Jose L. Mancilla-Aguilar

Jose L. Mancilla-Aguilar, Hernan Haimovich

We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive systems with inputs the well-established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the Lyapunov function on the system trajectories. This reduction is performed in such a way so that the resulting scalar system has no inputs. Novel sufficient conditions for ISS are provided, which generalize existing results for time-invariant and time-varying, switched and nonswitched, impulsive and nonimpulsive systems in several directions.

Hernan Haimovich, Jose L. Mancilla-Aguilar

Time-invariant finite-dimensional systems, under reasonable continuity assumptions, exhibit the property that if solutions exist for all future times, the set of vectors reachable from a bounded set of initial conditions over bounded time intervals is also bounded. This property can be summarized as follows: forward completeness implies bounded reachability sets. By contrast, this property does not necessarily hold for infinite-dimensional systems in general, and time-delay systems in particular. Sufficient conditions for this property to hold that can be directly tested on the function defining the system dynamics are only known in the case of systems with pointwise (or discrete) delays. This paper develops novel sufficient conditions for the boundedness of the reachability sets of time-delay systems involving mixed pointwise and distributed delays. Broad classes of systems satisfying these conditions are identified.