Upper semicontinuity of the solution map to a parametric elliptic optimal control problem
type: Item
instance of: scholarly article
title: Upper semicontinuity of the solution map to a parametric elliptic optimal control problem
zbMATH Open document ID: 1473.49031
author: Yanyan Li
published in: Set-Valued and Variational Analysis
publication date: 2021-08-12T00:00:00Z
review text: One investigates a parametric optimal control problem of the form: find a couple of control functions \((u,v)\), such that if \(y\) is the solution of a controlled Robin boundary value problem to an elliptic partial differential equation, an integral objective functional \(J (y, u, v, \mu)\) gets minimized. The coefficients in the elliptic model are assumed to satisfy some auxiliary constraints and depend on parameter \(\lambda\). The control variables act in the domain \(\Omega\) as well as on the boundary \(\partial \Omega\) = \(\Gamma\). The second section of the paper is devoted to basic definitions and results on \((s,\omega)\)-upper semicontinuity and \((s,s)\)-upper semicontinuity. In the third section one shows some sufficient conditions for the existence and uniqueness of a solution to the Robin boundary problem. Under some restrictive assumptions on coefficients, parameters and controls one proves necessary optimality conditions for the existence of optimal solutions. One writes the adjoint equation and shows the existence and uniqueness of an admissible solution to the stated control problem. The proofs can be found in the fourth section. In the fifth section some illustrative examples are presented.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49K40
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 7381142
zbMATH Keywords: parameteric optimal control
zbMATH Keywords: solution stability
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: continuity
zbMATH Keywords: mixed pointwise constraint
MaRDI profile type: Publication
full work available at URL: s11228-020-00546-0
OpenAlex ID: W3040957030
cites work: Q4073917
cites work: Set-valued analysis
cites work: Lipschitz stability for elliptic optimal control problems with mixed control-state constraints
cites work: Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
cites work: Boundary Control of Semilinear Elliptic Equations with Pointwise State Constraints
cites work: Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints
cites work: Direct methods in the calculus of variations
cites work: Perturbations, approximations and sensitivity analysis of optimal control systems
cites work: Lipschitz stability of solutions to some state-constrained elliptic optimal control problems
cites work: Q3470840
cites work: Lower Semicontinuity of the Solution Set to a Parametric Optimal Control Problem
cites work: Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints
cites work: Q5072164
cites work: Q4139197
cites work: Sufficient optimality conditions in stability analysis for state-constrained optimal control
cites work: Stability and sensitivity analysis for linear-quadratic optimal control subject to state constraints
cites work: Sensitivity analysis for optimal control problems subject to higher order state constraints
cites work: Stability Analysis for Nonlinear Optimal Control Problems Subject to State Constraints
cites work: Second-order conditions in stability analysis for state constrained optimal control
cites work: Q4789256
cites work: Hölder continuity of the solution map to an elliptic optimal control problem with mixed constraints
cites work: Second-order optimality conditions for boundary control problems with mixed pointwise constraints
cites work: On the semicontinuity of the solution map to a parametric boundary control problem
cites work: Q3558349
DOI: 10.1007/S11228-020-00546-0
description: scientific article; zbMATH DE number 7381142