Hybrid systems which are capable of exhibiting simultaneously several
kinds of dynamic behavior in different parts of a system (e.g., contin
uous-time dynamics, discrete-time dynamics, jump phenomena, switching
and logic commands, and the like) are of great current interest, In th
e present paper we first formulate a model for hybrid dynamical system
s which covers a very large class of systems and which is suitable for
the qualitative analysis of such systems, Next, we introduce the noti
on of an invariant set (e.g., equilibrium) for hybrid dynamical system
s and we define several types of (Lyapunov-like) stability concepts fo
r an invariant set, We then establish sufficient conditions for unifor
m stability, uniform asymptotic stability, exponential stability, and
instability of an invariant set of hybrid dynamical systems, Under som
e mild additional assumptions, we also establish necessary conditions
for some of the above stability types (converse theorems). In addition
to the above, we also establish sufficient conditions for the uniform
boundedness of the motions of hybrid dynamical systems (Lagrange stab
ility), To demonstrate the applicability of the developed theory, we p
resent specific examples of hybrid dynamical systems and we conduct a
stability analysis of some of these examples (a class of sampled-data
feedback control systems with a nonlinear (continuous-time) plant and
a linear (discrete-time) controller, and a class of systems with impul
se effects).