Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

Abstract

We study an optimal control problem for a hybrid system exhibiting several internal switching variables whose discrete evolutions are governed by some delayed thermostatic laws. By the dynamic programming technique we prove that the value function is the unique viscosity solution of a system of several Hamilton-Jacobi equations, suitably coupled. The method involves a contraction principle and some suitably adapted results for exit-time problems with discontinuous exit cost.

Hybrid control for optimal visiting problems for a single player and for a crowd

“…The use of a switching/discontinuous/hybrid memory, as in the present paper, was instead used for a one-dimensional optimal visiting problem on a network in [5]. For switching hybrid control problems, as well as for differential games, related to the model here presented, and in connection with Hamilton-Jacobi equations, we refer to [4] and to [6]. A more general discussion is done in [12] (similar formulations for the deterministic case have also been proposed in [15,21]).…”

Hybrid Thermostatic Approximations of Junctions for Some Optimal Control Problems on Networks

“…Assume (4),(5), and(8). The functionṼ is a viscosity solution of the Hamilton-Jacobi-Bellman problem(13) …”

“…Assume (4),(5), and(8). The function V * is a viscosity solution and the maximal subsolution of the HJB problem, in the sense of Definition 4.12.+ H 1 (x, V ) = 0 in int(R 1 ), λV + H 2 (x, V ) = 0 in int(R 2 ), λV + H 3 (x, V ) = 0 in int(R 3 ), min {λV + H 1 , λV + H 2 , λV + H 3 } ≤ 0 on x = 0, max {λV + H 1 , λV + H 2 , λV + H 3 } ≥ 0 on x = 0.…”

A Differential Game with Exit Costs

“…Hence, the switching evolution of the parameter w is not directly at disposal of the external controller, but it follows some internal switching rules which are intrinsic to the system. In [1,3], the value function is proven to be the unique viscosity solution of a suitably coupled system of HJB equations, where the coupling is given by the boundary conditions in the regions where the thermostat certainly assumes a constant value (cannot switch). This is done by splitting the optimal control problem in some problems of exit time kind: in every space-region where the thermostat is constant, the problem is equivalent to an exit-time problem with unknown exit-cost given by the value function itself evaluated in the other region of constancy for w. Then, an ad hoc fixed point procedure is applied.…”

“…In [1,3,4] some motivations and applications for studying optimal control problems with thermostatic dynamics are given. Similar motivations certainly suggest the study of differential games with thermostatic dynamics.…”