Linear-convex guarantee optimization problems with control delay

M. I. Gomoyunov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 45 (2015) no. 1, pp. 37-105.

Résumé

A control problem under condition of disturbances is considered for a linear dynamical system with control delay. Optimized quality index is nonterminal and contains the evaluation of the system motion by a set of deviations at given instants of time from given targets. Depending on the structure of the quality index the existence of optimal control strategies in appropriate classes of feedback strategies is established. For calculating the value of the optimal guaranteed result and finding the optimal control laws a procedure of backward construction of upper convex hulls of auxiliary functions is proposed. In the case of the positional quality index a reduction of this procedure is performed. This reduction significantly reduces the dimension of the domains of covexified functions. Results of numerical simulations are given.

Détail

Keywords: control theory, differential games, control delay, feedback strategies.

Bibliographie

[1] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1987, 517 pp. | MR

[2] Subbotin A. I., Chentsov A. G., Guarantee optimization in control problems, Nauka, M., 1981, 288 pp.

[3] Krasovskii N. N., Control of a dynamic system, Nauka, M., 1985, 516 pp.

[4] Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkhäuser, Berlin etc., 1995, 322 pp. | MR

[5] Subbotin A. I., Generalized solutions of first-order PDEs: the dynamical optimization perspective, Birkhäuser, Boston etc., 1995, 312 pp. | MR

[6] Berdyshev Yu. I., Chentsov A. G., “Optimization of a weighted criterion function in one control problem”, Cybernetics, 22:1 (1986), 67–74 | MR | MR | Zbl

[7] Berdyshev Yu. I., “Problem of successive optimization without time decomposition”, Cybernetics, 23:4 (1987), 474–479 | Zbl

[8] Berdyshev Yu. I., “A problem of the sequential approach to a group of moving points by a third-order non-linear control system”, J. Appl. Math. Mech., 66:5 (2002), 709–718 | MR | Zbl

[9] Krasovskii A. N., “A differential game for the positional functional”, Sov. Math., Dokl., 22:1 (1980), 251–255 | MR | Zbl

[10] Krasovskii A. N., “On positional minimax control”, J. Appl. Math. Mech., 44:4 (1980), 602–610 | MR | Zbl

[11] Krasovskii A. N., “Nonlinear differential games with integral payoffs”, Differential Equations, 18:2 (1982), 1306–1312 | MR | Zbl

[12] Krasovskii N. N., Lukoyanov N. Yu., “Problem of conflict control with hereditary information”, J. Appl. Math. Mech., 60:6 (1996), 869–882 | MR | Zbl

[13] Kharatishvili G. L., “A maximum principle in external problems with delays”, Mathematical Theory on Control, Academic Press, New York, 1967, 26–34 | MR

[14] Halanay A., “Optimal controls for systems with time lag”, SIAM J. Control, 6:2 (1968), 215–234 | MR | Zbl

[15] Verzhbitski A., “Principle of maximum for processes with non-trivial delay of control”, Avtomat. i Telemekh., 1970, no. 10, 13–20 (in Russian)

[16] Banks H. T., Jakobs M. Q., Latina M. R., “The synthesis of optimal controls for linear, time-optimal problems with retarded controls”, J. Optim. Theor. Appl., 8:5 (1971), 319–366 | MR

[17] Klamka J., “Relative controllability and minimum energy control of linear systems with distributed delays in control”, IEEE Trans. Autom. Contr., 21:4 (1976), 594–595 | MR | Zbl

[18] Osipov Yu. S., Pimenov V. G., “On the theory of differential games in systems with aftereffect”, J. Appl. Math. Mech., 42:6 (1978), 969–977 | MR

[19] Olbrot A. W., “Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays”, IEEE Trans. Autom. Contr., 23:5 (1978), 887–890 | MR | Zbl

[20] Artstein Z., “Linear systems with delayed controls: a reduction”, IEEE Trans. Autom. Contr., 27:4 (1982), 869–879 | MR | Zbl

[21] Kwon W., Pearson A., “Feedback stabilization of linear systems with delayed control”, IEEE Trans. Autom. Contr., 25:2 (1980), 266–269 | MR | Zbl

[22] Osipov Yu. S., Pimenov V. G., “On positional control under aftereffect in the controlling forces”, J. Appl. Math. Mech., 45:2 (1981), 223–229 | MR | Zbl

[23] Vinter R. B., Kwong R. H., “The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach”, SIAM J. Control and Optim., 19:1 (1981), 139–153 | MR | Zbl

[24] Pritchard A. J., Salamon D., “The linear-quadratic control problem for retarded systems with delays in control and observation”, IMA J. Math. Control Information, 2 (1985), 335–362 | Zbl

[25] Pimenov V. G., “A differential game with fixed terminal time for systems with aftereffect in control”, Zadachi pozitsionnogo modelirovaniya, Sverdlovsk, 1986, 103–118 (in Russian)

[26] Delfour M. C., Karrakchou J., “State space theory of linear time invariant systems with delays in state, control, and observation variables. I; II”, J. Math. Anal. Appl., 125:2 (1987), 361–399 ; 400–450 | MR | Zbl

[27] Pandolfi L., “Dynamic stabilization of systems with input delays”, Automatica, 27:6 (1991), 1047–1050 | MR

[28] Mirkin L., Tadmor G., “$H^\infty$ control of system with I/O delay: a review of some problem-oriented methods”, IMA J. Math. Control Information, 19 (2002), 185–199 | MR | Zbl

[29] Federico S., Tacconi E., “Dynamic programming for optimal control problems with delays in the control variable”, SIAM J. Control Optim., 52:2 (2014), 1203–1236 | MR | Zbl

[30] Fleming W. H., “The convergence problem for differential games”, J. Math. Anal. Appl., 1961, no. 3, 102–116 ; No 4, 764–766 | MR | Zbl | Zbl

[31] Pontryagin L. S., “Linear differential games. 1; 2”, Sov. Math., Dokl., 8 (1967), 769–771 ; 910–912 | Zbl | Zbl

[32] Pschenichnyi B. N., “The structure of differential games”, Sov. Math., Dokl., 10 (1969), 70–72 | MR | Zbl

[33] Pschenichnyi B. N., Sagaidak M. I., “Differential games of prescribed duration”, Cybernetics, 6:2 (1970), 72–80 | MR

[34] Friedman A., Differential games, Wiley Interscience, New York, 1971, 368 pp. | MR | Zbl

[35] Pontryagin L. S., “Linear differential games of pursuit”, Mathematics of the USSR-Sbornik, 40:2 (1981), 285–303 | MR | Zbl | Zbl

[36] Ushakov V. N., “On the problem of constructing stable bridges in a differential game of approach and avoidance”, Eng. Cybernetics, 18:4 (1980), 16–23 | MR | MR | Zbl

[37] Isakova E. A., Logunova G. V., Patsko V. S., “Computation of stable bridges for linear differential games with fixed time of termination”, Algoritmy i programmy resheniya lineinykh differentsial'nykh igr, Ural Scientific Center, Sverdlovsk, 1984, 127–158 (in Russian) | Zbl

[38] Grigorenko N. L., Kiselev Yu. N., Lagunova N. V., Silin D. B. et al., “Solution methods for differential games”, Computational Mathematics and Modeling, 7:1 (1996), 101–116 | MR

[39] Patsko V. S., Turova V. L., Numerical solution of two-dimensional differential games, IMM UrO RAN, Yekaterinburg, 1995, 78 pp.

[40] Ushakov V. N., Khripunov A. P., “Approximate construction of solutions in game-theoretic control problems”, J. Appl. Math. Mech., 61:3 (1997), 401–408 | MR | Zbl

[41] Polovinkin E. S., Ivanov G. E., Balashov M. V., Konstantinov R. V., Khorev A. V., “An algorithm for the numerical solution of linear differential games”, Sbornik: Mathematics, 192:10 (2001), 1515–1542 | MR | Zbl

[42] Kumkov S. S., Patsko V. S., “Construction of singular surfaces in linear differential games”, Adv. in Dynamic Games and Applications, Annals of the Intern. Soc. of Dynamic Games, 6, 2001, 185–202 | MR | Zbl

[43] Mikhalev D. K., Ushakov V. N., “Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit”, Automation and Remote Control, 68:11 (2007), 2056–2070 | MR | Zbl

[44] Dvurechensky P. E., Ivanov G. E., “Algorithms for computing Minkowski operators and their application in differential games”, Computational Mathematics and Mathematical Physics, 54:2 (2014), 235–264 | MR | Zbl

[45] Tarasyev A. M., “Approximation schemes for constructing minimax solutions of Hamilton–Jacobi equations”, J. Appl. Math. Mech., 58:2 (1994), 207–221 | MR | Zbl

[46] Tarasyev A. M., Uspenskii A. A., Ushakov V. N., “Approximation schemes and finite difference operators for constructing generalized solutions of the Hamilton–Jacobi equations”, Journal of Computer and System Sciences Inernational, 33:6 (1995), 127–139 | MR | MR

[47] Bardi M., Falcone M., Soravia P., “Numerical methods for pursuit-evasion games via viscosity solutions”, Stochastic and Differential Games, Birkhäuser, Boston, 1999, 105–175 | MR | Zbl

