We study solution concepts and their properties for hybrid systems that can flow and jump, affected by continuous-time inputs. While the solution concepts, the existence of solutions and (forward) completeness properties are extensively discussed in the absence of (external) inputs, there are surprisingly few results when inputs are present, certainly in the case where the flow and jump sets depend on the inputs. Given the relevance of this class of hybrid systems for many applications such as hybrid or networked control for plants subject to disturbances or measurement noise, we discuss in this paper notions of solutions in the presence of inputs and show through various examples the subtleties that can occur. Moreover, we provide tools to guarantee the existence of solutions and results to establish completeness properties.

In this paper, general conditions for set stabilization of (distributed) event-triggered control systems affected by measurement noises are presented. It is shown that, under these conditions, both static and dynamic triggers can be designed using a space-regularization approach such that the closed-loop system ensures an input-to-state practical stability property. Additionally, by proper choice of the tuning parameters, the system does not exhibit Zeno behavior. Contrary to various results in the literature, the noises do not have to be differentiable. The general results are applied to point stabilization and consensus problems as particular cases. Simulations illustrate our results.

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