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Optimal control of a non-smooth elliptic PDE with non-linear term acting on the control

Livia Betz

TL;DR

The paper addresses optimal control of a non-smooth elliptic PDE where the control enters nonlinearly through a regularized Heaviside term and the state nonlinearity $\beta$ is non-differentiable. It develops a rigorous first-order theory by establishing a B-stationarity condition via directional differentiability of the control-to-state map, and then deriving a strong stationary system under two constraint qualifications that relate to the nonlinearity and the control constraints. The main contributions are the explicit adjoint system with a Clarke-style multiplier, sign conditions on the adjoint in non-smooth regions, and precise CQ assumptions ensuring equivalence to the primal first-order condition. These results advance non-smooth PDE-constrained optimization and connect to fixed-domain shape-optimization approaches, enabling limit analyses in subsequent p3/p1 work.

Abstract

This paper continues the investigations from [7] and is concerned with the derivation of first-order conditions for a control constrained optimization problem governed by a non-smooth elliptic PDE. The control enters the state equation not only linearly but also as the argument of a regularization of the Heaviside function. The non-linearity which acts on the state is locally Lipschitz-continuous and not necessarily differentiable, i.e., non-smooth. This excludes the application of standard adjoint calculus. We derive conditions under which a strong stationary optimality system can be established, i.e., a system that is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. For this, two assumptions are made on the unknown optimizer. Some of the presented findings are employed in the recent contribution [8], where limit optimality systems for non-smooth shape optimization problems [7] are established.

Optimal control of a non-smooth elliptic PDE with non-linear term acting on the control

TL;DR

The paper addresses optimal control of a non-smooth elliptic PDE where the control enters nonlinearly through a regularized Heaviside term and the state nonlinearity is non-differentiable. It develops a rigorous first-order theory by establishing a B-stationarity condition via directional differentiability of the control-to-state map, and then deriving a strong stationary system under two constraint qualifications that relate to the nonlinearity and the control constraints. The main contributions are the explicit adjoint system with a Clarke-style multiplier, sign conditions on the adjoint in non-smooth regions, and precise CQ assumptions ensuring equivalence to the primal first-order condition. These results advance non-smooth PDE-constrained optimization and connect to fixed-domain shape-optimization approaches, enabling limit analyses in subsequent p3/p1 work.

Abstract

This paper continues the investigations from [7] and is concerned with the derivation of first-order conditions for a control constrained optimization problem governed by a non-smooth elliptic PDE. The control enters the state equation not only linearly but also as the argument of a regularization of the Heaviside function. The non-linearity which acts on the state is locally Lipschitz-continuous and not necessarily differentiable, i.e., non-smooth. This excludes the application of standard adjoint calculus. We derive conditions under which a strong stationary optimality system can be established, i.e., a system that is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. For this, two assumptions are made on the unknown optimizer. Some of the presented findings are employed in the recent contribution [8], where limit optimality systems for non-smooth shape optimization problems [7] are established.
Paper Structure (7 sections, 11 theorems, 123 equations)

This paper contains 7 sections, 11 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Standing assumptions and preliminary results
  3. Optimality conditions for \ref{['p10']}
  4. B-stationarity
  5. Strong stationarity
  6. A condition on the given data for which Assumption \ref{['assu:h_a']} is satisfied
  7. Proof of Lemma \ref{['lem:tool']}

Key Result

Lemma 2.5

\newlabellem:S For any right-hand side $g \in L^2(D)$, the state equation in p10 admits a unique solution $y \in H^1_0(D) \cap H^2(D)$. This satisfies where $c>0$ is independent of $\beta, f, \varepsilon$ and $g$. The control-to-state mapping $S:L^2(D) \to H^1_0(D) \cap H^2(D)$ is Lipschitz continuous on bounded sets, i.e., for every $M > 0$, there exists $L_{M} > 0$ so that

Theorems & Definitions (33)

  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5: Solvability of the state equation, p1
  • Lemma 2.6: Convergence properties, p1
  • Proposition 2.7: Existence of solution for \ref{['p10']}, p1
  • Definition 3.1: p1
  • Remark 3.2
  • Remark 3.4
  • Lemma 3.5
  • proof
  • ...and 23 more