Geodesic
On the structure of a~solution set of controlled initial-boundary value problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2016), pp. 75-86.

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For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As concerns boundedness and precompactness, we mean certain conditions of the majorant type without Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. In the capaсity of examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation and verification the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.
Keywords: reachability tube, convexity conditions, total preservation of solvability, functional-operator equation of the Hammerstein type, nonlinear distributed system
Mots-clés : parabolic equation, elliptic equation. 
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A. V. Chernov. On the structure of a~solution set of controlled initial-boundary value problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2016), pp. 75-86. http://geodesic.mathdoc.fr/item/IVM_2016_2_a9/

[1] Chernov A. V., “O vypuklosti mnozhestv dostizhimosti upravlyaemykh nachalno-kraevykh zadach”, Differents. uravneniya, 50:5 (2014), 702–712 | DOI | MR | Zbl

[2] Polyak B., “Convexity of the reachable set of nonlinear systems under $L_2$ bounded controls”, Dynamics of continuous, discrete and impulsive systems. Ser. A: Math. Anal., 11:2–3 (2004), 255–268 | MR

[3] Raisig G., “Vypuklost mnozhestv dostizhimosti sistem upravleniya”, Avtomatika i telemekhanika, 2007, no. 9, 64–78 | MR | Zbl

[4] Cannarsa P., Sinestrari C., “Convexity properties of the minimum time function”, Calc. Var. Partial Diff. Equat., 3:3 (1995), 273–298 | DOI | MR | Zbl

[5] Chernov A. V., “Ob odnom mazhorantnom priznake totalnogo sokhraneniya globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Izv. vuzov. Matem., 2011, no. 3, 95–107 | MR | Zbl

[6] Chernov A. V., “O mazhorantno-minorantnom priznake totalnogo sokhraneniya globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Izv. vuzov. Matem., 2012, no. 3, 62–73 | MR | Zbl

[7] Chernov A. V., “O volterrovom obobschenii metoda monotonizatsii dlya nelineinykh funktsionalno-operatornykh uravnenii”, Vestn. Udmurtsk. un-ta. Matem. Mekhan. Kompyuternye nauki, 2012, no. 2, 84–99 | Zbl

[8] Chernov A. V., “Ob odnom obobschenii metoda monotonnykh operatorov”, Differents. uravneniya, 49:4 (2013), 535–544 | MR | Zbl

[9] Potapov D. K., “Zadachi upravleniya dlya uravnenii so spektralnym parametrom i razryvnym operatorom pri nalichii vozmuschenii”, Zhurn. Sib. un-ta. Ser. matem. i fiz., 5:2 (2012), 239–245

[10] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972 | MR

[11] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988 | MR

[12] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, M., 1956 | MR

[13] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984 | MR

[14] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[15] Chernov A. V., “O neotritsatelnosti resheniya pervoi kraevoi zadachi dlya parabolicheskogo uravneniya”, Vestn. Nizhegorodsk. un-ta im. N. I. Lobachevskogo, 2012, no. 5(1), 167–170

[16] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[17] Vorobev A. Kh., Diffuzionnye zadachi v khimicheskoi kinetike, MGU, M., 2003

[18] Tröltzsch F., Optimal control of partial differential equations: theory, methods and applications, Graduate Stud. Math., 112, American Math. Soc., Providence, R.I., 2010 | MR | Zbl

[19] Lubyshev F. V., Manapova A. R., “Raznostnye approksimatsii zadach optimizatsii dlya polulineinykh ellipticheskikh uravnenii v vypukloi oblasti s upravleniyami v koeffitsientakh pri starshikh proizvodnykh”, Zhurn. vychisl. matem. i matem. fiz., 53:1 (2013), 20–46 | DOI | MR | Zbl

[20] Vakhitov I. S., “Obratnaya zadacha identifikatsii starshego koeffitsienta v uravnenii diffuzii-reaktsii”, Dalnevostochn. matem. zhurn., 10:2 (2010), 93–105 | MR

[21] Karchevskii M. M., Pavlova M. F., Uravneniya matematicheskoi fiziki. Dopolnitelnye glavy, KGU, Kazan, 2012

[22] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR