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Robust stability theory for stochastic dynamical systems


In this work, we focus on developing analysis tools related to stability theory for

certain classes of stochastic dynamical systems that permit non-unique solutions. The

non-unique nature of solutions arise primarily due to the system dynamics that are

modeled by set-valued mappings. There are two main motivations for studying such

classes of systems. Firstly, understanding such systems is crucial to developing a robust

stability theory. Secondly, such system models allow flexibility in control design problems.

We begin by developing analysis tools for a simple class of discrete-time stochastic

system modeled by set-valued maps and then extend the results to a larger class of

stochastic hybrid systems. Stochastic hybrid systems are a class of dynamical systems

that combine continuous-time dynamics, discrete-time dynamics and randomness. The

analysis tools are established for properties like global asymptotic stability in probability

and global recurrence. We focus on establishing results related to sufficient conditions for stability, weak sufficient conditions for stability, robust stability conditions and converse Lyapunov theorems. In this work a primary assumption is that the stochastic system satisfies some mild regularity properties with respect to the state variable and random input. The regularity properties are needed to establish the existence of random solutions and results on sequential compactness for the solution set of the stochastic system.

We now explain briefly the four main types of analysis tools studied in this work.

Sufficient conditions for stability establish conditions involving Lyapunov-like functions

satisfying strict decrease properties along solutions that are needed to verify stability

properties. Weak sufficient conditions relax the strict decrease nature of the Lyapunov like function along solutions and rely on either knowledge about the behavior of the

solutions on certain level sets of the Lyapunov-like function or use multiple nested non-strict Lyapunov-like functions to conclude stability properties. The invariance principle

and Matrosov function theory fall in to this category. Robust stability conditions determine

when stability properties are robust to sufficiently small perturbations of the

nominal system data. Robustness of stability is an important concept in the presence

of measurement errors, disturbances and parametric uncertainty for the nominal system.

We study two approaches to verify robustness. The first approach to establish robustness

relies on the regularity properties of the system data and the second approach is

through the use of Lyapunov functions. Robustness analysis is an area where the notion

of set-valued dynamical systems arise naturally and it emphasizes the reason for our

study of such systems. Finally, we focus on developing converse Lyapunov theorems for

stochastic systems. Converse Lyapunov theorems are used to illustrate the equivalence

between asymptotic properties of a system and the existence of a function that satisfies

a decrease condition along the solutions. Strong forms of the converse theorem imply

the existence of smooth Lyapunov functions. A fundamental way in which our results

differ from the results in the literature on converse theorems for stochastic systems is

that we exploit robustness of the stability property to establish the existence of a smooth

Lyapunov function.

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