Based on recent results on the exploitation of \drift criteria" for general state space Markov processes, we derive rates of convergence for (moments of) processes associated with a renewal process with common inter-renewal time distribution F. Some of the results are classical and some are new, but the proofs are novel and, we believe, useful if one needs to derive convergence results based on...

Momentum methods play a central role in optimization. Several momentum methods are provably optimal, and all use a technique called estimate sequences to analyze their convergence properties. The technique of estimate sequences has long been considered difficult to understand, leading many researchers to generate alternative, “more intuitive” methods and analyses. In this paper we show there is...

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-stat...

This paper proposes several Converse Lyapunov Theorems for nonlinear dynamical systems defined on smooth connected Riemannian manifolds and characterizes properties of corresponding Lyapunov functions in a normal neighborhood of an equilibrium. We extend the methods of constructing of Lyapunov functions for ordinary differential equations on R to dynamical systems defined on Riemannian manifold...

In this note, we study the relationship between the sample and moment Lyapunov exponents for jump linear systems. Using a large deviation theorem, a modified version of Arnold’s formula for connecting sample path and moment Lyapunov exponents for continuous-time linear stochastic systems is extended to discrete-time jump linear systems. Sample path stability properties of linear stochastic syst...

Time-delay naturally appears in many control systems, and it is frequently a source of instability. However, for some systems, the presence of delay can have a stabilizing effect. Therefore, stability and control of time-delay systems is of theoretical and practical importance. Modern control systems usually employ digital technology for controller implementation, i.e. sampled-data control. A t...

Periodic Lyapunov differential equations can be used to formulate robust optimal periodic control problems for nonlinear systems. Typically, the added Lyapunov states are discretized in the same manner as the original states. This straightforward technique fails to guarantee conservation of positive-semidefiniteness of the Lyapunov matrix under discretization. This paper describes a discretizat...