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Second-order conditions for bang-bang control of elliptic equations in arbitrary dimensions

Gerd Wachsmuth

TL;DR

This work develops a dimension-free second-order analysis for bang-bang controls in semilinear elliptic PDEs by employing dual Bessel potential spaces $L^{\alpha,p}(\mathbb{R}^d)^*$. It proves a key norm-transport inequality for measures and derives a growth inequality that replaces the classical $L^1$ estimate with a dual Bessel norm, enabling quadratic growth and second-order optimality conditions in arbitrary dimensions. The abstract SSC framework is then instantiated with two choices of the dual space (measure space $\mathcal{M}(\Omega)$ and $L_0^{\alpha,p}(\Omega)^*$), showing the equivalence of second subderivatives and providing explicit bang-bang SSC conditions. Consequently, the main bang-bang optimality condition $F''(\bar{u})h^2+\int_{\{\bar{\varphi}=0\}}( |\nabla\bar{\varphi}|/2 ) h^2 \,d\mathcal{H}^{d-1} >0$ characterizes quadratic growth in arbitrary dimensions, extending prior $d\le3$ results and implying dimension-agnostic convergence implications for semismooth Newton methods.

Abstract

We consider an optimal control problem governed by a semilinear PDE in cases where the optimal control is of bang-bang type. By utilizing the theory of Bessel potential space, we characterize quadratic growth of the objective via a second-order optimality condition. In contrast to previous contributions, our method of proof works in arbitrary spatial dimensions.

Second-order conditions for bang-bang control of elliptic equations in arbitrary dimensions

TL;DR

This work develops a dimension-free second-order analysis for bang-bang controls in semilinear elliptic PDEs by employing dual Bessel potential spaces . It proves a key norm-transport inequality for measures and derives a growth inequality that replaces the classical estimate with a dual Bessel norm, enabling quadratic growth and second-order optimality conditions in arbitrary dimensions. The abstract SSC framework is then instantiated with two choices of the dual space (measure space and ), showing the equivalence of second subderivatives and providing explicit bang-bang SSC conditions. Consequently, the main bang-bang optimality condition characterizes quadratic growth in arbitrary dimensions, extending prior results and implying dimension-agnostic convergence implications for semismooth Newton methods.

Abstract

We consider an optimal control problem governed by a semilinear PDE in cases where the optimal control is of bang-bang type. By utilizing the theory of Bessel potential space, we characterize quadratic growth of the objective via a second-order optimality condition. In contrast to previous contributions, our method of proof works in arbitrary spatial dimensions.
Paper Structure (7 sections, 5 theorems, 126 equations)

This paper contains 7 sections, 5 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Bessel potential spaces
  3. Growth inequalities
  4. Application to optimal control
  5. Abstract second-order conditions via subderivatives
  6. Second subderivatives in dual Bessel potential spaces
  7. Second-order conditions for bang-bang control

Key Result

Lemma 3

Let $\alpha > 0$, $p \in (1, \infty)$ and $\mu \in \mathcal{M}^+(\mathbb{R}^d)$ be given. Then, $\mu \in L^{\alpha,p}(\mathbb{R}^d)^\star$ if and only if $g_\alpha \mathbin{*} \mu \in L^{p'}(\mathbb{R}^d)$, where $p' \in (1, \infty)$ is the conjugate exponent. Moreover, we have

Theorems & Definitions (15)

  • Lemma 3
  • proof
  • proof
  • Lemma 7
  • proof
  • proof : Proof of \ref{['thm:main_inequality']}
  • Definition 10
  • Definition 11
  • Lemma 13
  • proof
  • ...and 5 more