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On a majorant-minorant criterion for the total preservation of global solvability of distributed controlled systems. (English. Russian original) Zbl 1336.93084
Differ. Equ. 52, No. 1, 111-121 (2016); translation from Differ. Uravn. 52, No. 1, 112-122 (2016).
Summary: For distributed controlled systems that can be represented by a functional-operator equation of the Hammerstein type with an additional term on the right-hand side in the form of a linear operator (an extended equation of the Hammerstein type), we prove a criterion for the total (over the entire set of admissible controls) preservation of global solvability. In this connection, we develop an earlier-proved majorant-minorant criterion for the total preservation of global solvability. As an example, we study a controlled semilinear integro-differential equation describing radiation transport with Compton scattering diagram. The choice of this example is explained, in particular, by the fact that the preceding version of the majorant-minorant criterion is not applicable to it.

93C25 Control/observation systems in abstract spaces
93B28 Operator-theoretic methods
93C05 Linear systems in control theory
Full Text: DOI
[1] Kalantarov, V.K.; Ladyzhenskaya, O.A., Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Nauchn. Sem. LOMI, 69, 77-102, (1977) · Zbl 0354.35054
[2] Sumin, V.I., The features of gradient methods for distributed optimal-control problems, USSR Comput. Math. Math. Phys., 30, 1-15, (1990) · Zbl 0719.49003
[3] Sumin, V.I., Funktsional’nye vol’terrovy uravneniya v teorii optimal’nogo upravleniya raspredelennymi sistemami (Volterra functional equations in optimal control theory for distributed systems), (1992)
[4] Chernov, A.V., On the convergence of the conditional gradient method in distributed optimization problems, Comput. Math. Math. Phys., 51, 1510-1523, (2011) · Zbl 1274.49037
[5] Chernov, A.V., Smooth finite-dimensional approximations of distributed optimization problems via control discretization, Comput. Math. Math. Phys., 53, 1839-1852, (2013) · Zbl 1313.49034
[6] Chernov, A.V., On the smoothness of an approximated optimization problem for a Goursat-Darboux system on a varied domain, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, 305-321, (2014)
[7] Chernov, A.V., Sufficient conditions for the controllability of nonlinear distributed systems, Comput. Math. Math. Phys., 52, 1115-1127, (2012) · Zbl 1274.93032
[8] Chernov, A.V., On Volterra functional operator games on a given set, Autom. Remote Control, 75, 787-803, (2014)
[9] Chernov, A.V., On e-equilibrium in noncooperative functional operator n-person games, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, 316-328, (2013)
[10] Chernov, A.V., On existence of e-equilibrium in noncooperative n-person games associated with elliptic partial differential equations, Mat. Teor. Igr Prilozhen., 6, 91-115, (2014) · Zbl 1309.91025
[11] Chernov, A.V., A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Math., 55, 85-95, (2011) · Zbl 1244.47064
[12] Chernov, A.V., A majorant-minorantcriterion for the total preservation of global solvability of a functional operator equation, Russian Math., 56, 55-65, (2012) · Zbl 1345.39013
[13] Chernov, A.V., On a generalization of the method of monotone operators, Differ. Equ., 49, 517-527, (2013) · Zbl 1279.47079
[14] Yarovenko, I.P., On the solvability of a boundary value problem for the radiation transport equation with regard of Compton scattering, Dalnevost. Mat. Zh., 14, 109-121, (2014) · Zbl 1335.35245
[15] Chernov, A.V., On controllability of nonlinear distributed systems on a set of discretized controls, Vestn. Udmurt. Univ. Mat. Mekh., 1, 83-98, (2013) · Zbl 1299.93028
[16] Sumin, V.I.; Chernov, A.V., Operators in spaces of measurable functions: the Volterra property and quasinilpotency, Differ. Equ., 34, 1403-1411, (1998) · Zbl 0958.47014
[17] Pugachev, V.S., Lektsii po funktsional’nomu analizu (Lectures on Functional Analysis), Moscow, 1996.
[18] Mordukhovich, B.Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya (Approximation Methods in Problems of Optimization and Control), Moscow: Nauka, 1988. · Zbl 0643.49001
[19] Chernov, A.V., On Volterra type generalization of monotonization method for nonlinear functional operator equations, Vestn. Udmurt. Univ. Mat. Mekh., 2, 84-99, (2012) · Zbl 1299.47120
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