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 Trudy Inst. Mat. i Mekh. UrO RAN, 2015, Volume 21, Number 1, Pages 264–279 (Mi timm1163)

On piecewise constant approximation in distributed optimization problems

A. V. Chernovab

a N. I. Lobachevski State University of Nizhni Novgorod
b Institute of Radio Engineering and Information Technologies, Nizhniy Novgorod State Technical University

Abstract: 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.

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

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UDC: 517.957+517.988+517.977.56

Citation: A. V. Chernov, “On piecewise constant approximation in distributed optimization problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 1, 2015, 264–279

Citation in format AMSBIB
\Bibitem{Che15} \by A.~V.~Chernov \paper On piecewise constant approximation in distributed optimization problems \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2015 \vol 21 \issue 1 \pages 264--279 \mathnet{http://mi.mathnet.ru/timm1163} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3407900} \elib{http://elibrary.ru/item.asp?id=23137995} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Andrei V. Chernov, “O suschestvovanii ravnovesiya po Neshu v differentsialnoi igre, svyazannoi s ellipticheskimi uravneniyami: monotonnyi sluchai”, MTIP, 7:3 (2015), 48–78
2. F. V. Lubyshev, M. E. Fairuzov, “Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives”, Comput. Math. Math. Phys., 56:7 (2016), 1238–1263
3. A. V. Chernov, “On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator”, Russian Math. (Iz. VUZ), 61:6 (2017), 72–81
4. A. V. Chernov, “Ob ispolzovanii kvadratichnykh eksponent s variruemymi parametrami dlya approksimatsii funktsii odnogo peremennogo na konechnom otrezke”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 267–282
5. A. V. Chernov, “O primenenii kvadratichnykh eksponent dlya diskretizatsii zadach optimalnogo upravleniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 558–575
6. A. V. Chernov, “O totalnom sokhranenii odnoznachnoi globalnoi razreshimosti operatornogo uravneniya pervogo roda s upravlyaemoi dobavochnoi nelineinostyu”, Izv. vuzov. Matem., 2018, no. 11, 60–74
7. A. V. Chernov, “On application of Gaussian functions to numerical solution of optimal control problems”, Autom. Remote Control, 80:6 (2019), 1026–1040
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