We study control design methods to endow hybrid systems under disturbances with safety guarantees as an inverse-optimality problem. First, we provide sufficient conditions to guarantee input-to-state safety of a hybrid system with disturbance inputs only. Next, given a nominal feedback law, we show that a hybrid system, with inputs and disturbances, can be rendered input-to-state controlled safe under the existence of a control barrier function (CBF) using pointwise min-norm safeguarding feedback laws. Finally, we demonstrate that every CBF is a meaningful value function for a two-player zero-sum hybrid game in the context of safety, and that every pointwise min-norm safeguarding feedback law is optimal for such a game, even though its design is independent of any cost functional. The main results are illustrated in an example.

In this paper, we propose a learning-based algorithm for hybrid systems with a twofold purpose: first, to design Lyapunov functions and, second, to upper bound the cost of solutions to the system. Via enforcing conditions at finitely many points of a set of interest and leveraging regularity properties of the maps defining the dynamics of the system and the stage costs associated to solutions, we extend the conditions to the entire set of interest. The method employs neural networks to learn a Lyapunov function and a value-like function to guarantee the extended pointwise conditions at all points in the set of interest and thus, guarantee practical asymptotic stability of a set or provide an upper bound on the cost of solutions, respectively. The approach is illustrated in a hybrid oscillator system.

In this paper, we formulate a two-player zero-sum game under dynamic constraints given in terms of hybrid dynamical systems. We present sufficient conditions with Hamilton-Jacobi-Isaacs-like equations to guarantee attaining a solution to the game. It is shown that when the players select the optimal strategy, the value function can be evaluated without the need of computing solutions. Under additional conditions, we show that the optimal feedback laws render a set of interest asymptotically stable. Using this framework, we address an optimal control problem under the presence of an adversarial action in which the decision-making agents have dynamics that might exhibit both continuous and discrete behavior. Applications of this problem, as presented here, include disturbance rejection and security scenarios, for which the effect of the worst-case adversarial action is minimized.

In this work, we formulate a two-player zero-sum game under dynamic constraints given in terms of hybrid dynamical systems. Find the full version in [8], including the main results and outlines of the corresponding proofs. We propose sufficient conditions to guarantee attaining a solution to the game. When the players select the optimal strategy, the value function can be evaluated without the need of computing solutions. Under additional conditions, the optimal feedback laws render a set of interest asymptotically stable. Using this framework, we address an optimal control problem under the presence of an adversarial action in which the decision-making agents have dynamics that might exhibit both continuous and discrete behavior.