Infinite Horizon Optimal Control Problems with Discount Factors

TL;DR

This paper analyzes infinite horizon optimal control problems with semilinear parabolic equations, focusing on differing discount factors, deriving optimality conditions, and establishing convergence of finite horizon approximations.

Contribution

It introduces the analysis of problems with different discount factors for cost and state, deriving new optimality conditions and convergence results.

Findings

Derived first- and second-order optimality conditions.

Demonstrated the importance of different discount factors for second-order analysis.

Proved convergence and rate of convergence for finite horizon approximations.

Abstract

This paper is dedicated to the analysis of infinite horizon optimal control problems subject to semilinear parabolic equations with constraints on the controls and discounted cost functionals. The discount factors on the cost and the state components are allowed to differ from each other. First-order as well as second-order optimality conditions are derived and the importance of allowing different discount factors for the second-order analysis for the class of nonlinearities under consideration is demonstrated. Finally convergence and rate of convergence for the approximation of the infinite horizon problem by a family of finite horizon problems is proven.