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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received: 25.06.2014

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
\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

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    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  mathnet
    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  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  isi
    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  mathnet  crossref  elib
    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  mathnet  crossref  elib
    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  mathnet
    7. A. V. Chernov, “On application of Gaussian functions to numerical solution of optimal control problems”, Autom. Remote Control, 80:6 (2019), 1026–1040  mathnet  crossref  crossref  isi  elib
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