Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems

The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we f...

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Published inJournal of global optimization Vol. 84; no. 3; pp. 783 - 805
Main Authors Cen, Jinxia, Haddad, Tahar, Nguyen, Van Thien, Zeng, Shengda
Format Journal Article
LanguageEnglish
Published New York Springer US 01.11.2022
Springer
Springer Nature B.V
Subjects
Applied mathematics
Approximation
Asymptotic properties
Boundary conditions
Computer Science
Control theory
Convergence
Differential equations
Mathematics
Mathematics and Statistics
Nonlinear systems
Operations Research/Decision Theory
Operators (mathematics)
Optimal control
Optimization
Partial differential equations
Real Functions
Uniqueness
47J20
Simultaneous optimal control problem
49J20
49J40
90C33
Asymptotic behavior
Nonhomogeneous partial differential operator
Complementarity problem
Mixed boundary condition
35J25
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Summary:The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-022-01155-x