponding trajectory contains a constrained arc wherein the path constraint associated with the Game of Engagement becomes active.},
  author = {Von Moll, Alexander and Fuchs, Zachariah},
  date = {2020-08-24},
  doi = {10.1109/CCTA41146.2020.9206388},
  location = {Montreal, Canada},
  booktitle = {Conference on Control Technology and Applications},
  title = {Optimal Constrained Retreat within the Turret Defense Differential Game},
  url = {https://avonmoll.github.io/files/turret-ocr.pdf},
  year = {2020},
}

@inproceedings{vonmoll2020attacker,
  abstract = {The characteristics for the solution to the Turret Defense Differential Game are explored over the parameter space. We collapse the five natural parameters of capture radius, attacker speed, turret turn rate, time penalty constant, and look-angle penalty constant into two composite parameters in order to facilitate the analysis. There exist three singular surfaces in the game, two of which have an analytic form, and one which can only be obtained numerically. We focus on the latter: the Attacker Dispersal Surface, wherein the attacker can choose between an indirect or direct route to capture. For certain parameter settings, the Attacker Dispersal Surface is present, while for others, the surface is absent. These regions in the parameter space are identified, and the numerical procedures to do so are detailed. Backwards shooting of the optimal state and adjoint dynamics features prominently in the procedures. Two pieces of numerical evidence are utilized to indicate the presence or absence of the Attacker Dispersal Surface.},
  author = {Von Moll, Alexander and Fuchs, Zachariah},
  date = {2020-07-31},
  doi = {10.1016/j.ifacol.2020.12.2549},
  issue = {2},
  pages = {15659--15666},
  location = {Berlin, Germany},
  booktitle = {IFAC World Congress on Automatic Control},
  title = {Attacker Dispersal Surface in the Turret Defense Differential Game},
  url = {https://avonmoll.github.io/files/attacker-dispersal-surface.pdf},
  volume = {53},
  year = {2020},
}

@inproceedings{garcia2019strategies,
  abstract = {A scenario is considered where a coastline or border is coming under attack by two aircraft. The border is guarded by a faster defender bent on intercepting both aircraft in sequence. A differential game is formulated where the defender sequentially pursues both aircraft and tries to capture them as far as possible from the border. The two aircraft execute cooperative maneuvers and try to minimize their combined distance to the border at the moment when each one is intercepted by the defender. In this paper, the regular solution of this differential game is obtained and the game singular surfaces are identified.},
  author = {Garcia, Eloy and Von Moll, Alexander and Casbeer, David and Pachter, Meir},
  date = {2019-12-31},
  doi = {10.1109/CDC40024.2019.9029340},
  location = {Nice, France},
  booktitle = {IEEE Conference on Decision and Control},
  title = {Strategies for Defending a Coastline Against Multiple Attackers},
  year = {2019},
}

@inproceedings{weintraub2020maximum,
  abstract = {This paper considers a two agent scenario containing an observer and a non-maneuvering target. The observer is maneuverable but is slower than the course-holding target. In this scenario, the observer is endowed with a nonzero radius of observation within which he strives at keeping the target for as long as possible. Using the calculus of variations we pose and solve the optimal control problem, solving for the heading of the observer which maximizes the amount of time the target remains inside the radius of observation. Utilizing the optimal observer heading we compute the exposure time based upon the angle by which the target is initially captured. Presented, along with an example, are the zero-time of exposure heading, maximum time of observation heading, and proof that observation is persistent under optimal control.},
  author = {Weintraub, Isaac E. and Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Demers, Zachary J. L. and Pachter, Meir},
  date = {2020-07-05},
  doi = {10.23919/ACC45564.2020.9147340},
  location = {Denver, CO},
  booktitle = {American Control Conference},
  title = {Maximum Observation of a Faster Non-Maneuvering Target by a Slower Observer},
  year = {2020},
}

@inproceedings{pachter2019singular,
  abstract = {The Two Cutters and Fugitive Ship game posed by Isaacs is revisited again. We discuss and analyze the singular configuration of this two-pursuer one-evader differential game. This paper addresses the question of whether or not either player has the ability to exploit the dispersal surface. Specifically, we in- vestigate the case where the Evader effectively stands still (e.g., by dithering in a small neighborhood). We show that the canonical optimal pursuit policy yields chattering in the discrete-time version of the game. As the timestep approaches zero, the capture time approaches the Value of the game, and thus the Evader is not penalized for standing still. Implications on related scenarios are discussed.},
  author = {Pachter, Meir and Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Milutinovi\'{c}, Dejan},
  date = {2019-06-15},
  doi = {10.1109/ICUAS.2019.8798244},
  location = {Atlanta, GA},
  booktitle = {2019 International Conference on Unmanned Aircraft Systems},
  title = {Singular Trajectories in the Two Pursuer One Evader Differential Game},
  url = {https://avonmoll.github.io/files/2p1e-singular.pdf},
  year = {2019},
}

@article{vonmoll2019robust,
  abstract = {Analysis of the pursuit-evasion differential game consisting of multiple pursuers and single evader with simple motion is difficult due to the wellknown curse of dimensionality. Policies have been proposed for this scenario, and we show that these policies are Global Stackelberg equilibrium strategies.  However we also show that they are not saddle point equilibria in the feedback sense. The argument is twofold: there are cases where the saddle point condition is violated, and cases where the strategy profiles are not time consistent (subgame perfect). The issue of capturability is explored, and sufficient conditions for guaranteed capture are provided. A new pursuit policy is proposed which guarantees capture while also providing an upper bound for capture time. The Evader policy corresponding to the Global Stackelberg equilibrium is shown to provide a lower bound for capture time. Thus these policies are robust from the Pursuer and Evader perspectives, respectively, should they implement them. Several other interesting pursuit and evasion policies are explored and compared with the robust policies in a series of experiments.},
  author = {Von Moll, Alexander and Pachter, Meir and Garcia, Eloy and Casbeer, David and Milutinovi\'{c}, Dejan},
  date = {2019-05-04},
  doi = {10.1007/s13235-019-00313-3},
  issue = {10},
  journal = {Dynamic Games and Applications},
  pages = {202-221},
  title = {Robust Policies for a Multiple Pursuer Single Evader Differential Game},
  url = {https://avonmoll.github.io/files/mp1e-robust.pdf},
  year = {2019},
}

