Existence of the optimal controls for a controlled elliptic system with an \(L^0\) term in the cost functional
type: Item
instance of: scholarly article
title: Existence of the optimal controls for a controlled elliptic system with an \(L^0\) term in the cost functional
zbMATH Open document ID: 1542.49004
author: Hongwei Lou
author: Ying Zhao
published in: Mathematical Control and Related Fields
publication date: 2024-06-17T00:00:00Z
review text: The primary aim of this paper is to establish the existence, regularity, and necessary conditions for optimal control in Problem (P). Demonstrating the existence of optimal controls is fundamental for characterizing and deriving them. Typically, a Cesari-type condition, which naturally generalized optimal control problems involving linear state equations with convex cost functionals, is required to ensure the existence of optimal (classical) controls.\N\NSeveral difficulties arose when considering the existence of the optimal controls for Problem (P). First, the control \( u(\cdot) \) is allowed to take values in \( (-\infty, +\infty) \), an open and unbounded range. To address this, a series of approximation problems, where the control took values in a compact set, is studied first. Second, since the map \( u \mapsto \Vert u \Vert_{0} \) is not convex and therefore not weakly lower semicontinuous from \( L_{p}(\Omega) \) to \( \mathbb{R} \), proving the existence of optimal controls in the usual way is infeasible. Consequently, a relaxed version is studied initially, where the existence and necessary conditions for the optimal relaxed controls are derived. It is proven that under our assumptions, every optimal relaxed control is classical. Additionally, \( F \) in the cost functional is an abstract strictly convex function, necessitating classification and detailed discussion.\N\NTo characterize the optimal controls, a necessary condition of Pontryagin's type is given. For the approximation problems, each optimal control coincide with an optimal relaxed control, whose necessary conditions are obtained using convex variational methods. For Problem (P), due to the non-smooth nature of the functional \( u \mapsto \Vert u \Vert_{0} \), spike perturbations are used to deduce the necessary condition. It is then shown that the optimal control is bounded and achieve \( C^{1,\lambda} \)-regularity of the optimal state.
reviewed by: Jaarat Gasimov
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7866363
zbMATH Keywords: optimal control
zbMATH Keywords: existence
zbMATH Keywords: elliptic equation
zbMATH Keywords: sparse control
zbMATH Keywords: relaxed control
MaRDI profile type: Publication
cites work: On a useful compactification for optimal control problems
cites work: A General Approach to Lower Semicontinuity and Lower Closure in Optimal Control Theory
cites work: New Existence Results for Optimal Controls in the Absence of Convexity: The Importance of Extremality
cites work: Comment on an existence result for a noncoercive nonconvex variational problem
cites work: Optimal control theory
cites work: First and Second Order Conditions for Optimal Control Problems with an $L^0$ Term in the Cost Functional
cites work: Q4743352
cites work: Characterization of maximum hands-off control
cites work: A convex analysis approach to optimal controls with switching structure for partial differential equations
cites work: Multi-bang control of elliptic systems
cites work: Nonconvex Variational Problems Related to a Hyperbolic Equation
cites work: Q5187597
cites work: Optimal Control with $L^p(\Omega)$, $p\in [0,1)$, Control Cost
cites work: Existence of optimal controls for semilinear elliptic equations without Cesari-type conditions
cites work: Existence of optimal controls for semilinear parabolic equations without Cesari-type conditions
cites work: Existence of optimal controls in the absence of Cesari-type conditions for semilinear elliptic and parabolic systems
cites work: The existence of optimal controls in the absence of convexity conditions
cites work: Existence theorems without convexity assumptions for optimal control problems governed by parabolic and elliptic systems
cites work: Existence theorems for optimization problems concerning linear, hyperbolic partial differential equations without convexity conditions
cites work: Iterative Hard-Thresholding Applied to Optimal Control Problems with $L^0(\Omega)$ Control Cost
cites work: Q5667133
DOI: 10.3934/MCRF.2023016
description: scientific article; zbMATH DE number 7866363