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

¹ Mathematical Institute, University of Bayreuth, 95447 Bayreuth, Germany.
² Institute for Mathematics, Technische Universität Ilmenau, 98693 Ilmenau, Germany.

^** Corresponding author: manuel.schaller@tu-ilmenau.de

Received: 29 August 2020
Accepted: 15 March 2021

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.