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

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.

An aspect of the turnpike property. Long time horizon behavior

“…Due to (7), our variations δu of the control generates a variation δz of the state in the sense that the function δz is given by…”

“…Measure and integral turnpike properties have been studied in [21]. The turnpike property for systems that are governed by semilinear partial differential equations is studied in [7]. The relation of the turnpike property and the receding-horizon method has been studied in [2].…”

Turnpike in Lipschitz—nonlinear optimal control

“…This is illustrated in theorem 3.1, corollary 3.1 and theorem 3.2 in the context of the semilinear wave and heat equation with globally Lipschitz-only nonlinearity, once again under the assumption that the running target is a steady control-state pair. We make no smallness assumptions neither on it, nor on the initial data, thus covering some cases where results from [21,36,40,54] are not applicable.…”

“…The semilinear heat equation is a commonly used benchmark for nonlinear turnpike results, thus this example serves to compare with existing results. For instance, while we assume that the running targets are steady states, we make no smallness assumptions on the targets or on the initial data, unlike [21,40]. Furthermore, since we do not use (or thus linearize) the optimality system, we may work with solely globally Lipschitz nonlinearities, in which case the techniques of [21,36,40] do not apply.…”

The turnpike property for mean-field optimal control problems

“…Measure and integral turnpike properties have been studied in [38]. A turnpike analysis for systems that are governed by semilinear partial differential equations is presented in [28], while the relation between the turnpike property and the receding-horizon method is investigated in [8]. In [19], manifold turnpikes are also studied.…”