Non-monotonicity of Lyapunov functions for functional differential equations with enlightenments for related research methods

In this paper, a particular property of Lyapunov functions for functional differential equations (FDEs) is developed, that is the direct dependence of the signs of the derivatives of the Lyapunov functions on the initial data. This property implies that the derivatives of the Lyapunov functions for FDEs cannot be guaranteed to be negative definite generally, and then makes the FDEs differ from the ordinary differential equations constitutionally. With this property, we give some enlightenments for the research methods for establishing stability theorems or criteria for FDEs, which may help us to form a common view about the choice of the investigation methods on the stability of FDEs. The conclusion is stated in both the deterministic and stochastic versions. Two illustrative examples are given to show and verify our conclusion through the paper.     

Unified type non-stationary Lyapunov matrix equation — Simultaneous eigenvalue bounds

Simultaneous eigenvalue bounds for the solution of the unified non-stationary Lyapunov matrix equation are presented. When the solution becomes stationary, the results reduce to bounds of the unified type algebraic Lyapunov equation. In the limiting cases, the results reduce to bounds for the solution of the differential and difference Lyapunov equations. The bounds given in this paper are a generalization of some existing bounds obtained separately for the continuous and discrete type stationary and non-stationary Lyapunov equations.     

Moment exponential stability of stochastic nonlinear delay systems with impulse effects at random times

In this paper, we investigate the pth moment exponential stability for a class of impulsive stochastic functional differential equations with impulses at random times. The impulsive times considered in this paper are random times that are different from those investigated in the existing literature. By using the stochastic process theory, stochastic analysis theory, Razumikhin technique, and Lyapunov method, we obtain some new criteria of the pth moment exponential stability for the related system. Finally, some examples are provided to show the effectiveness of the theoretical results.     

Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations

In this paper, we propose lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations. We give comparisons between the parallel estimates. Finally, we give examples showing that our bounds can be better than the previous results for some cases.     

ASYMPTOTIC BEHAVIOURS OF STOCHASTIC DIFFERENTIAL DELAY EQUATIONS

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochastic differential delay equations. Moreover, from them follow many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well‐known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results. Two examples are also given for illustration.     

On the Lyapunov theorem for singular systems

In this paper, we revisit the Lyapunov theory for singular systems. There are basically two well-known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of the work by Lewis. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clarified. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of Takaba et al.     

On application of the implemented semigroup to a problem arising in optimal control

In this paper we show that the concept of an implemented semigroup provides a natural mathematical framework for analysis of the infinite-dimensional differential Lyapunov equation. Lyapunov equations of this form arise in various system-theoretic and control problems with a finite time horizon, infinite-dimensional state space and unbounded operators in the mathematical model of the system. The implemented semigroup approach allows us to derive a necessary and sufficient condition for the differential Lyapunov equation with an unbounded forcing term to admit a bounded solution in a suitable space. Whilst our focus is on the differential Lyapunov equation, we show that the same framework is also appropriate for the algebraic version of this equation. As an application we show that the approach can be used to solve a simple decoupling problem arising in optimal control. The problem of infinite time admissibility of the control operator and an infinite-dimensional version of the Lyapunov theorem serve as additional illustrations.     

Feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance

In this paper, we consider the feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance. Due to unknown time-delay coefficient, the system might be exponentially increasing at the lack of control. We design the feedback control law based on Lyapunov function method. Different from usual use of Lyapunov function method, our approach is to combine the construction of Lyapunov functionals with the controller design, which will guarantee the system energy function decays exponentially. In this procedure, we deduce the inequality equations satisfied by the system parameters. We prove the well-posedness of the corresponding closed-loop system by using semigroup theory and the inequality equations are solvable. Moreover, the exponential decay rate of the system is estimated. In addition, some numerical simulations are also presented to support the obtained results.     

Bounds for the solution of the Lyapunov matrix equation — A unified approach

In recent years, several bounds have been reported for the solution of the continuous and the discrete Lyapunov equations. Using the unified Lyapunov equation, we give in this paper bounds for the solution of this equation. In the limiting cases, the bounds reduce to existing bounds for both the continuous and discrete Lyapunov equations.     

Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays

In this paper, a cellular neural network whose state variables are governed by stochastic non-linear integro-differential equations is investigated. The considered delays are distributed continuously over unbounded intervals. By applying the Lyapunov functional method, the semimartingale convergence theorem, and some inequality technique, we obtain some sufficient criteria to check the almost sure exponential stability of the system, which generalizes and improves some earlier publications. Two examples are also given to demonstrate our results.     

Monotonicity of algebraic Lyapunov iterations for optimal control of jump parameter linear systems

In this paper we show that the sequences of the solutions of the decoupled algebraic Lyapunov equations are monotonic under proper initialization. These sequences converge from above to the positive-semidefinite stabilizing solutions of the system of coupled algebraic Riccati equations of the optimal control problem of jump parameter linear systems.     

Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays

In this paper, we study cellular neural networks with almost periodic variable coefficients and time-varying delays. By using the existence theorem of almost periodic solution for general functional differential equations, introducing many real parameters and applying the Lyapunov functional method and the technique of Young inequality, we obtain some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution. The results obtained in this paper are new, useful, and extend and improve the existing ones in previous literature.     

时滞随机大系统的分散次优镇定

本文基于Ｒｉｃａａｔｉ矩阵方程、采用Ｌｙａｐｕｎｏｖ泛函研究具有时滞的Ｉｔｏ型随机大系统的分散次优镇定，建立了分散次优镇定的判据，给出了确定分散次优控制器的一个算法，这个算法以求解Ｌｙａｐｕｎｏｖ矩阵方程与Ｒｉｃｃａｔｉ矩阵方程为基础。文中命同了数值算例以说明本文算法的用法。     

On asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks

In this paper, we obtain some sufficient conditions for determining the asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks by utilizing the Lyapunov functional method. An example is given to show the validity of the results.     

Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity

The purpose of this paper is to develop a systematic method for global asymptotic stabilisation in probability of nonlinear control stochastic systems with stable in probability unforced dynamics. The method is based on the theory of passivity for nonaffine stochastic differential systems combined with the technique of Lyapunov asymptotic stability in probability for stochastic differential equations. In particular, we prove that a nonlinear stochastic differential system whose unforced dynamics are Lyapunov stable in probability is globally asymptotically stabilisable in probability provided some rank conditions involving the affine part of the system coefficients are satisfied. In this framework, we show that a stabilising smooth state feedback law can be designed explicitly. A dynamic output feedback compensator for a class of nonaffine stochastic systems is constructed as an application of our analysis.     

Lyapunov iterations for optimal control of jump linear systems atsteady state

In this paper we construct a sequence of Lyapunov algebraic equations,whose solutions converge to the solutions of the coupled algebraic Riccati equations of the optimal control problem for jump linear systems. The obtained solutions are positive semidefinite, stabilizing, and unique. The proposed algorithm is extremely efficient from the numerical point of view since it operates only on the reduced-order decoupled Lyapunov equations, Several examples are included to demonstrate the procedure     

Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching

This paper is devoted to study the well-known Razumikhin-type theorem for a class of stochastic functional differential equations with Lévy noise and Markov switching. In comparison to the standard Gaussian noise, Lévy noise and Markov switching make the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. By using the Razumikhin method and Lyapunov functions, we obtain several Razumikhin-type theorems to prove the pth moment exponential stability of the suggested system. Based on these results, we further discuss the pth moment exponential stability of stochastic delay differential equations with Lévy noise and Markov switching. In particular, the results obtained in this paper improve and generalise some previous works given in the literature. Finally, an example is provided to illustrate the effectiveness of the theoretical results.     

A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching

In this paper, we consider neutral stochastic delay differential equations with Markovian switching. Our key aim is to establish LaSalle-type stability theorems for the underlying equations. The key techniques used in this paper are the method of Lyapunov functions and the convergence theorem of nonnegative semi-martingales. The key advantage of our new results lies in the fact that our results can be applied to more general non-autonomous equations.