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Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities....
It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector...
Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid...
We give a simple, explicit example of a two-dimensional polynomial vector field that is globally asymptotically stable but does not admit a polynomial Lyapunov function.
This note is concerned with a class of differential inequalities in the literature that involve higher order derivatives of Lyapunov functions and have been proposed to infer asymptotic stability of a dynamical system without requiring the first derivative of the Lyapunov function to be negative definite. We show that whenever a Lyapunov function satisfies these conditions, we can explicitly construct...
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