9. Infinite-Dimensional Dynamical Systems

We address the Lyapunov stability and the boundedness of motions (Lagrange stability) of infinite-dimensional dynamical systems determined by differential equations defined on Banach spaces and by semigroups with an emphasis on the qualitative properties of equilibria. We consider continuous as well as discontinuous dynamical systems (DDS). Most of the results involve monotonic Lyapunov functions. However, some of the stability results for DDS involve non-monotonic Lyapunov functions as well.

We present the Principal Stability and Boundedness Results (sufficient conditions) and some Converse Theorems (necessary conditions) for dynamical systems determined by general differential equations defined on Banach spaces. Most of these results are consequences of corresponding results established in Chapter 3 for dynamical systems defined on metric spaces. We demonstrate the applicability of these results in the analysis of several specific classes of differential equations defined on different Banach spaces. For autonomous differential equations defined on Bansch spaces we present invariance results and we apply these results in the analysis of specific classes of systems. We develop a comparison theory for general differential equations defined on Banach spaces and we apply these results in the stability analysis of a point kinetics model of a multicore nuclear reactor described by Volterra integrodifferential equations. Finally, we present stability results for composite systems defined on Banach spaces described by a mixture of different differential equations and we apply these results in the analysis of a specific class of systems.

Special important differential equations in Banach spaces include retarded functional differential equations. For dynamical systems determined by such equations, some of the preceding results can be improved. We present stability and boundedness results for dynamical systems determined by retarded functional differential equations, including Razumikhin-type theorems, and invariance results for dynamical systems determined by retarded functional differential equations. We apply some of these results in the qualitative analysis of the Cohen–Grossberg neural network model endowed with multiple time delays.

Finally, we present stability and boundedness results for discontinuous dynamical systems determined by differential equations in Banach spaces (involving non-monotonic Lyapunov functions) and by linear and nonlinear semigroups defined on Banach spaces. We demonstrate the applicability of these results by means of several classes of infinite-dimensional dynamical systems.