On piecewise constant approximation in distributed optimization problems

A. V. Chernov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 264-279

Résumé

The paper is devoted to optimal control problems for distributed parameter systems representable by functional operator equations of Hammerstein type in a Banach space compactly embedded in a Lebesgue space. The problem of minimizing an integral functional on a set of “state-control” pairs satisfying a control equation of the mentioned type is considered. We prove that this problem is equivalent to an optimization problem obtained from the original one by passing to a description of the control system in terms of V.I. Sumin's functional operator equation in a Lebesgue space. The equivalent optimization problem is called S-dual. For an S-dual optimization problem, we investigate a piecewise constant approximation for the “state-control” pair. For this approximation method, we state the following results: (1) convergence of piecewise constant approximations with respect to the functional and the equation for the S-dual optimization problem; (2) existence of a global solution of an approximating finite-dimensional mathematical programming problem; (3) convergence with respect to the functional of solutions of an approximating optimization problem to a solution of the original problem. As an auxiliary result of independent interest, we prove a theorem on the total (over the whole set of admissible controls) preservation of solvability for a control equation of Hammerstein type. The Dirichlet problem for a semilinear elliptic equation of diffusion-reaction type is considered as an example of reducing a distributed parameter control system to such an equation.

Détail

Keywords: piecewise constant approximation; optimal control; equation of Hammerstein type; convergence by functional; total preservation of solvability; semilinear stationary diffusion-reaction equation.

Bibliographie

[1] Vasilev F.P., Ishmukhametov A.Z., Potapov M.M., Obobschennyi metod momentov v zadachakh optimalnogo upravleniya, Izd-vo MGU, M., 1989, 142 pp. | MR

[2] Potapov M.M., Razgulin A.V., “Raznostnye metody v zadachakh optimalnogo upravleniya statsionarnym samovozdeistviem svetovykh puchkov”, Zhurn. vychisl. matematiki i mat. fiziki, 30:8 (1990), 1157–1169 | MR

[3] Ishmukhametov A.Z., “Usloviya ustoichivosti i approksimatsii v zadachakh optimalnogo upravleniya giperbolicheskimi sistemami”, Zhurn. vychisl. matematiki i mat. fiziki, 34:1 (1994), 12–28 | MR | Zbl

[4] Vasilev F.P., Metody optimizatsii, Faktorial Press, M., 2002, 824 pp.

[5] Potapov M.M., “Raznostnaya approksimatsiya zadach dirikhle-nablyudeniya slabykh reshenii volnovogo uravneniya s kraevymi usloviyami III roda”, Zhurn. vychisl. matematiki i mat. fiziki, 47:8 (2007), 1323–1339 | MR

[6] Troltzsch F., “Semidiscrete Ritz-Galerkin approximations of nonlinear parabolic boundary control problems - strong convergence of optimal controls”, Appl. Math. Optimization, 29:3 (1994), 309–329 | DOI | MR | Zbl

[7] Chryssovergchi I., “Mixed discretization-optimization methods for relaxed optimal control of nonlinear parabolic systems”, Proc. of the 6th WSEAS International Conf. on Simulation, Modelling and Optimization, Lisbon, 2006, 41–47

[8] Fu H., “A characteristic finite element method for optimal control problems governed by convection-diffusion equations”, J. Comput. Appl. Math, 235:3 (2010), 825–836 | DOI | MR | Zbl

[9] Tsepelev I.A., “Approksimatsiya negladkikh reshenii retrospektivnoi zadachi dlya modeli konvektsii-diffuzii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:2 (2012), 281–290

[10] Isaev V.I., Shapeev V.P., “Razvitie metoda kollokatsii i naimenshikh kvadratov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:1 (2008), 41–60 | MR

[11] Chernov A.V., “O gladkikh konechnomernykh approksimatsiyakh raspredelennykh optimizatsionnykh zadach s pomoschyu diskretizatsii upravleniya”, Zhurn. vychisl. matematiki i mat. fiziki, 53:12 (2013), 2029–2043 | DOI | MR | Zbl

[12] Golubev Yu.F., Seregin I.A., Khayrullin R.Z., “The floating nodes method”, Sov. J. Comput. Syst. Sci, 30:2 (1992), 71–76 | MR

[13] Chernov A.V., “O gladkosti approksimirovannoi zadachi optimizatsii sistemy Gursa - Darbu na variruemoi oblasti”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:1 (2014), 305–321

[14] Chernov A.V., “O priblizhennom reshenii zadach optimalnogo upravleniya so svobodnym vremenem”, Vestn. Nizhegorod. un-ta im. N.I. Lobachevskogo, 2012, no. 6(1), 107–114 | MR

[15] Volkov Yu.S., Subbotin Yu.N., “50 let zadache Shenberga o skhodimosti splain-interpolyatsii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:1 (2014), 52–67

[16] Su Laiping, Abe Kenichi, “Optimal control for linear periodic systems by using multirate piecewise constant sampled state feedback”, Int. J. Syst. Sci., 24:2 (1993), 355–371 | DOI | MR | Zbl

[17] Ladyzhenskaya O.A., Uraltseva N.N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973, 576 pp. | MR

[18] Vorobev A.Kh., Diffuzionnye zadachi v khimicheskoi kinetike, Izd-vo MGU, M., 2003, 98 pp.

[19] Troltzsch F., Optimal control of partial differential equations: theory, methods and applications, Graduate Studies in Mathematics, 112, American mathematical society, Providence, 2010, 399 pp. | DOI | MR | Zbl

[20] 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. matematiki i mat. fiziki, 53:1 (2013), 20–46 | DOI | MR | Zbl

[21] Sumin V.I., “Funktsionalno-operatornye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami”, Dokl. AN SSSR, 305:5 (1989), 1056–1059 | MR | Zbl

[22] Sumin V.I., “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zhurn. vychisl. matematiki i mat. fiziki, 30:1 (1990), 3–21 | MR | Zbl

[23] Sumin V.I., Funktsionalnye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami. Ch.I, Izd-vo Nizhegorod. gos. un-ta, Nizhnii Novgorod, 1992, 110 pp.

[24] Vainberg M.M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972, 416 pp. | MR

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

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

[27] Chernov A.V., “O dostatochnykh usloviyakh upravlyaemosti nelineinykh raspredelennykh sistem”, Zhurn. vychisl. matematiki i mat. fiziki, 52:8 (2012), 1400–1414 | MR | Zbl

[28] Chernov A.V., “Ob upravlyaemosti nelineinykh raspredelennykh sistem na mnozhestve konechnomernykh approksimatsii upravleniya”, Vestn. Udm. un-ta. Matematika. Mekhanika. Kompyuternye nauki, 2013, no. 1, 83–98 | Zbl

[29] Krasnoselskii M.A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, M., 1956, 392 pp. | MR

[30] Kantorovich L.V., Akilov G.P., Funktsionalnyi analiz, Nauka, M., 1984, 752 pp. | MR

[31] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989, 464 pp. | MR

[32] Karchevskii M.M., Pavlova M.F., Uravneniya matematicheskoi fiziki. Dopolnitelnye glavy, Izd-vo Kazan. gos. un-ta, Kazan, 2012, 228 pp.