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On the convexity of reachability sets of controlled initial-boundary value problems. (English. Russian original) Zbl 1294.93018
Differ. Equ. 50, No. 5, 700-710 (2014); translation from Differ. Uravn. 50, No. 5, 702-712 (2014).
Summary: For a functional-operator equation describing a broad class of controlled initial-boundary value problems, we introduce the notion of abstract reachability set. We obtain sufficient conditions for the convexity and precompactness of that set. The situation of a Nash $$\varepsilon$$-equilibrium is justified in the sense of program strategies in noncooperative functional-operator games with many players. As an example of reduction of a controlled initial-boundary value problem to the equation under study, we consider the Cauchy problem for a semilinear wave equation with two space variables.

##### MSC:
 93B03 Attainable sets, reachability 93C25 Control/observation systems in abstract spaces 91A10 Noncooperative games
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##### References:
 [1] Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov (Estimation of the Phase State of Dynamical Systems. Method of Ellipsoids), Moscow, 1988. · Zbl 1124.93300 [2] Zuazua, E., Controllability and Observability of Partial Differential Equations: Some Results and Open Problems. Handbook of Differential Equations: Evolutionary Equations, vol. III. Amsterdam, 2007, pp. 527-621. · Zbl 1193.35234 [3] Unsolved Problems in Mathematical Systems and Control Theory, Blondel, V.D. and Megretski, A., Eds., Princeton; Oxford, 2004. · Zbl 1052.93002 [4] Petrosyan, L.A. and Zakharov, V.V., Vvedenie v matematicheskuyu ekologiyu (Introduction to Mathematical Ecology), Leningrad: Leningrad. Univ., 1986. [5] Vorob’ev, N.N., Teoriya igr dlya ekonomistov-kibernetikov (Game Theory for Economists and Cybernetics), Moscow: Nauka, 1985. · Zbl 0592.90098 [6] Egorov, A.I., Osnovy teorii upravleniya (Foundations of the Control Theory), Moscow, 2005. [7] Vakhrameev, SA, A remark on the convexity in smooth nonlinear systems, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Optim. Upravl. 1, 60, 42-73, (1999) [8] Topunov, MV, Convexity of reachable sets of a smooth linear control system in phase variables, Automation and Remote Control, 65, 1761-1766, (2004) · Zbl 1095.93003 [9] Polyak, B; 2, Convexity of the reachable set of nonlinear systems under $$L$$_{2} bounded controls, Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal., 11, 255-268, (2004) · Zbl 1050.93007 [10] Reißig, G, Convexity of reachable sets of nonlinear ordinary differential equations, Automation and Remote Control, 68, 1527-1543, (2007) · Zbl 1145.93007 [11] Cannarsa, P; Sinestrari, C, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations, 3, 273-298, (1995) · Zbl 0836.49013 [12] Krabs, W; Sklyar, GM; Wozniak, J, On the set of reachable states in the problem of controllability of rotating Timoshenko beams, J. Anal. Appl., 22, 215-228, (2003) · Zbl 1097.93004 [13] Djebali, S; Gorniewicz, L; Ouahab, A, First-order periodic impulsive semilinear differential inclusions: existence and structure of solution sets, Math. Comput. Modelling, 52, 683-714, (2010) · Zbl 1202.34110 [14] Tolstonogov, A.A., Differentsial’nye vklyucheniya v banakhovom prostranstve (Differential Inclusions in a Banach Space), Novosibirsk: Nauka, 1986. · Zbl 0689.34014 [15] Chernov, AV, On Volterra functional operator games on a given set, Automation and Remote Control, 75, 787-803, (2014) [16] Chernov, AV, A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Math., 55, 85-95, (2011) · Zbl 1244.47064 [17] Chernov, AV, A majorant-minorantcriterion for the total preservation of global solvability of a functional operator equation, Russian Math., 56, 55-65, (2012) · Zbl 1345.39013 [18] Chernov, AV, Sufficient conditions for the controllability of nonlinear distributed systems, Comput. Math. Math. Phys., 52, 1115-1127, (2012) · Zbl 1274.93032 [19] Chernov, AV, On the convergence of the conditional gradient method in distributed optimization problems, Comput. Math. Math. Phys., 51, 1510-1523, (2011) · Zbl 1274.49037 [20] Chernov, AV, On the existence of an $$ɛ$$-equilibrium in Volterra functional-operator games without discrimination, Mat. Teor. Igr Prilozh., 4, 74-92, (2012) · Zbl 1273.91062 [21] Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Conditions of Uncertainty), Moscow: Nauka, 1977. · Zbl 0461.93001 [22] Gurman, VI; Trushkova, EA, Estimates for attainability sets of control systems, Differential Equations, 45, 1636-1644, (2009) · Zbl 1187.93009 [23] Chernov, AV, On the convexity of global solvability sets for controlled initial-boundary value problems, Differential Equations, 48, 586-595, (2012) · Zbl 1248.49036 [24] Mordukhovich, B.Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya (Approximation Methods in Problems of Optimization and Control), Moscow: Nauka, 1988. · Zbl 0643.49001 [25] Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1984. · Zbl 0555.46001 [26] Fedorov, V.M., Kurs funktsional’nogo analiza (The Course of Functional Analysis), St. Petersburg, 2005. [27] Sukharev, A.G., Timokhov, A.V., and Fedorov, V.V., Kurs metodov optimizatsii (Course of Optimization Methods), Moscow, 2005. · Zbl 0602.90091 [28] Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1976. · Zbl 0235.46001 [29] Krasnosel’skii, M.A., Topologicheskie metody v teorii nelineinykh integral’nykh uravnenii (Topological Methods in the Theory of Nonlinear Integral Equations), Moscow: Gosudarstv. Izdat. Tekhn.-Teor. Lit., 1956. [30] Kleimenov, AF, Universal solution in a nonantogonistic positional differential game with vector criteria, Tr. Inst. Mat. Mekh. Ural. Otdel. RAN, 1, 97-105, (1992) · Zbl 0815.90144 [31] Mikhailov, V.P., Differentsial’nye uravneniya v chastnykh proizvodnykh (Partial Differential Equations), Moscow: Nauka, 1976. [32] Ladyzhenskaya, O.A., Smeshannaya zadacha dlya giperbolicheskogo uravneniya (Mixed Problem for a Hyperbolic Equation), Moscow: Gosudarstv. Izdat. Tekhn.-Teor. Lit., 1953.
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