Discrete-time minimax hybrid control with cost constraints

This paper addresses a discrete-time optimal control problem under the worst-case scenario for the controller, where its control policies are subject to specific restrictions at defined costs. The system dynamics follows a hybrid evolution, incorporating both regular and impulsive sub-dynamics that drive the state through time. The goal is to minimize a total cost criterion in the worst-case scenario, with costs discounted over time by a state-action-dependent discount factor that can reach the values of zero or one at certain points. The problem is approached through an auxiliary family of unrestricted minimax control problems, each indexed by a particular parameter. For each member of this family, we determine the optimal solution, and then seek the 'best' parameter that aligns the unrestricted problem with the original one. This alignment ensures that the optimal solutions for the unrestricted problems also apply to the restricted version. To illustrate our theory, we provide an example based on a transboundary pollution accumulation problem.