Remarks on finite and infinite time-horizon optimal control problems

TL;DR

This paper investigates the relationship between finite and infinite time-horizon optimal control problems, demonstrating that solutions to finite horizon problems approximate the infinite horizon solution even without prior knowledge of the latter.

Contribution

It establishes a theoretical link showing finite horizon solutions converge to the infinite horizon solution for a broad class of nonlinear and possibly unstable systems.

Findings

Finite horizon solutions approximate infinite horizon solutions.

Numerical simulations validate theoretical convergence.

Applicable to systems governed by ODEs and PDEs.

Abstract

The minimization of energy-like cost functionals is addressed in the context of optimal control problems. For a general class of dynamical systems, with possibly unstable and nonlinear free dynamics, it is shown that a sequence of solutions of finite time-horizon optimal control problems approximates a solution of the analog infinite time-horizon problem. The latter solution and corresponding optimal cost value function are not assumed to be known a-priori. Numerical simulations are presented validating the theoretical findings for several examples, including systems governed by both ordinary and partial differential equations.