Verification and synthesis of optimal decision strategies for complex systems

Complex systems make a habit of disagreeing with the mathematical models strategically designed to capture their behavior. A recursive process ensues where data is used to gain insight into the disagreement. A simple model may give way to a model with hybrid dynamics. A deterministic model may give way to a model with stochastic dynamics. In many cases, the modeling framework that sufficiently characterises the system is both hybrid and stochastic; these systems are referred to as stochastic hybrid systems. This dissertation considers the stochastic hybrid system framework for modeling complex systems and provides mathematical methods for analysing, and synthesizing decision laws for, such systems. We first propose a stochastic reach-avoid problem for discrete time stochastic hybrid systems. In particular, we present a dynamic programming based solution to a probabilistic reach-avoid problem for a controlled discrete time stochastic hybrid system. We address two distinct interpretations of the reach-avoid problem via stochastic optimal control. In the first case, a sum-multiplicative cost function is introduced along with a corresponding dynamic recursion that quantifies the probability of hitting a target set at some point during a finite time horizon, while avoiding an unsafe set at all preceding time steps. In the second case, we introduce a multiplicative cost function and a dynamic recursion that quantifies the probability of hitting a target set at the terminal time, while avoiding an unsafe set at all preceding time steps. In each case, optimal reach-avoid control policies are derived as the solution to an optimal control problem via dynamic programming. We next introduce an extension of the reach-avoid problem where we consider the verification of discrete time stochastic hybrid systems when there exists uncertainty in the reachability specifications themselves. A summultiplicative cost function is introduced along with a corresponding dynamic recursion that quantifies the probability of hitting a target set at some point during a finite time horizon, while avoiding an obstacle set during each time step preceding the target hitting time. In contrast with the general reach-avoid formulation, which assumes that the target and obstacle sets are constant and deterministic, we allow these sets to be both time-varying and probabilistic. An optimal reach-avoid control policy is derived as the solution to an optimal control problem via dynamic programming. A framework for analyzing probabilistic safety and reachability problems