The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms

TL;DR

This paper demonstrates that for certain optimal control problems with non-smooth tracking terms, the system state can be driven exactly to a desired stationary state in finite time, applicable to both finite and infinite-dimensional systems.

Contribution

It establishes finite-time stabilization results for optimal control problems with non-smooth tracking terms, covering both ODE and PDE systems under controllability assumptions.

Findings

Optimal control solutions reach the desired state in finite time.

Applicable to systems governed by ODEs and PDEs like the wave equation.

Shows exact stabilization despite non-smooth tracking terms.

Abstract

In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example for boundary control systems governed by the wave equation, under certain assumptions the optimal system state is steered exactly to the desired state after finite time.