FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints

被引：0

作者：

Kien, Bui Trong ^{[1
]}

Roesch, Arnd ^{[2
]}

Son, Nguyen Hai ^{[3
]}

Tuyen, Nguyen Van ^{[4
]}

机构：

[1] Vietnam Acad Sci & Technol, Inst Math, Dept Optimizat & Control Theory, 18 Hoang Quoc Viet Rd, Hanoi, Vietnam

[2] Univ Duisbug Essen, Fac Math, Thea Leymann Str 9, D-45127 Essen, Germany

[3] Hanoi Univ Sci & Technol, Sch Appl Math & Informat, 1 Dai Co Viet, Hanoi, Vietnam

[4] Hanoi Pedag Univ, Dept Math, 2 Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

来源：

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS | 2023年 / 197卷 / 01期

关键词：

Finite element method; Optimal control; Semilinear elliptic equation; First-and second-order optimality conditions; Convergence; Error estimate; FINITE-ELEMENT APPROXIMATION; NUMERICAL APPROXIMATION; POINTWISE CONTROL; STATE; DISCRETIZATION;

D O I：

10.1007/s10957-023-02187-3

中图分类号：

C93 [管理学]; O22 [运筹学];

学科分类号：

070105 ; 12 ; 1201 ; 1202 ; 120202 ;

摘要：

This paper studies the convergence and error estimates of approximate solutions to an optimal control problem governed by semilinear elliptic equations with non-convex cost function and non-convex mixed pointwise constraints, and unbounded constraint set. We discretize the optimal control problems by the finite element method in order to obtain a sequence of mathematical programming problems in finite-dimensional spaces. We show that under certain conditions, the optimal solutions of the obtained mathematical programming problems converge to an optimal solution of the original problem. In particular, if the original problem satisfies the so-called no-gap second-order conditions, then some error estimates of approximate solutions are obtained.

引用

页码：130 / 173

页数：44

共 26 条

[1] Error estimates for the numerical approximation of a semilinear elliptic control problem
Arada, N
Casas, E
Tröltzsch, F
[J]. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, 2002, 23 (02) : 201 - 229
[2] Brezis H., 2011, FUNCTIONAL ANAL SOBO, DOI DOI 10.1007/978-0-387-70914-7
[3] Casas E, 2002, CONTROL CYBERN, V31, P695
[4] Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations
Casas, Eduardo
Raymond, Jean-Pierre
[J]. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 2006, 45 (05) : 1586 - 1611
[5] Casas E, 2002, COMPUT APPL MATH, V21, P67
[6] A general theorem on error estimates with application to a quasilinear elliptic optimal control problem
Casas, Eduardo
Troeltzsch, Fredi
[J]. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, 2012, 53 (01) : 173 - 206
[7] NUMERICAL ANALYSIS OF SOME OPTIMAL CONTROL PROBLEMS GOVERNED BY A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS
Casas, Eduardo
Troeltzsch, Fredi
[J]. ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS, 2011, 17 (03) : 771 - 800
[8] FIRST- AND SECOND-ORDER OPTIMALITY CONDITIONS FOR A CLASS OF OPTIMAL CONTROL PROBLEMS WITH QUASILINEAR ELLIPTIC EQUATIONS
Casas, Eduardo
Troeltzsch, Fredi
[J]. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 2009, 48 (02) : 688 - 718
[9] Error estimates for the discretization of elliptic control problems with pointwise control and state constraints
Cherednichenko, S.
Roesch, A.
[J]. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS, 2009, 44 (01) : 27 - 55
[10] Ciarlet P. G., 1978, FINITE ELEMENT METHO

← 1 2 3 →