Pre-orders for reasoning about stability properties with respect to input of hybrid systems

Pre-orders on systems are the basis for abstraction based verification of systems. In this paper, we investigate pre-orders for reasoning about stability with respect to inputs of hybrid systems. First, we present a superposition type theorem which gives a characterization of the classical incremental input-to-state stability of continuous systems in terms of the traditional ε-δ definition of stability. We use this as the basis for defining a notion of incremental input-to-state stability of hybrid systems. Next, we present a pre-order on hybrid systems which preserves incremental input-to-state stability, by extending the classical definitions of bisimulation relations on systems with input, with uniform continuity constraints. We show that the uniform continuity is a necessary requirement by exhibiting counter-examples to show that weaker notions of input bisimulation with just continuity requirements do not suffice to preserve stability. Finally, we demonstrate that the definitions are useful, by exhibiting concrete abstraction functions which satisfy the definitions of pre-orders.