  There are recent shifts in demand for design controllers from simplified to
complex model-based. Although simplification approaches are successful in many
areas of engineering control systems, high-fidelity simulation-based control
design, for example, reinforcement learning, has been rising in robotics areas.
On the other hand, the lack of assurances about the stability and robustness of
simulation-based control design restricts its applications to safety-critical
systems. We develop computational methods to verify the stability and
robustness of safety-critical systems. By extending the inverse Lyapunov
theorem, we present a practical method to compute the constants required to
check the exponential stability conditions of dynamic systems implemented in a
numerical simulator. It is shown that the norm-bound of the propagated states
is a function of the numerical integration steps, where the numerical simulator
may include discontinuous jumps of states. The energy bounds for the transition
states are obtained based on the exponential stability assumption of the
inverse Lyapunov theorem. Finally, a finite sampling algorithm provides the
deterministic stability guarantee for the continuous state space.
