On the controlled eigenvalue problem for stochastically perturbed multi-channel systems

TL;DR

This paper investigates how to minimize the long-term exit rates of stochastically perturbed multi-channel systems by linking invariant sets, eigenvalues, and Hamilton-Jacobi-Bellman equations, and provides conditions for Pareto optimality.

Contribution

It establishes a novel connection between invariant sets, eigenvalues, and HJB equations for controlled multi-channel systems with small random perturbations.

Findings

Connection between invariant sets and eigenvalues for stability.

Conditions for existence of Pareto equilibrium in controlled eigenvalue problems.

Sufficient criteria for optimal exit rates in stochastic multi-channel systems.

Abstract

In this brief paper, we consider the problem of minimizing the asymptotic exit rate of diffusion processes from an open connected bounded set pertaining to a multi-channel system with small random perturbations. Specifically, we establish a connection between: (i) the existence of an invariant set for the unperturbed multi-channel system w.r.t. certain class of state-feedback controllers; and (ii) the asymptotic behavior of the principal eigenvalues and the solutions of the Hamilton-Jacobi-Bellman (HJB) equations corresponding to a family of singularly perturbed elliptic operators. Finally, we provide a sufficient condition for the existence of a Pareto equilibrium (i.e., a set of optimal exit rates w.r.t. each of input channels) for the HJB equations -- where the latter correspond to a family of nonlinear controlled eigenvalue problems.