Differential game with a ``life-line'' for nonlinear motion dynamics of players

We prove the existence of a value for pursuit games with state constraints. We also prove that this value is lower semicontinuous.

A linear two player zero-sum pursuit-evasion differential game is considered. The control functions of players are subject to integral constraints. In the game, the first player, the Pursuer, tries to force the state of the system towards the origin, while the aim of the second player, the Evader, is the opposite. We construct the optimal strategies of the players when the control resource of the Pursuer is greater than that of the Evader. The case where the control resources of the Pursuer are less than or equal to that of the Evader is studied to prove the main theorem. For this case a new method for solving of the evasion problem is proposed. We assume that the instantaneous control employed by the Evader is known to the Pursuer. For construction, the strategy of the Evader information about the state of the system and the control resources of the players is used. References R. Isaacs. Differential games. John Wiley and Sons, New York, 1965. L. S. Pontryagin. Collected works. Nauka, Moscow, 1988. (Russian) L. D. Berkovitz. Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM Journal on Control , 5 , 1--24, 1967. L. D. Berkovitz. A survey of differential games. Mathematical Theory of Control , Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, 373--385, 1967. N. N. Krasovskii and A. I. Subbotin. Game-theoretical control problems. New York, Springer, 1988. W. H. Fleming. The convergence problem for differential games. Journal of Mathematical Analysis and Applications . 3 , 102--116, 1961. W. H. Fleming. The convergence problem for differential games, Part 2. Advances in Game Theory, Annals of Mathematics Studies , (52), Princeton University Press, Princeton, New Jersey,195--210, 1964. A. Friedman. Differential games. Wiley-Interscience, New York, 1971. R. J. Elliott and N. J. Kalton. The existence of value in differential games. Memoirs of the American Mathematical Society , 126 , 1--67, 1972. L. A. Petrosyan. Differential games of pursuit. World Scientific, Singapore, London, 1993. O. Hajek. Pursuit games. Academic Press, New York, San Francisco, 1975. A. Ya. Azimov. Linear differential pursuit game with integral constraints on the control. Differentsial'nye Uravneniya, 11 (10), 1975, 1723--1731; English transl. in Differential Equations 11 , 1283--1289, 1975. A. Ya. Azimov. A linear differential evasion game with integral constraints on the controls. USSR Computational Mathematics and Mathematical Physics, 14 (6), 56--65, 1974. M. S. Nikolskii. The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, (2), 49--59, 1969. A. I. Subbotin and V. N. Ushakov. Alternative for an encounter-evasion differential game with integral constraints on the playersi controls. PMM 39 (3), 387--396, 1975. V. N. Ushakov. Extremal strategies in differential games with integral constraints. PMM , 36 (1), 15--23, 1972. B. N. Pshenichnii and Yu. N. Onopchuk. Linear differential games with integral constraints. Izvestige Akademii Nauk SSSR, Tekhnicheskaya Kibernetika , (1), 13--22, 1968. A. A. Azamov, B. Samatov. $\pi $-strategy. An elementary introduction to the theory of differential games. National University of Uzbekistan. Tashkent, Uzbekistan, 2000. G. I. Ibragimov. A game problem on a closed convex set. Siberian Advances in Mathematics . 12 (3), 16--31, 2002. G. I. Ibragimov. A problem of optimal pursuit in systems with distributed parameters. J. Appl. Math. Mech, 66 (5), 719--724, 2003. E. B. Lee and and L. Markus. Foundations of optimal control theory, John Wiley and Sons Inc., New York, 1967.

Plane Pursuit with Curvature Constraints

The main purpose of this work is to solve one of the main problems of Isaacs, i.e., a game with a “lifeline” for Pontryagin’s control example when both players have the same movement dynamics. To solve this problem, the pursuer is offered a strategy of parallel pursuit (briefly, $\Pi$-strategy), which ensures the fastest convergence of the players and the capture of the evader within a certain closed ball. In addition, for the differential game under consideration, an explicit analytical formula for the players' attainability domain is given and the main lemma is generalized (L.A. Petrosjan's lemma on monotonicity of the players' attainability domain with respect to embedding for a game of simple pursuit). Using this main lemma, we find conditions for the solvability of the game with a “lifeline” for Pontryagin's control example as well. For clarity, at the end of the work, examples are given for some special cases.

