Related papers: Computation of Lyapunov Functions under State Cons…
This paper proposes novel approaches for designing control Lyapunov functions (CLFs) for constrained linear systems. We leverage recent configuration-constrained polyhedral computing techniques to devise piecewise affine convex CLFs.…
Let $f:\mathbb{R}^n \to \mathbb{R}$ be a continuously differentiable convex function with its minimizer denoted by $x_*$ and optimal value $f_* = f(x_*)$. Optimization algorithms such as the gradient descent method can often be interpreted…
We study optimization-based criteria for the stability of switching systems, known as Path-Complete Lyapunov Functions, and ask the question "can we decide algorithmically when a criterion is less conservative than another". Our…
Lyapunov functions play a fundamental role in analyzing the stability and convergence properties of optimization methods. In this paper, we propose a novel and straightforward approach for constructing Lyapunov functions for first-order…
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of…
This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic…
This paper is concerned with the problem of finding a quadratic common Lyapunov function for a family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and…
A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate…
In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the…
Computer assisted procedures of Lyapunov functions defined in given neighborhoods of fixed points for flows and maps are discussed. We provide a systematic methodology for constructing explicit ranges where quadratic Lyapunov functions…
In this paper, by using a characterization of functions having fractional derivative, we propose a rigorous fractional Lyapunov function candidate method to analyze stability of fractional-order nonlinear systems. First, we prove an…
This paper presents a novel method to synthesize stochastic control Lyapunov functions for a class of nonlinear, stochastic control systems. In this work, the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation is…
This work makes several contributions on stability and performance verification of nonlinear dynamical systems controlled by neural networks. First, we show that the stability and performance of a polynomial dynamical system controlled by a…
This paper provides a novel definition for Lyapunov functions for difference inclusions defined by convex processes. It is shown that this definition reflects stability properties of nonstrict convex processes better than previously used…
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…
While ensuring stability for linear systems is well understood, it remains a major challenge for nonlinear systems. A general approach in such cases is to compute a combination of a Lyapunov function and an associated control policy.…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the…
A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as…
In this manuscript, we study the properties of a family of second-order differential equations with damping, its discretizations and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth…