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Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov…
The objective of the research is to develop a general method of constructing Lyapunov functions for non-linear non-autonomous differential inclusions described by ordinary differential equations with parameters. The goal has been attained…
This contribution presents two exponential stability criteria for linear systems with multiple pointwise and distributed delays. These results (necessary and sufficient conditions) are given in terms of the delay Lyapunov matrix and the…
The paper investigates sufficient conditions on a differential inclusion which guarantee that the origin is a finite time stable equilibrium, namely a weak local one, a weak global one or a strong local one. The analysis relies on the…
For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal…
In this article we introduce new possibilities of bounding the stability constants that play a vital role in the reduced basis method. By bounding stability constants over a neighborhood we make it possible to guarantee stability at more…
This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property,…
We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization…
We establish the existence, uniqueness, and stability of the stationary solution of the one-dimensional viscous Burgers equation with the Dirichlet boundary conditions on a finite interval. We obtain explicit formulas for solutions and…
This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed…
This paper considers the problem of characterizing the stability region of a large-scale networked system comprised of dissipative nonlinear subsystems, in a distributed and computationally tractable way. One standard approach to estimate…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing…
We introduce a new class of quadratic functions based on a hierarchy of linear time-varying (LTV) dynamical systems. These quadratic functions in the higher order space can be also seen as a non-homogeneous polynomial Lyapunov functions for…
We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the…
This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it…
Assuming the existence of a general nonuniform dichotomy for the evolution operator of a non-autonomous ordinary linear differential equation in a Banach space, we establish the existence of invariant stable manifolds for the semiflow…
This paper presents an automated algorithm to analyze the stability of piecewise affine (PWA) dynamical systems due to their broad applications. We parametrize the Lyapunov function as a PWA function, with polytopic regions defined by the…
Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS)…
This paper deals with asymptotic stability of a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical…