Une fonction scalaire V(x) continue et différenciable sur un voisinage U de xe, est 0 ∀ ∈ D fonction de Lyapunov stricte pour. Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamicsSolutions bornées et fonction de Lyapunov pour un système. Dans cette Note on propose une nouvelle fonction de Lyapunov pour l’étude de la stabilité asymptotique globale dans un modèle mathématique de compétition.

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Stability of the interconnected systems is then analyzed by the small gain theorem in comparison form and the small gain theorem in dissipative form, respectively. A Lyapunov function for an autonomous foncfion system.

Inria – Computation of Lyapunov functions and stability of interconnected systems

Monday, June 8, – 5: University of Bayreuth, Saturday, March 7, – Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.

We propose a new approach of computing Lyapunov functions for dynamic systems without perturbations with an asymptotically stable equilibrium at the origin in Chapter 2. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.

The proposed method constructs a continuous and piecewise affine CPA function on a compact subset of state space with the origin in its interior based on functions from classical converse Lyapunov theorems originally due to Yoshizawa, and then verifies if the vertex values satisfy linear inequalities for vertices in the subset excluding a small neighborhood of the origin.


Furthermore, stability of two interconnected iISS systems is investigated. From Wikipedia, the free encyclopedia.

Lyapunov function – Wikipedia

Huijuan Li 1 AuthorId: An academic example is shown to illustrate how this method is applied. In Chapter 4, we design a numerical algorithm for computing ISS Lyapunov functions for dynamic systems with perturbations. We prove that the linear optimization problem has a feasible solution if the system is ISS. Monday, March 21, – 5: Have you forgotten your login? For certain classes of ODEs, the existence of Fnction functions is a necessary and sufficient condition for stability.

For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly asymptotically stable.

Furthermore, we study stability of interconnected ISS systems. Wikipedia articles incorporating text from PlanetMath. This page was last edited on 6 Octoberat Fonctiob functions arise in the study of equilibrium points of dynamical systems. If the linear inequalities are satisfied, then the CPA function is a CPA Lyapunov function on the subset excluding a small neighborhood of the origin.

Since a maximal robust Lyapunov function for uniformly asymptotically stable systems can be obtained using Zubov’s method, we present a new way of computing integral input-to- state foncction iISS Lyapunov functions by Zubov’s method and auxiliary systems in Chapter 3. A similar concept appears in the theory of general state space Markov chainsusually under the name Foster—Lyapunov functions.

Since the interpolation errors are incorporated in the linear constraints, as in Chapter 2 the computed ISS Lyapunov function is a true ISS Lyapunov function rather than a numerical approximation. Then a robust Lyapunov function for the auxiliary system is computed by Zubov’s method.

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Since the obtained ISS Lyapunov functions satisfy linear inequalities, the stability of interconnected systems can be analyzed by the small gain theorem in linear form. This algorithm relies on a linear optimization problem. By the chain rule, for any function, H: Computation of Lyapunov functions and stability of interconnected systems.


Several examples are presented to show the feasibility of the approach. If the linear optimization problem has a feasible solution, then the solution is proved to be a CPA ISS Lyapunov function on a spatial grid covering the given compact set excluding a small neighborhood of the origin.

We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation. Huijuan Li 1 Details. We further state that the iISS Lyapunov function is a local input to state stable ISS Lyapunov function for fonvtion considered dynamic system with perturbations on a subset of the domain of attraction for the auxiliary system.

Optimization and Control [math. Views Read Edit View history. Retrieved from ” https: Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. The converse is also true, and was proved by J.

Create your web page Haltools: In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. By using this site, you agree to the Terms of Use and Privacy Policy.

In the theory of ordinary differential equations ODEsLyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. For each subsystem, an ISS Lyapunov function is computed by our proposed method.