Maximum Principle for Linear-Convex Boundary Control Problems applied to Optimal Investment with Vintage Capital

TL;DR

This paper develops a maximum principle for boundary control problems in infinite dimensions, linking it to dynamic programming, with applications to optimal investment models involving vintage capital.

Contribution

It establishes necessary and sufficient optimality conditions for infinite horizon boundary control problems and connects the maximum principle with the value function's gradient.

Findings

Necessary and sufficient conditions for open loop controls.

Co-state variable equals the spatial gradient of the value function.

Application to optimal investment with vintage capital.

Abstract

The paper concerns the study of the Pontryagin Maximum Principle for an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. The optimal control model has already been studied both in finite and infinite horizon with Dynamic Programming methods in a series of papers by the same author, or by Faggian and Gozzi. Necessary and sufficient optimality conditions for open loop controls are established. Moreover the co-state variable is shown to coincide with the spatial gradient of the value function evaluated along the trajectory of the system, creating a parallel between Maximum Principle and Dynamic Programming. The abstract model applies, as recalled in one of the first sections, to optimal investment with vintage capital.