Stability of viscoelastic systems under stochastic excitation

An effective method for the investigation of the stability for viscoelastic systems under a parametric stochastic excitation is proposed. The parametric force is assumed in the form of a Gaussian stationary colored noise. The method is based on the simulation of random processes, the numerical solution of differential equations, describing the perturbed motion of the considered system, and the calculation of top Liapunov exponents. The considered method makes it possible to estimate the almost sure stability and the stability with respect to statistical moments of the different order. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of singularly perturbed nonlinear stochastic hybrid systems

The problem of exponential mean-square stability of nonlinear singularly perturbed, stochastic hybrid systems is studied in this article. Two groups of nonlinear systems are considered separately. To obtain the sufficient conditions of stability, two basic approaches of stability analysis for hybrid systems with a given Markovian switching rule and any Markovian switching rule and singularly perturbed non–hybrid systems were combined. The Lyapunov techniques were used in both approaches. The obtained results are illustrated by examples.     

Stability conditions of nonlinear distributed processes under parametric excitation

We derive a stability criterion relative to a given measure on a finite time interval for distributed processes under parametric excitation. The corresponding theorem is proved by the comparison method combined with Lyapunov second method. Treating time as a parameter, we use the extremal properties of the Rayleigh quotient for self-adjoint operators in a Hilbert space, which in turn involves solving the eigenvalue problem generated by the linear operators corresponding to the original problem. The results are applied to establish sufficient conditions of technical stability relative to a given measure in the nonlinear problem of a hinged pole under the action of a continuous longitudinal force.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 107–116, 1986.     

Stability properties of nonlinear stochastic impulsive systems with time delay

This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.     

Robust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching

This paper is concerned with the problem of robust reliable control for a class of uncertain stochastic switched nonlinear systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system. A design scheme for the reliable controller is proposed to guarantee almost surely exponential stability for stochastic switched systems with actuator failures, and the dwell time approach is utilized for the stability analysis. Then the approach is extended to take into account stochastic switched system with Lipschitz nonlinearities and structured uncertainties. Finally, a numerical example is employed to verify the proposed method.     

Feedback stabilization of stochastic bilinear systems and of some nonlinear stochastic systems

The purpose of this note, is to derive sufficient conditions for the existence of stabilizing feedback laws for control stochastic bilinear systems and to apply these results to the stabilization of a class of nonlinear stochastic differential systems. The method used in this paper rely on the stochastic Lyapunov machinery     

Stability of interconnected stochastic delay systems

A decomposition of a large-scale stochastic functional differential system into a family of isolated subsystems is made. A set of criteria is obtained under which the uniform asymptotic stability with probability one of the trivial solution process of the large-scale system is preserved. These criteria are expressed in terms of the stability of isolated subsystems and bounds on interconnection components. As a tool of our analysis, Razumikhin-type stability conditions for stochastic functional differential systems are also obtained. Several examples are given to show the applicability of our results.     

Stability estimates of linear stochastic systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1267–1273, September, 1989.     

Optimal control over nonlinear stochastic systems

The synthesis of optimal control over nonlinear stochastic systems that are described by the Itô equations is reduced to the solution of recurrence relations derived from the Bellman stochastic equation.     

Controllability of perturbed nonlinear stochastic integrodifferential systems

The purpose of this paper is to establish the sufficient conditions concerning the complete controllability of stochastic integrodifferential systems in finite dimensional spaces by using the Banach fixed point theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toward a theory of nonlinear stochastic systems

The inequalities of Bihari, Langenhop, and Chebyshev are applied to a system of stochastic nonlinear differential equations. As a result, a one-parametric family of determinate estimates from above and below are obtained for the norm of the solution of the system. Each of these estimates is valid with a probability not less than some quantity depending upon the parameter of the family.Translated from Matematicheskii Zametki, Vol. 5, No. 5, pp. 607–614, May, 1969.The author expresses his appreciation to G. Ya. Lyubarski for his interest in the study and his valuable comments.     

Sample controllability of nonlinear stochastic integrodifferential systems

Sufficient conditions are established for the sample controllability of nonlinear stochastic integrodifferential systems by using the notion of comparison principle and a random fixed point theorem. Examples are provided to illustrate the result.     

Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

Active controlled structural systems under delta-correlated random excitation: Linear and nonlinear case

Reduction of structural vibration in active controlled dynamical system is usually performed by means of convenient control forces dependent of the dynamic response. In this paper the existent studies will be extended to dynamical systems subjected to non-Gaussian random process accounting for the time delay involved in the application of active control actions. Control forces acting with time-delay effects will be expanded in Taylor series evaluating response statistics by means of the extended Itô differential rule to consider the effects of the non-normality of the input processes. Numerical application provided shows the feasibility of the proposed method to analyze stochastic dynamic systems with delayed actions under delta-correlated process contrasting statistics of response with estimates from Monte Carlo simulation.     

Stability of infinite systems of stochastic equations

A study is presented of processes with local interaction and infinite systems of Itô equations, which are small perturbations of independent processes at each point of a lattice with noncompact set of values. A survey of results and a complete exposition are devoted to the technique of cluster expansions, which uses estimates of the Lyapunov function type. This technique permits total control over the temporal evolution of the system and, in particular, enables one to prove that the temporal evolution is exponentially convergent.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 27, pp. 79–128, 1990.     

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.