Simulation and bisimulation relations define pre-orders on processes which serve as the basis for approximation based verification techniques, and have been extended towards the design of continuous and hybrid systems with complex logic specifications. We study pre-orders between hybrid systems which preserve stability properties with respect to input. We show that these properties are not bisimulation invariant, and hence propose stronger notions which strengthen simulation and bisimulation relations with uniform continuity constraints. We show that uniform continuity is necessary on the relations corresponding to both the state-space and the input-space, and continuity itself does not suffice. Finally, we demonstrate the satisfiability of our definitions by casting the well-known Lyapunov function based techniques for stability analysis as constructing a simple one-dimensional system which is stable and uniformly continuously simulates the original system.

URL: https://doi.org/10.1007/s10626-017-0262-9