2015/01/06 by Getachew K. Befekadu, Befekadu, Getachew K.
Economics, Econometrics and Finance · Engineering · Mathematics · #34D05 #34D10 #34D15 #34H05 #37C75 #37H10 #49L25 #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Material Science and Thermodynamics #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1501.01256
openalex publication_date 2015/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.