Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces

TL;DR

This paper investigates the exponential turnpike property in infinite-dimensional optimal control problems, showing that optimal trajectories stay close to steady or periodic solutions over long time horizons, with applications to PDEs.

Contribution

It establishes the exponential turnpike property for control problems in Hilbert spaces using a novel dichotomy transformation and Riccati equations, extending previous finite-dimensional results.

Findings

Turnpike property holds for large time horizons in Hilbert space control problems.

The approach applies to linear PDEs like heat and wave equations with periodic tracking.

The method involves algebraic Riccati and Lyapunov equations for stability analysis.

Abstract

In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large time, the optimal state, control and adjoint vector remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.