Finite Time Stability, Semistability, and Optimality of Discrete-Time Nonlinear Deterministic and Stochastic Dynamical Systems with Application to Network Consensus

In this dissertation, we develop Lyapunov theorems for finite time and fixed time stability of discrete-time nonlinear dynamical systems. The regularity properties of the Lyapunov functions satisfying the sufficient conditions for finite and fixed time stability are shown to be strongly dependent on the regularity properties of the settling-time function. In addition, the optimal control problem is extended to address finite time and fixed time stabilization for nonlinear discrete-time dynamical systems. Specifically, an optimal finite time and fixed time state feedback control problem is formulated and solved using Hamilton-Jacobi-Bellman theory and connections to an inverse optimal control problem is provided. For systems having a continuum of equilibria, asymptotic and finite time stability are not the appropriate notion of stability since every neighborhood of a nonisolated equilibrium contains another equilibrium. For such problems, the additional requirement that all solutions converge to limit points that are Lyapunov stable is required. We address semistability and finite time semistability, which guarantees that every point of a set of system equilibria is Lyapunov stable and the system trajectories converge to one of the Lyapunov stable point in the set of equilibria. This analysis framework is used to design control protocols for multiagent dynamical systems using a thermodynamic-based framework. In network systems communication uncertainty and attenuation errors between the agents in the network can be modeled as a stochastic dynamical systems. We develop a rigorous framework for semistability of discrete-time stochastic dynamical systems by providing sufficient Lyapunov conditions and the first converse Lyapunov theorem for semistability in probability. In addition, we develop consensus protocols for multiagent systems with stochastic nonlinear discrete-time dynamics using stochastic thermodynamic principles.Ph.D