[48] Ivanov G. E., Kazeev V. A., “Minimax algorithm for constructing an optimal control strategy in differential games with a lipschitz payoff”, Computational Mathematics and Mathematical Physics, 51:4 (2011), 550–574 | MR | Zbl

[49] Botkin N. D., Hoffmann K.-H., Turova V. L., “Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations”, SIAM J. Sci. Comput., 33:2 (2011), 992–1007 | MR | Zbl

[50] Chentsov A. G., “On a game problem of converging at a given instant of time”, Mathematics of the USSR-Sbornik, 28:3 (1976), 353–376 | MR | Zbl

[51] Chistyakov S. V., “On solving pursuit game problems”, J. Appl. Math. Mech., 41:5 (1977), 845–852 | MR

[52] Cardaliaguet P., Quincampoix M., Saint-Pierre P., “Set-valued numerical analysis for optimal control and differential games”, Stochastic and differential games, Birkhäuser, Boston, 1999, 177–247 | MR | Zbl

[53] Krasovskii A. N., “Construction of mixed strategies on the basis of stochastic programs”, J. Appl. Math. Mech., 51:2 (1987), 144–149 | MR | Zbl

[54] Lukoyanov N. Yu., “The problem of computing the value of a differential game for a positional functional”, J. Appl. Math. Mech., 62:2 (1998), 177–186 | MR | Zbl

[55] Krasovskii N. N., Tretiakov V. E., “Stochastic program synthesis for a positional differential game”, Dokl. Akad. Nauk SSSR, 259:1 (1981), 24–27 (in Russian) | MR | Zbl

[56] Krasovskii N. N., Reshetova T. N., “On the program synthesis of a guaranteed control”, Probl. Contr. Inform. Theory, 17:6 (1988), 333–343 | MR

[57] Kovrizhnykh A. Yu., “On the problem of conflict control with a quasipositional functional”, Proceedings of the Steklov Institute of Mathematics, 2010, 79–93

[58] Lokshin M. D., “Differential games with integral restrictions on the controlling actions”, Differential Equations, 28:11 (1992), 1602–1610 | MR | Zbl

[59] Lukoyanov N. Yu., “The problem of conflicting control with mixed constraints”, J. Appl. Math. Mech., 59:6 (1995), 911–919 | MR | Zbl

[60] Lukoyanov N. Yu., “How to compute the value of a positional differential game”, Differential Equations, 37:1 (2001), 17–26 | MR | Zbl

[61] Gomoyunov M. I., Lukoyanov N. Yu., “On the stability of a procedure for solving a minimax control problem for a positional functional”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, no. 1, 2014, 68–82 (in Russian)

[62] Kornev D. V., “On numerical solution of positional differential games with nonterminal payoff”, Automation and Remote Control, 73:11 (2012), 1808–1821 | MR | Zbl

[63] Gomoyunov M. I., Kornev D. V., Lukoyanov N. Yu., “On the numerical solution of a minmax control problem with a positional functional”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, no. 3, 2014, 58–75 (in Russian)

[64] Isaacs R., Differential games, John Wiley and Sons, Inc., New York, 1965, 384 pp. | MR | Zbl

[65] Gomoyunov M. I., Kornev D. V., “On calculating the value of a differential game in the class of counterstrategies”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, no. 1, 2013, 59–68 (in Russian)

[66] Kornev D. V., Lukoyanov N. Yu., “On numerical solution of differential games with nonterminal payoff in classes of mixed strategies”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 3, 34–48 (in Russian)

[67] Krasovskii N. N., Some problems of the theory of stability of motion, Fizmatgiz, M., 1959, 211 pp.

[68] Gomoyunov M. I., “On the problem of optimizing the guarantee in a system with delay in control”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 3, 21–36 (in Russian)

[69] Gomoyunov M. I., Lukoyanov N. Yu., “Guarantee optimization in functional-differential systems with a control aftereffect”, J. Appl. Math. Mech., 76:4 (2012), 369–377 | MR

[70] Gomoyunov M. I., “The optimization of a guaranteed result with a delay in the control”, J. Appl. Math. Mech., 77:5 (2013), 459–469 | MR

[71] Gomoyunov M., Kornev D., Lukoyanov N., “Game theory applications to guarantee optimization in dynamical systems with control delays”, International Game Theory Review, 16:2 (2014), 1440010, 19 pp. | MR | Zbl

[72] Lukoyanov N. Yu., “A differential game with integral performance criterion”, Differential Equations, 30:11 (1994), 1759–1766 | MR | Zbl

[73] Ioffe A. D., Tikhomirov V. M., Theory of extremal problems, Nauka, M., 1974, 480 pp.

[74] Fan K., “Minimax theorems”, Proc. Nat. Acad. Sci. USA, 39:1 (1953), 42–47 | MR