Stability analysis of discontinuous dynamical systems

The qualitative analysis of dynamical systems, with emphasis on stability and boundedness studies, has been an active field of inquiry in engineering science and applied mathematics for over a century. Advances in technology over the past few decades; involving increasingly complex systems, have motivated abstractions, resulting in the qualitative analysis of general dynamical systems defined on abstract spaces. The models for such systems are general enough to include finite dimensional and infinite dimensional systems whose motions may evolve along discrete; continuous, or generalized time, resulting, respectively, in discrete-time, continuous-time; or hybrid dynamical systems. In the case of continuous-time dynamical systems, a distinction is made between continuous dynamical systems (whose motions are continuous with respect to time) and discontinuous dynamical systems (DDS) (whose motions need not be continuous with respect to time). It turns out that the qualitative analysis of many of the hybrid dynamical systems considered in the literature may be accomplished by studying appropriate associated DDS.; Although the stability theory for general dynamical systems described above is fairly complete, there are notable missing pieces. In the present dissertation, we will try to address some of these: (A) We will establish a theory for the partial stability and partial boundedness of general dynamical systems (discrete-time, as well as continuous-time) and we will apply our results in the analysis of a class of discrete event systems (including a computer load balancing problem). (B) We will establish a theory for the partial stability and partial boundedness of DDS and we will apply our results in the analysis of a class of finite dimensional dynamical systems subjected to impulsive forces. (C) We will establish a theory for partial stability under arbitrary initial z-perturbations and partial boundedness under arbitrary initial z-perturbations of DDS and we will again apply our results in the analysis of the dynamical systems subjected to impulse effects. We emphasis that although the results in (B) and (C) are similar in form, their underlying essence is different. (D) Most of the existing stability and boundedness results for DDS concern finite dimensional systems (e.g., switched systems, etc.). We will establish stability and boundedness results for several important classes of infinite dimensional DDS, including: (1) DDS determined by functional differential equations; (2) DDS determined by linear and nonlinear semigroups; and (3) DDS determined by Cauchy problems defined on Hilbert and Banach spaces. We will apply the above results in the analysis of several important specific classes of dynamical systems determined by ordinary differential equations, delay differential equations, Volterra integrodifferential equations, partial differential equations and the like.