Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

TL;DR

This paper develops an abstract framework to analyze the sensitivity of extremal equations in nonlinear optimal control, applies it to semilinear PDEs, and demonstrates exponential turnpike properties and error decay, enhancing control scheme efficiency.

Contribution

It introduces an abstract implicit function approach for sensitivity analysis and proves exponential turnpike results for semilinear PDE control problems.

Findings

Perturbations decay exponentially over time.

Discretization errors can be efficiently controlled.

Theoretical results are validated with nonlinear heat equation examples.

Abstract

We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.