Optimal growth for linear processes with affine control

TL;DR

This paper investigates optimal control of linear systems with affine dependence, establishing the existence of eigenvalues for constant, periodic, and optimal controls, and revealing surprising numerical equivalences among them.

Contribution

It introduces a comprehensive analysis of eigenvalues in affine-controlled linear systems, connecting constant, periodic, and optimal controls with novel theoretical insights.

Findings

Existence of an optimal Perron eigenvalue for constant controls

Existence of a generalized eigenvalue for time-periodic controls

Numerical evidence suggests the three eigenvalues are the same

Abstract

We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider $\dot x_\alpha(t) = (G + \alpha(t) F)x_\alpha(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. In the case of constant control $\alpha(t)\equiv \alpha$, we show the existence of an optimal Perron eigenvalue with respect to varying $\alpha$ under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls $\alpha(t)$. Finally we prove the existence of an eigenvalue (in the generalized sense) for the optimal control problem. The proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e] concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the relations between the three eigenvalues. Surprisingly enough, the three eigenvalues appear to be numerically the same.