# Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints

## TL;DR

This paper addresses an optimal control problem for a 3D energy-critical semilinear wave equation, establishing existence and optimality conditions while incorporating sparsity-promoting regularization and control constraints.

## Contribution

It introduces a framework for optimal control of a critical wave equation with pointwise control constraints and nonsmooth regularization, providing existence and optimality conditions.

## Key findings

- Existence of globally optimal solutions is proven.
- First- and second-order optimality conditions are derived.
- A sparsity-promoting regularization term is effectively incorporated.

## Abstract

This paper is concerned with an optimal control problem subject to the $H^1$-critical defocusing semilinear wave equation on a smooth and bounded domain in three spatial dimensions. Due to the criticality of the nonlinearity in the wave equation, unique solutions to the PDE obeying energy bounds are only obtained in special function spaces related to Strichartz estimates and the nonlinearity. The optimal control problem is complemented by pointwise-in-time constraints of Trust-Region type $\|u(t)\|_{L^2(\Omega)} \leq \omega(t)$. We prove existence of globally optimal solutions to the optimal control problem and give optimality conditions of both first- and second order necessary as well as second order sufficient type. A nonsmooth regularization term for the natural control space $L^1(0,T;L^2(\Omega))$, which also promotes sparsity in time of an optimal control, is used in the objective functional.