A fuzzy differential game theory is proposed to solve the n-person (or n-player) nonlinear differential noncooperative game and cooperative game (team) problems, which are not easily tackled by the conventional methods. In the paper, both noncooperative and cooperative quadratic differential games are considered. First, the nonlinear stochastic system is approximated by a fuzzy model. Based on the fuzzy model, a fuzzy controller is proposed to deal with the noncooperative differential game in the sense of Nash equilibrium strategies or with the cooperative game in the sense of Pareto-optimal strategies. Using a suboptimal approach, the outcomes of the fuzzy differential games for both the noncooperative and the cooperative cases are parameterized in terms of an eigenvalue problem. Since the state variables are usually unavailable, a suboptimal fuzzy observer is also proposed in this study to estimate the states for these differential game problems. Finally, simulation examples are given to illustrate the design procedures and to indicate the performance of the proposed methods.

The paper deals with the application of attainability domains for the solution of missile control problems. Methods of calculating the attainability domains are analyzed and two examples of calculating the attainability domains of a missile are given. A set of problems is presented for the solution of which the attainability domains were used. A conflicting problem of approach-evasion of two missiles with and without taking account of errors of measurements of motion parameters is discussed. The problem is considered as a differential game of two players with opposite interests. Controls of the players are selected at discrete points in time, based on the analysis of the relative position of attainability domains constructed for a number of the future meeting time points. If the errors of measuring are taken into account the attainability domains are constructed not from the current position but from the information domains that contain the exact values of the motion parameters. The approach problem of two missiles with a maneuvering object is presented in the form of a coalition differential game. In this case the combined attainability domains of the missiles are constructed. Minimax attainability domains constructed taking into account the action of disturbances are used in the problem of synthesis of normal acceleration of the missile under the action of disturbances. In the final part of the paper we discuss the problem of minimax filtration of the missile motion parameters in which information domains including the attainability domain of the missile are approximated by the parallelepipeds in the phase space under consideration.

In this chapter, a new optimal coordination control for the consensus problem of heterogeneous multi-agent differential graphical games by iterative ADP is developed. The main idea is to use iterative ADP technique to obtain the iterative control law which makes all the agents track a given dynamics and simultaneously makes the iterative performance index function reach the Nash equilibrium. In the developed heterogeneous multi-agent differential graphical games, the agent of each node is different from the one of other nodes. The dynamics and performance index function for each node depend only on local neighbor information. A cooperative policy iteration algorithm for graphical differential games is developed to achieve the optimal control law for the agent of each node, where the coupled Hamilton–Jacobi (HJ) equations for optimal coordination control of heterogeneous multi-agent differential games can be avoided. Convergence analysis is developed to show that the performance index functions of heterogeneous multi-agent differential graphical games can converge to the Nash equilibrium. Simulation results will show the effectiveness of the developed optimal control scheme.

Stochastic Differential Game Techniques

We study a fixed duration pursuit-evasion differential game problem of one pursuer and one evader with Grönwall-type constraints (recently introduced in the work of Samatov et al. (Ural Math J 6:95–107, 2020b)) imposed on all players’ control functions. The players’ dynamics are governed by a generalized dynamic equation. The payoff is the greatest lower bound of the distances between the evader and the pursuers when the game is terminated. The pursuers’ goal, which contradicts that of the evader, is to minimize the payoff. We obtained sufficient conditions for completion of pursuit and evasion as well. To this end, players’ attainability domain and optimal strategies are constructed.

The purpose of this work is to study the pursuit-evasion problem and the “Life-line” game for two objects (called Pursuer and Evader) with simple harmonic motion dynamics of the same type in the Euclidean space. In this case, the objects move by controlled acceleration vectors. The controls of the objects are subject to geometrical constraints. In the pursuit problem, the strategy of parallel pursuit (in brief, the [Formula: see text]-strategy) is suggested for the Pursuer, and by this strategy a capture condition is achieved. In the evasion problem, a constant control function is offered for the Evader, and an evasion condition is derived. Employing the [Formula: see text]-strategy we generate an analytic formula for the attainability domain of the Evader (the set of all the meeting points of the objects), and we prove the Petrosjan type theorem describing that the attainability domain is monotonically decreasing with respect to the inclusion in time. In the “Life-line” problem, first, by virtue of the [Formula: see text]-strategy solvability conditions to the advantage of the Pursuer are achieved and next, in constructing a reachable domain of the Evader by a control function, solvability conditions to the advantage of the Evader are identified. Differential games under harmonic motions are more complex owing to some troubles in determining optimal strategies and in building the meeting domain of objects. Accordingly, such types of games have not been fairly investigated than the simple motion games. From this point of view, studying the pursuit, evasion, and “Life-line” problems for oscillated motions arouses a special interest.

