Hybrid Models and Switching Control with Constraints

Abstract

We consider a class of hybrid control, relative to Markov-Feller processes, where the discrete and the continuous type variables exchange information when a signal arrives. These problems can also be studied as optimal stopping and impulse control problems for a Markov-Feller process where the controls are allowed only when a signal arrives. There are a few references of the authors in the last years, where the HJB equation was solved and an optimal control (for the optimal stopping problem and impulse control problem) was obtained, under suitable conditions, including a setting on a (locally) compact metric state space, a strictly positive cost-per-impulse, and without multiple simultaneous impulses. In this work, we use these results to discuss optimal switching problems for Markov-Feller processes on a locally compact state-space under weaker conditions, as a particular case of optimal hybrid control problems.

On Some Quasi-Variational Inequalities and Other Problems with Moving Sets

“…, n, i.e., the operator A becomes A i and a simple expression for M takes the form M v(x, i) = inf v(x, j) + k(i, j)} : j = i , k(i, j) ≥ 0, which can be combined with previous forms of M . There is a vast literature on these problems, as recent books, the reader may consult Arapostathis et al [5], Yin and Zhu [80], among others; and for instance, a relative complex situations is discussed in Menaldi and Robin [59].…”