Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state

TL;DR

This paper establishes existence and optimality conditions for controlling quasilinear parabolic PDEs with gradient-dependent nonlinearities and pointwise or averaged gradient constraints, extending regularity analysis.

Contribution

It extends regularity analysis and derives first-order optimality conditions for control problems with gradient constraints in quasilinear parabolic PDEs.

Findings

Existence results for control problems with gradient constraints.

First-order necessary optimality conditions derived.

Extension of regularity analysis for quasilinear PDEs.

Abstract

We derive existence results and first order necessary optimality conditions for optimal control problems governed by quasilinear parabolic PDEs with a class of first order nonlinearities that include for instance quadratic gradient terms. Pointwise in space and time or averaged in space and pointwise in time constraints on the gradient of the state control the growth of the nonlinear terms. We rely on and extend the improved regularity analysis for quasilinear parabolic PDEs on a whole scale of function spaces from [Hoppe et al, 2023]. In case of integral in space gradient-constraints we derive first-order optimality conditions under rather general regularity assumptions for domain, coefficients, and boundary conditions, similar to e.g. [Bonifacius and Neitzel, 2018]. In the case of pointwise in time and space gradient-constraints we use slightly stronger regularity assumptions leading to a classical smoother $W^{2,p}$-setting similar to [Casas and Chrysafinos, 2018].