We study a pursuit differential game with many Pursuers when the Evader moves on the surface of a given cylinder. Maximal speeds of all players are equal. We consider two cases: in the first case, the Pursuers move arbitrarily without phase constraints; and in the second case, the Pursuers move on the surface of the cylinder. In both cases, we give necessary and sufficient conditions to complete the pursuit. In addition, in the second case, we show that pursuit differential game on a cylinder are equivalent to a differential game on the plane with many groups of Pursuers where each group consists of infinite number of pursuers having the same control parameter. References Isaacs, R. Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization. New York: Wiley, 1963. Petrov, N. N. A problem of group pursuit with phase constraints. J. Appl. Math. Mech. 1988, 52, No 6, 1030--1033. Petrosyan, L. A. Survival differential game with many participants. Dokl. Akad. Nauk USSR. 1965, 161, No 2, 285--287. Petrosyan, L. S. Differentsial'nye igry presledovaniya (Pursuit Differential Games). Leningrad (SPb): Leningr.State Univ(SPbSU), 1977. Pshennichnyi, B. N. Simple pursuit of several targets. Kibernetika. 1976, No 3, 145--146. Chernous'ko, F. L. A problem of evasion of several pursuers. J. Appl. Maths Mechs.1976, 40, No 1, 14--24. Ivanov R. P. Simple pursuit-evasion on a compact. Dokl. Akad. Nauk SSSR. 1980, 254, No 6, 1318--1321. Melikyan, A. A. and Ovakimyan, N. V. Singular trajectories in the problem of simple pursuit on a manifold. J. Appl. Math. Mech. 1991, 55, No 1, 42--48. Melikyan, A. A. and Ovakimyan, N. V. Differential Games of Simple Pursuit and Approach on Manifolds. Institute of Mechanics, National Academy of Sciences of Armenia. Yerevan. Preprint, 1993. Kuchkarov, A. Sh. The problem of optimal approach in locally euclidean spaces. Automation and Remote Control. 2007, 68, No 6, 974-978. Kuchkarov, A. Sh. A simple pursuit--evasion problem on a ball of a Riemannian manifold. Mathematical Notes. 2009, 85, No 2, 190--197. Azamov, A. On a problem of escape along a prescribed curve. J. Appl. Math. Mech. 1982, 46, No 4, 553--555. Kuchkarov, A. Sh. and Rikhsiev, B. B. on the solution of a pursuit problem with phase constraints. Automation and Remote Control. 2001, 62, No 8, 1259--1262. Ibragimov G. I. A game problem on a closed convex set. Siberian advances in mathematics., 2002, 12, No 3, 16--31. Nikulin, V. V. and Shafarevich, I. R. Geometriya i Gruppy (Geometric and Groups). Moscow: Nauka, 1983.

Optical manipulation techniques have been widely applied in the biomedical field. However, the key issues limiting the efficiency of optical manipulation techniques are the weak driving force of optical scattering and the small working range of optical gradient forces. The optothermal Marangoni convection enables effective control of flow fields through optical means, and particle manipulation based on this mechanism offers advantages such as a wide working range, strong driving force, and high flexibility. In recent years, it has been applied in fields such as biological cell manipulation and micro/nanomaterial assembly. However, current research predominantly focuses on particle manipulation in static environments, overlooking the potential applications of this method in dynamic and complex flow fields. In this study, we investigate particle manipulation methods based on optothermal Marangoni convection in dynamic flow fields. Through combined simulation and experiment, we systematically characterized flow field profile and particle trajectories under coupled "optothermal-flow" control, developed manipulation schemes with extended working range (>20 μm) and multiparticle capacity for trapping, assembly, and migration. Through laser spot positioning, we achieved real-time flow field modulation in microchannels, enabling versatile multimodal particle control. These findings demonstrate the substantial potential of optothermal Marangoni convection in microfluidic applications, offering a novel methodology for dynamic flow field regulation and high-efficiency on-chip particle manipulation.

