Optimal Control on Positive Cones

TL;DR

This paper studies optimal control problems on positive cones, showing that under certain conditions, the Bellman equation simplifies to a linear form solvable via convex optimization, with applications to various structured control systems.

Contribution

It introduces a critical assumption linking cone properties to Bellman equation solutions, unifying several control scenarios under a common framework.

Findings

Bellman equation reduces to a linear function under the cone assumption.

Convex optimization can be used to compute the Bellman function.

Special cases include positive semi-definite matrices, polyhedral cones, and structured LQ control.

Abstract

An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption on the cone to the existence of minimal elements in certain subsets of the dual cone. Three special cases are derived as examples. The first one, where the positive cone is the set of positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems. The third special case corresponds to linear quadratic control with additional structure, such as spatial invariance.