The extension of a solution to a hybrid system beyond its Zeno time is a simple exercise in modeling. However, assuring that the extended system is well-posed in a certain sense, in particular, that the extension of a solution depends reasonably on initial, pre-Zeno, conditions, has not been addressed. In this paper it is shown that these results hold for hybrid systems that exhibit Zeno behavior when the set of Zeno equilibria forms a continuum that has certain stability properties. Several scenarios of going past Zeno are presented. Dependence of limits of Zeno solutions, of Zeno times, and of reachable sets on initial conditions is also discussed.

Convex analysis and the theory of differential inclusions with maximal monotone right-hand sides suggests casting consensus algorithms as systems involving switching between such differential inclusions. Convergence of solutions to such switching systems is shown and applications to consensus are presented. Robustness of pointwise asymptotic stability for a single differential inclusion which has some monotonicity-related properties, but needs not be monotone, is shown.

Hybrid dynamical systems, modeled by hybrid inclusions - a combination of differential equations or inclusions, of difference equations or inclusions, and of constraints on the resulting motions - are considered. Pointwise asymptotic stability, a property of a set of equilibria in a hybrid system where every equilibrium is Lyapunov stable and solutions from near the equilibria converge to some equilibrium, is studied. Sufficient conditions, relying on set-valued Lyapunov functions with strict or weak decrease, on invariance arguments, or on standard Lyapunov functions that also limit the lengths of solutions, are given. Structural properties of sets of solutions to a hybrid system, of reachable sets, and of limits of solutions are investigated in the presence of a pointwise asymptotically stable set of equilibria, and also under further uniform Zeno assumptions. Many of these results are extended to the case of partial pointwise asymptotic stability. The results are then used to extend Zeno solutions to hybrid systems beyond their Zeno times, in a way preserving reasonable dependence of solutions on initial conditions and enabling the analysis of convergence of extended solutions to a compact attractor.

Switching between finitely many continuous-time autonomous steepest descent dynamics for convex functions is considered. Convergence of complete solutions to common minimizers of the convex functions, if such minimizers exist, is shown. The convex functions need not be smooth and the dynamics may be subject to constraints. Since the common minimizers may represent consensus in a multi-agent system modeled by an undirected communication graph, several known results about asymptotic consensus are deduced as special cases. Extension to a time-varying case is included.

Pointwise asymptotic stability is a property of a set of equilibria of a dynamical system, where every equilibrium is Lyapunov stable and every solution converges to some equilibrium. Hybrid systems are dynamical systems which combine continuous-time and discrete-time dynamics. In this paper, they are modeled by a combination of differential equations or inclusions, of difference equations or inclusions, and of constraints on the resulting motions. Sufficient conditions for pointwise asymptotic stability of a closed set are given in terms of set-valued Lyapunov functions: they require that the values of the Lyapunov function shrink along solutions. Cases of strict and weak decrease are considered. Lyapunov functions, not set-valued, which imply that solutions have finite length are used in sufficient conditions and related to the set-valued Lyapunov functions. Partial pointwise asymptotic stability is also addressed.