@article{pachter2019two-on-one,
  abstract = {A three agent pursuit-evasion differential game in the Euclidean plane where two pursuers engage an Evader is considered. The Pursuers are faster than the Evader and the three player have simple motion/are holonomic. Coincidence of the Evader with either one, or both Pursuers, is capture, and time of capture is the payoff/cost. In this paper Isaacs' Two Cutters and Fugitive Ship differential game is revisited and the solution of the Game of Kind is provided. Using Isaacs' geometric method, explicit formulae for the players' optimal state feedback strategies and the Value of the game are derived. The merits of the suboptimal Evader tactic of reducing speed, or opting to stay stationary, are examined.},
  author = {Pachter, Meir and Von Moll, Alexander and Garcia, Eloy and Casbeer, David and Milutinovi\'{c}, Dejan},
  date = {2019-07-08},
  doi = {10.2514/1.G004068},
  number = {7},
  journal = {Journal of Guidance, Control, and Dynamics},
  title = {Two-on-One Pursuit},
  url = {https://avonmoll.github.io/files/TwoCutters.pdf},
  volume = {42},
  year = {2019},
}

@inproceedings{vonmoll2020optimalevasion,
  abstract = {The Pure Pursuit strategy is ubiquitous both in the control literature but also in real-world implementation. In this paper, we pose and solve a variant of Isaacs' Two Cutters and Fugitive Ship problem wherein the Pursuers' strategy is fixed to Pure Pursuit, thus making it an optimal control problem. The Pursuers are faster than the Evader and are endowed with a finite capture radius. All agents move with constant velocity and can change heading instantaneously. Although capture is inevitable, the Evader wishes to delay capture as long as possible. The optimal trajectories cover the entire state space. Regions corresponding to either solo capture or isochronous (dual) capture are computed and both types of maximal time-to-capture optimal trajectories are characterized.},
  author = {Von Moll, Alexander and Fuchs, Zachariah and Pachter, Meir},
  date = {2020-07-05},
  doi = {10.23919/ACC45564.2020.9147776},
  location = {Denver, CO},
  booktitle = {American Control Conference},
  publisher = {IEEE},
  title = {Optimal Evasion Against Dual Pure Pursuit},
  url = {https://avonmoll.github.io/files/optimal-evasion-against-dual-pure-pursuit.pdf},
  year = {2020},
}

@inproceedings{vonmoll2018pursuit-evasion,
  abstract = {In this paper, we extend the well-studied results of the two-pursuer, single-evader differential game to any number of pursuers. The main objective of this investigation is to exploit the benefits of cooperation amongst the pursuers in order to reduce the capture time of the evader. Computational complexity is a chief concern as this problem would need to be solved in an online fashion, e.g. in the case of autonomous unmanned aerial vehicles. A new geometric approach to solving the game is introduced and analyzed, which changes the problem of optimizing over continuous domains to a discrete combinatoric optimization. While past efforts at solving multiple pursuer problems have suffered from the curse of dimensionality, the geometric algorithms put forth here are shown to be scalable. Categorization and removal of redundant pursuers is the primary means by which scalability is achieved. The solution of this problem serves as a stepping stone to more complex problems such as the M-pursuer N-evader differential game.},
  author = {Von Moll, Alexander and Casbeer, David W. and Garcia, Eloy and Milutinovi\'{c}, Dejan},
  date = {2018-06-01},
  doi = {10.1109/ICUAS.2018.8453470},
  pages = {133-142},
  location = {Dallas, TX},
  booktitle = {2018 International Conference on Unmanned Aircraft Systems (ICUAS)},
  publisher = {IEEE},
  title = {Pursuit-evasion of an Evader by Multiple Pursuers},
  url = {https://avonmoll.github.io/files/mp1e.pdf},
  year = {2018},
}

@article{vonmoll2019multi-pursuer,
  abstract = {We consider a general pursuit-evasion differential game with three or more pursuers and a single evader, all with simple motion. It is shown that traditional means of differential game analysis is difficult for this scenario. But simple motion and min-max time to capture plus the two-person extension to Pontryagin's maximum principle imply straight-line motion at maximum speed which forms the basis of the solution using a geometric approach. Safe evader paths and policies are defined which guarantee the evader can reach its destination without getting captured by any of the pursuers, provided its destination satisfies some constraints. A linear program is used to characterize the solution and subsequently the saddle-point is computed numerically. We replace the numerical procedure with a more analytical geometric approach based on Voronoi diagrams after observing a pattern in the numerical results. The solutions derived are open-loop optimal, meaning the strategies are a saddle-point equilibrium in the open-loop sense.},
  author = {Von Moll, Alexander and Casbeer, David and Garcia, Eloy and Milutinovi\'{c}, Dejan and Pachter, Meir},
  date = {2019-01-02},
  doi = {10.1007/s10846-018-0963-9},
  issue = {2},
  pages = {193-207},
  journal = {Journal of Intelligent and Robotic Systems},
  title = {The Multi-Pursuer Single-Evader Game: A Geometric Approach},
  url = {https://avonmoll.github.io/files/mp1e-journal.pdf},
  volume = {96},
  year = {2019},
}