Vladimir Turetsky

Increasing pursuer capturability by using hybrid dynamics

Abstract A robust interception of a maneuverable target (evader) by an interceptor (pursuer) with hybrid dynamics is considered. The controls of the pursuer and the evader are bounded. The duration of the engagement is prescribed. The pursuer has two possible dynamic modes, which can be switched once during the engagement, while the dynamics of the evader are fixed. The case where for both dynamic modes there exists an unbounded capture zone was analyzed in our previous work. The conditions under which the pursuer can increase its capturability by utilizing the hybrid dynamics were established and the new robust capture zone was constructed. In the present paper, we extend this result to the cases where at least for one dynamic mode of the pursuer the capture zone is bounded. For these instances, conditions of increasing the pursuer’s hybrid capturability are derived. Respective capture zones are constructed. Illustrative examples and results of extensive simulation for a realistic non-linear engagement model in the presence of a random wind are given.

Proficiency testing: binary data analysis

A method for evaluating qualitative proficiency testing (PT) of laboratories conducting binary tests is proposed. The method is based on the scale-invariant item response model proposed by the authors in earlier publications. We consider the case where the laboratories under the PT conduct test consisting of a set of test items/species presenting different, but unknown beforehand levels of difficulty when trying to detect a particular property of theirs, and we need to evaluate/compare both the intrinsic abilities of the participating laboratories and the level of difficulty of the test items. We assume that the responses to different test items do not affect one another and discuss how to get and interpret the most likely estimates/scores. The method is illustrated by the example presented in a recent publication by our colleagues from QuoData GmbH and can be considered as an alternative to that proposed in their publication method of scoring.

Pursuit-Evasion Game of Kind Between Hybrid Players

A pursuit-evasion differential game of kind with bounded controls and prescribed duration is considered. Both players have two possible dynamics and both can switch between them once during the game. Each player knows the two possible dynamics of the other, but not the actual one. The optimal strategies of the players in this game include the order of the two modes and the time of the mode change between them. The optimal use of the mode change enlarges the winning zone of the respective player, compared to its winning zone when using fixed dynamics. An algorithmic example illustrates the complexity of the game with hybrid players.

Open-loop solution of a defender–attacker–target game: penalty function approach

A defender–attacker–target problem with non-moving target is considered. This problem is modelled by a pursuit-evasion zero-sum differential game with linear dynamics and quadratic cost functional....

Cheap control robust tracking: Insight by solving the problem with simple motions

A robust trajectory tracking problem is considered. Based on the solvability conditions, as well as the tracking and the control boundedness conditions, derived in a previous paper of the authors, the case of tracking with simple motions is analyzed. This case is studied both analytically and numerically, giving the insight for the cheap control approach to the robust tracking problem.

Book Review: Mathematical Game Theory and Applications Vladimir Mazalov, Wiley 2014

The book, written by Vladimir Mazalov, gives a fascinating view on the state-ofthe-art of a classical and modern Game Theory. It addresses a wide variety of game types (from matrix games to differential games with all conceivable stopovers), providing a great collection of examples (from economics to sport). The examples are not only illustrative but in many cases reflect the research results of the author (optimal stopping games, arbitration games, bioresource sharing models, fish wars, location problem, poker model, etc.) All the examples, presented in the book, are thoroughly elaborated, beginning with a rather simple, understandable, model, and gradually becoming more complicated. This also features a pedagogical value of the book: a reader, not so familiar with the mathematical game theory, an undergraduate student, for example, can use it as a helpful textbook, guiding her/him from the basics of the theory to its various complex applications. Numerous examples produce for a reader a “stable bridge” between mathematics and a real life of individuals and groups. The author emphasizes this close connection by animating the mathematical objects: sets in this book “enjoy” convexity, functions “suffer” from being discontinuous, and so on. The book consists of 10 chapters. First seven chapters are devoted to the noncooperative games: Chapters 1–3 deal with the normal-form games, while Chapters 4–7 address the extensive-form (positional) games. The essentials of the cooperative games are given in Chapter 8. In Chapter 9, the author discusses the network games. The last but not the least, Chapter 10 contains the theory of dynamic (discrete-time and differential) games. Each chapter is equipped by the set of the exercises. Chapter 1 deals with a normal-form two-players game, where the players choose their strategies from prescribed sets in the beginning of the game and their payoffs functions are given. Fundamental theorems on the existence of Nash equilibrium in

Optimal quadratic control of linear time delay systems: One approach to numerical solution

A finite-horizon linear-quadratic optimal control problem for systems with point-wise and distributed state delays is considered. Based on well known control optimality conditions, the solution of this problem is reduced to solution of the set of three Riccati-type matrix differential equations: an ODE and two first-order PDEs with two and three independent variables. In the paper, this set of Riccati-type equations is further reduced to a set of two equations. One of these equations is the ODE, while the other is a partial integro-differential equation. A numerical procedure of solution of this new set of equations is proposed. An illustrative example is presented.