On stability analysis for neural networks with interval time-varying delays via some new augmented Lyapunov–Krasovskii functional

This paper is concerned with the problem of stability analysis of neural networks with interval time-varying delays. It is assumed that the lower bound of time-varying delays is not restricted to be zero. By constructing a newly augmented Lyapunov–Krasovskii functional which has not been proposed yet and utilizing some integral information on activation function as elements of augmented vectors, an improved stability criterion with the framework of linear matrix inequalities (LMIs) is introduced in Theorem 1. Based on the result of Theorem 1 and utilizing the property of the positiveness of Lyapunov–Krasovskii functional, a further relaxed stability condition will be proposed in Theorem 2. The effectiveness and less conservatism of the proposed theorems will be illustrated via three numerical examples.     

Moser–Trudinger inequalities for singular Liouville systems

Mathematische Zeitschrift - In this paper we prove a Moser–Trudinger inequality for the Euler–Lagrange functional of general singular Liouville systems on a compact surface. We...     

Razumikhin–Lyapunov functional method for the stability of impulsive switched systems with time delay

This paper investigates the uniform stability and the uniform asymptotical stability of impulsive switched systems with time delay. By employing the method of Razumikhin–Lyapunov functional, several Razumikhin-type theorems of uniform stability and uniform asymptotical stability are established, which improve some of the existing results. Several examples are also given to illustrate the results.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Construction of the Lyapunov–Krasovskii Functionals for some Classes of Positive Delay Systems

A criterion is established for the diagonal stability of positive linear difference-differential systems with discrete and distributed delay. The problem is studied of existence of a common diagonal Lyapunov–Krasovskii functional for a family of these systems. The approach developed is used to obtain conditions of the absolute stability and estimates of the time of transient processes for nonlinear difference-differential systems with sector type nonlinearities.     

Note on the stability of reaction–diffusion systems with delays by Lyapunov functional

This paper aims to investigate the stability of reaction–diffusion equations with delays. We extend a stability theorem on FDEs introduced by Hale to reaction–diffusion equations with time delay.     

On the use of a general quadratic Lyapunov function for studying the stability of Takagi–Sugeno systems

We study the stability of the zero solution to a nonlinear system of ordinary differential equations on the base of its Takagi–Sugeno (TS) representation. As is known, the most constructive stability and stabilization conditions for TS systems stated as linear matrix inequalities are established with the help of a general quadratic Lyapunov function (GQLF). However, such conditions are often too rigid. Using a modification of the Lyapunov direct method, we propose asymptotic stability conditions with weaker requirements to GQLF. They allow an application to a wider class of systems. We also give some illustrative examples.     

Minimax solutions of Hamilton–Jacobi functional equations for neutral-type systems

A functional Hamilton–Jacobi equation with covariant derivatives which corresponds to neutral-type dynamical systems is obtained. The definition of a minimax solution of this equation is given. Conditions under which such a solution exists and is unique and well defined are found.     

Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations

We consider Lyapunov exponents and Sacker–Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the reduction to finite dimension of the evolution family associated to the system, to which a classic discrete QR method is then applied. The discretization of the evolution family is accomplished by a combination of collocation and generalized Fourier projection. A rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra. The efficacy of the results is illustrated with some numerical examples.     

A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate

We consider the dynamics of a general stage-structured predator–prey model which generalizes several known predator–prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.     

On singular limit for the compressible Navier–Stokes system with non-monotone pressure law

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.     

On the Cauchy problem for systems —Parabolic systems—

The author proposes the definition of thep-parabolic system, which is stable under the similar transformations, using thep-determinant of the matrix of differential operators. The relation between thep-parabolic systems andH ^∞ well-posedness is considered. To the memory of Lamberto Cattabriga     

A novel feedforward–feedback suboptimal control of linear time-delay systems

This paper presents a new approach for solving the optimal control problem of linear time-delay systems with a quadratic cost functional. In this approach, a method of successive substitution is employed to convert the original time-delay optimal control problem into a sequence of linear time-invariant ordinary differential equations (ODEs) without delay and advance terms. The obtained optimal control consists of a linear state feedback term and a forward term. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving recursively the above-mentioned sequence of linear non-delay ODEs. A fast-converging iterative algorithm for this purpose is presented which provides a promising possible reduction of computational efforts. Numerical examples demonstrating the efficiency, simplicity and high accuracy of the suggested technique have been included. Simulation results reveal that just a few iterations of the proposed algorithm are required to find an accurate enough feedforward–feedback suboptimal control.     

Mittag–Leffler stability of nonlinear fractional neutral singular systems

This paper studies Mittag–Leffler stability of nonlinear fractional neutral singular systems under Caputor and Riemann–Liouville derivatives. Several sufficient conditions are derived by extending Lyapunov direct method to such systems. Our theoretical results can also be applied to general fractional retarded, neutral and singular systems.     

A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional

The Swift–Hohenberg equation is a central nonlinear model in modern physics. Originally derived to describe the onset and evolution of roll patterns in Rayleigh–Bénard convection, it has also been applied to study a variety of complex fluids and biological materials, including neural tissues. The Swift–Hohenberg equation may be derived from a Lyapunov functional using a variational argument. Here, we introduce a new fully-discrete algorithm for the Swift–Hohenberg equation which inherits the nonlinear stability property of the continuum equation irrespectively of the time step. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and stability of our new algorithm. We also compare our method to other existing schemes, showing that is feasible alternative to the available methods.