ON CALCULATING THE VALUE OF A DIFFERENTIAL GAME IN THE CLASS OF COUNTER STRATEGIES

Mikhail I. Gomoyunov     (Department of Dynamical Systems, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russian Federation)
Dmitry V. Kornev     (Institute of Mathematics and Computer Sciences, Ural Federal University; Department of Dynamical Systems, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russian Federation)

Abstract


For a linear dynamic system with control and disturbance, a feedback control problem is considered, in which the Euclidean norm of a set of deviations of the system’s motion from given targets at given times is optimized. The problem is formalized into a differential game in “strategy-counterstrategy” classes. A game value computing procedure, which reduces the problem to a recursive construction of upper convex hulls of auxiliary functions, is justified. Results of numerical simulations are presented. 


Keywords


Differential games, Value of the game, Saddle point, Counterstrategies

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References


Krasovskii N.N. Control of a dynamic system. Problem about the minimum of the guaranteed result. Moscow: Nauka, 1985. [in Russian]

Isaacs R. Differential Games. New York: John Wiley, 1965.

Krasovskii N.N. On the problem of unification of differential games // Doklady AN SSSR, 1976. Vol. 226, no. 6. P. 1260–1263.

Krasovskii A.N. Construction of mixed strategies on the basis of stochastic programs // J. Appl. Math. Mech., 1987. Vol. 51, no. 2. P. 144–149.

Krasovskii A.N., Krasovskii N.N. Control under lack of information. Birkhäuser, Berlin etc., 1995.

Subbotin A.I. Minimax inequalities and Hamilton-Jacobi equations. Moscow: Nauka, 1991. [in Russian]

Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Birkhäuser, Boston, 1995.

Subbotin A.I. Existence and Uniqueness Results for Hamilton-Jacobi Equations // Nonlinear Anal., 1991. Vol. 16, no. 7/8, P. 683–699.

Lukoyanov N.Yu. One differential game with nonterminal payoff // Izvestiya akademii nauk. Teoriya i sistemi upravleniya, 1997. No. 1, P. 85–90.

Lukoyanov N.Yu. The problem of computing the value of a differential game for a positional functional // J. Appl. Maths Mechs, 1998. Vol. 62, no. 2, P. 177–186.

Blagodatskikh V.I., Filippov A.F. Differential inclusions and optimal control // Proc. Steklov Inst. Math., 1986. No. 4, P. 199–259.

Krasovskii A.N., Reshetova T.N. Control under information deficiency: Study guide. UrGU, Sverdlovsk, 1990. [in Russian]

Kornev D.V. On Numerical Solution of Positional Differential Games with Nonterminal Payoff // Automation and Remote Control, 2012. Vol. 73, no. 11, P. 1808–1821.




DOI: http://dx.doi.org/10.15826/umj.2016.1.004

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