Lyapunov modular functions

We prove a Lyapunov type theorem for modular functions on complemented lattices.     

Quantization of Lyapunov functions

For a state of equilibrium of an autonomous system of differential equations, we propose discrete conditions for stability and asymptotic stability in the sense of Lyapunov.     

Robustness via Lyapunov functions

For nonautonomous linear equations x^′=A(t)x, we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic Lyapunov functions. Nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies. In particular, we construct explicitly strict Lyapunov functions for each exponential dichotomy. As a nontrivial application, we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. This represents a considerable simplification of former work.     

Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications

In this note we provide sufficient conditions for the existence of a Lyapunov function for a class of parabolic feedback control problems. The results are applied to the long-time behavior of state functions for the following problems: (i) a model of combustion in porous media; (ii) a model of conduction of electrical impulses in nerve axons; and (iii) a climate energy balance model.     

Projection methods preserving Lyapunov functions

In this paper we consider ordinary differential equations with a known Lyapunov function. We study the use of Runge–Kutta methods provided with a dense output and a projection technique to preserve any given Lyapunov function. This approach extends previous work of Grimm and Quispel (BIT 45, 2005), allowing the use of Runge–Kutta methods for which the associated quadrature formula does not need to have positive or zero coefficients. Some numerical experiments show the good performance of the proposed technique.     

Lyapunov functions in the attractors dimension theory

The effectiveness of constructing Lyapunov functions in the attractors dimension theory is theory of the dimension demonstrated. Formulae for the Lyapunov dimension of the Lorenz, Hénon and Chirikov attractors are derived and proved. A hypothesis regarding the formula for the dimension of the Rössler attractor is formulated.     

Lyapunov functions versus exponential contractions

For nonautonomous linear equations in a Banach space admitting a nonuniform version of exponential contraction, we give an optimal characterization of the exponential behavior in terms of strict Lyapunov sequences. In particular, we construct explicitly strict Lyapunov sequences for each exponential contraction. We also consider the particular case of quadratic Lyapunov functions, and we use the corresponding characterization of the exponential behavior in terms of these functions to show that the stability of an exponential contraction persists under sufficiently small perturbations.     

Application of the method of Lyapunov periodic functions

The paper is concerned with asymptotic behavior of continuous and discrete phase control systems involving periodic differentiable nonlinear vector functions and featuring nonunique equilibrium state.     

Non-coercive Lyapunov functions for infinite-dimensional systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.     

Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.     

An equation of Lyapunov type

A new method of finding explicit solutions of Lyapunov equations is described based on a lemma on one-dimensional perturbations of invertible operators. If Y satisfies the equation $\text{Y?CYC}{}^1\text{= bb}{}^1$ for an appropriate vector b, then X = Y^-1 satisfies $\text{X? C}{}^1\text{XC = aa}{}^1$ for a given vector a. A concrete example [with a=(1,0,…,0)^T] is given.     

Infinitesimal Lyapunov functions for singular flows

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative $DX$ of the vector field $X$ together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.     

Generalized Lyapunov functions and functional equations

Summary Some results of Minty and Browder on the existence of solutions of functional equations are generalized by replacing the notion of monotony by one involving a Lyapunov function. In the last section, analogous arguments are used to obtain an existence theorem for an initial value problem belonging to an ordinary differential equation on Hilbert space. This work was supported by Air Force Office of Scientific Research.