Hybrid systems with continuous-time inputs: subtleties in solution concepts and existence results

International audience; 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.