PIV experimental study on dynamic and static interference flow field of multi-operating centrifugal pump under the influence of impeller wake

Pontryagin’s first and second (direct) methods and the so-called third pursuit method are the basic methods of the theory of differential games. We present various modifications of these methods. We analyze linear differential pursuit games for a delay equation under distinct constraints on the players’ control parameters. We give sufficient conditions for the solvability of the pursuit problem in finite time.

Identification - Inverse problems for partial differential equations: A stochastic formulation.- Key problems in the theory of centralized interaction of economic systems.- Some statements and ways of solving dynamic optimization problems under uncertainty.- A new algorithm for Gauss Markov identification.- On optimality criteria in identification problems.- Nonstationary processes for mathematical programming problems under the conditions of poorly formalized constraints and incomplete defining information.- Dynamic models of technological changes.- Identification and control for linear dynamic systems of unknown order.- Group choice and extremal problems in the analysis of qualitative attributes.- Studies in modelling and identification of distributed parameter systems.- Recursive solutions to indirect sensing measurement problems by a generalized innovations approach.- A system of models of output renewal.- Bilinear social and biological control systems.- A problem from the theory of observability.- Some questions of the modelling of complex chemical systems.- Evaluation model of humanistic systems by fuzzy multiple integral and fuzzy correlation.- Penalty function method and necessary optimum conditions in optimal control problems with bounded state variables.- Multilevel optimal control of interconnected distributed parameter systems.- Time optimal control problem for differential inclusions.- Application of maximum principle for optimization of pseudo-stationary catalytic processes with changing activity.- Approximate solution of optimal control problems using third order hermite polynomial functions.- Optimal stabilization of the distributed parameter systems.- About one problem of synthesis of optimum control by thermal conduction process.- About the problem of synthesis of optimum control by elastic oscillations.- On the partitioning problem in the synthesis of multilevel optimization structures.- On the problem of an optimal three-dimensional aircraft motion.- A general stochastic equation for the non-linear filtering problem.- On sufficiency of the necessary optimality of L.S. pontryagin's maximum principle analogues type.- On final stopping time problems.- Equilibrium and perturbations in plasma-vacuum systems.- Sufficient conditions for absolute minimum of the maximal functional in the multi - Criterial problem of optimal control.- Stratified universal manifolds and turnpike theorems for a class of optimal control problems.- On the numerical approximation of problems of impulse controls.- Satisficing.- On approximate solution of the problem with point and boundary control.- A-stable method for the solution of the cauchy problem for stiff systems of ordinary differential equations.- Some methods for numerical solution of optimal models in spatial-production planning.- An extension of the method of feasible directions.- A numerical method for solving linear control problems with mixed restrictions on control and phase coordinates.- Dual direction methods for function minimization.- Implementation of variable metric methods for constrained optimization based on an augmented lagrangian functional.- Limit extremum problems.- Algorithms for solving non-linear programming problems.- Structural optimization.- On the solution of a class of non linear dirichlet problems by a penalty-duality method and finite elements of order one.- Adaptive monte carlo method for solving constrained minimization problem in integer non-linear programming.- Application of the quadratic minimization method to the problem of simulated system characteristics representation.- Mathematical programming approach to a minimax theorem of statistical discrimination applicable to pattern recognition.- penalty methods and some applications of mathematical programming.- Numerical analysis of artificial enzyme membrane - Hysteresis, oscillations and spontaneous structuration.- The stability of optimal values in problems of discrete programming.- Optimal control with minimum problems and variational inequalities.- Dual minimax problems.- On the type of a polynomial relative to a circle - An open problem.- On bayesian methods for seeking the extremum.- Riemannian integral of set-valued function.- Characteristics of saturation of the class of convex functions.- A new heuristic method for general mixed integer linear programs: A report on work in progress (abstract).- Closed - Loop differential games.- A programmed construction for the positional control.- An extremal control in differential games.- Some differential games with incomplete information.- Some properties of nonzero-sum multistage games.- Equilibrium situations in games with a hierarchical structure of the vector of criteria.- A class of linear differential evasion games.- Analytical study of a case of the homicidal chauffeur game problem.- An informational game problem.- The pursuit game with the information lack of the evading player.- On constructing invariant sets in linear differential games.- A non cooperative game in a distributed parameter system.