Unbounded Viscosity Solutions of Hybrid Control Systems

TL;DR

This paper investigates hybrid control systems with both discrete and continuous controls, characterizing the value function as a unique viscosity solution to associated inequalities, even with unbounded costs and dynamics.

Contribution

It introduces a framework for unbounded viscosity solutions in hybrid control systems with jumps and unbounded costs, extending existing theory.

Findings

Value function characterized as unique viscosity solution.

Extension to unbounded cost functionals and dynamics.

Results for both autonomous and controlled jumps.

Abstract

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set $A$ or a controlled jump set $C$ where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.