On the controlled eigenvalue problem for stochastically perturbed 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.
Cite
@article{arxiv.1501.01256,
title = {On the controlled eigenvalue problem for stochastically perturbed multi-channel systems},
author = {Getachew K. Befekadu},
journal= {arXiv preprint arXiv:1501.01256},
year = {2016}
}
Comments
A short paper with 9 pages (a continuation of our previous paper arXiv:1408.6260)