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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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.

In Chapter 4, we design a numerical algorithm for computing ISS Lyapunov functions for dynamic systems with perturbations. In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are presented. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable. 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.

Create your web page Haltools: If the linear inequalities are satisfied, then the CPA function is a CPA Fonctio function on the subset excluding a small neighborhood of the origin.

## Lyapunov function

An academic example is shown to illustrate how this method is applied. 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.

Optimization and Control [math.

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. Wikipedia articles incorporating text from PlanetMath.

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This algorithm relies on a linear optimization problem. A Lyapunov function for an autonomous dynamical system. For an iISS dynamic system fknction perturbation, we introduce an auxiliary system which is uniformly asymptotically stable. University of Bayreuth, Saturday, March 7, – For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. We prove that the linear optimization fonctiob has a feasible solution if the system is ISS.

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. A similar concept appears in the theory of general state space Markov chainsusually under the name Foster—Lyapunov functions.

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For each subsystem, an ISS Lyapunov function is computed by our proposed method. Monday, March 21, – 5: 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 stable iISS Lyapunov functions by Zubov’s method and auxiliary systems in Chapter 3.

From Wikipedia, the free encyclopedia. We further state that the iISS Lyapunov function is a local fonctikn to state stable ISS Lyapunov function for the considered dynamic system with perturbations on a subset of the domain of attraction for the auxiliary system. Several examples are presented to show the feasibility of the approach. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known.

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. Furthermore, we study stability of interconnected ISS 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.

### Lyapunov function – Wikipedia

Huijuan Li 1 AuthorId: In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. In this thesis, we investigate the problems of computation of Lyapunov functions and stability analysis of interconnected systems. 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.

The converse is also true, and was proved by J. By the chain rule, for any function, H: Stability of nonlinear systems Lyapunov functions Interconnected systems. We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation.

Views Read Edit View history. This page was last edited on 6 Octoberat Furthermore, stability of two interconnected iISS systems is investigated. Computation of Lyapunov functions and stability of interconnected systems. Lyapunov functions arise in the study of equilibrium points of dynamical systems.