Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks. These models are a special form of compartmental systems, which involve mass- and energy-balance relations. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. Because of physical considerations, however, mass- action kinetics have special properties, such as nonnegative solutions, that are useful for analyzing their behavior. With this motivation in mind, this article has several objectives. First, a general construction of the kinetic equations based on the reaction laws is provided in a state-space form. Next, the nonnegativity of solutions to the kinetic equations is considered. The realizability problem, which is concerned with the inverse problem of constructing a reaction network having specified essentially nonnegative dynamics, is also considered. In particular, an explicit construction of a reaction network for essentially nonnegative polynomial dynamics involving a scalar state is provided. Next, the reducibility of the kinetic equations is considered as well as the stability of the equilibria of the kinetic equations. Lyapunov methods are applied to the kinetic equations, and semistability is guaranteed through the convergence to a Lyapunov- stable equilibrium that depends on the initial concentrations. Semistability is the appropriate notion of stability for compartmental systems in general, and reaction networks in particular, where the limiting concentration maybe nonzero and may depend on the initial concentrations. Finally, the zero deficiency result for mass-action kinetics in standard matrix terminology is presented and semistability is proven.

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems. Furthermore, we generalize dissipativity theory to non-linear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics and dissipated energy over the resetting instants. Furthermore, extended Kalman‐Yakubovich‐Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and non-expansive impulsive systems to provide a generalization of the classical notions of passivity and non-expansivity for non-linear impulsive systems. These results are used in the second part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systems as well as to develop optimal hybrid feedback controllers.

A partial differential equation (PDE) model for a flexible two-link manipulator is derived and transformed to a form appropriate for the development of stable control designs. Stable control of this nonlinear infinite dimensional two-link system is then achieved by a novel control design developed using passivity and Lyapunov-based methods. A two-link hardware experimental setup is used to validate the analytical PDE model and the proposed stable control design scheme.