Nonpathological Lyapunov functions and discontinuous Carathéodory systems

Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense are considered. For these equations, sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven.     

Potentially global stabilizability

In this note we consider a particular class of nonlinear systems with a controllable linear part. We prove that by an appropriate choice of a feedback law, it is possible to transform the system in such a way that the origin is asymptotically stable and the region of attraction is arbitrarily large.     

Stabilizability of closed orbits

The main contribution of this paper is a version of the Jurdjevic-Quinn method for feedback stabilization of closed orbits of autonomous affine control systems.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.     

Lyapunov functions and stability of dynamical polysystems

This paper is concerned with the characterization of asymptotic stability of a dynamical polysystem in the vicinity of a compact set in terms of a properly defined Lyapunov function. It turns out that Lyapunov functions correspond to a stronger type of asymptotic stability, which is specified via higher-order prolongational extensions of the reachable map of the system.     

Stability control and recurrent switching rules

Recently, it has been enlightened the interest of a class of switching rules with good properties, which are called eventually periodic: more precisely, it has been proven that a finite family of linear vector fields of can be stabilized by means of eventually periodic switching rules provided that it is asymptotically controllable and satisfies an additional finite time controllability condition. Unfortunately, simple examples point out that in general, eventually periodic switching rules are not robust with respect to state measurement errors. In this paper, we introduce a new type of switching rules with improved robustness properties, which are called recurrent switching rules. They are subject to the construction of a finite sequence of complete cones Γ₁, … ,Γ_H of . We shown that, if a stabilizing eventually periodic switching rule for is known, then Γ₁, … ,Γ_H can be constructed in such a way that is stabilized by any recurrent switching rule subject to Γ₁, … ,Γ_H. Copyright © 2012 John Wiley & Sons, Ltd.     

External stabilizability of nonlinear systems with some applications

The aim of this paper is to discuss the existence of suitable feedback laws which provide stability (in a technical sense to be specified) with respect to inputs and initial conditions of the state response of a nonlinear system. © 1998 John Wiley & Sons, Ltd.     

On the relationship between global and local controllability

Given a pair(X ⁰,D), whereX ⁰ is a vector field andD is a family of vector fields on a manifoldM, we can define a system affine in the control, i.e., the new family of all the vector fields of the formX ⁰ +uX, whereX ∈D andu is a real parameter. Such a system will be called globally controllable if each state is reachable from each other, whenever unbounded values foru are allowed. It is proved that a system affine in the control is globally controllable if and only if in any set of a suitable partition ofM there exist points locally controllable with bounded values foru. Further, it is proved that, under more restrictive assumptions, global controllability implies the existence of points locally controllable at a fixed time with bounded values foru. In the case of simply connected manifolds, a full equivalence among all the forms of controllability considered here is obtained.     

Stabilization by means of state space depending switching rules

In this paper we consider switched systems defined by a pair of linear systems , such that there exists a neutrally stable linear combination of the matrices A₁, A₂. Under an additional assumption, we prove that there exists a state space depending switching rule which stabilizes the system in a very general sense. The result is illustrated by some simulations and examples.