The interval turnpike property for adjoints

TL;DR

This paper establishes an interval turnpike property for adjoint variables in nonlinear optimal control problems, extending classical results to more complex dynamics and providing numerical validation.

Contribution

It introduces an interval turnpike result for adjoints in nonlinear control problems with semigroup dynamics, including analytic cases and a numerical example.

Findings

Turnpike property holds for adjoints under spectral decomposition conditions.

Stronger estimates are derived for analytic semigroups.

Numerical example demonstrates applicability to boundary controlled heat equations.

Abstract

In this work we derive an interval turnpike result for adjoints of finite- and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral decomposition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further give stronger estimates for analytic semigroups and provide a numerical example with a boundary controlled semilinear heat equation to illustrate the results.