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Classical dynamical systems can be clearly categorized into continuous-time systems, parameterized by real numbers, and discrete-time systems, parameterized by integers. However, a plethora of systems --- mechanical systems with impacts, those switching between modes like on/off, or those including both analog and digital devices, as well as systems resulting from implementation of hybrid feedback control --- may exhibit behaviors typical of both continuous-time and discrete-time dynamical systems. In short, there are systems where the state may both flow and jump. Such systems are called hybrid dynamical systems. The talk will give basic examples of hybrid phenomena, motivate the use of hybrid algorithms in control engineering, and present a particular framework for modeling and analysis of hybrid dynamical systems. The framework will feature hybrid time domains and hybrid inclusions: a combination of differential inclusions, difference inclusions, and of state constraints on the resulting flows and jumps. Considering inclusions, rather than equations, and abandoning the classical time domains will be shown to reflect the effect of noise on a hybrid system. The talk will then present a sample of results regarding the properties of solutions to hybrid systems, like the dependence on initial conditions, and regarding the asymptotic stability of hybrid systems, like a converse Lyapunov characterization. Implications for the design of hybrid feedback control will be discussed. The role of various elements of set-valued analysis, say graphical convergence, semicontinuity concepts for set-valued mappings, tangent cones to sets, etc., in the study of hybrid systems will be underlined.

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