Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

The generalized Cauchy problem for singularly perturbed impulsive systems in the critical case

We construct an asymptotic expansion for solutions to nonlinear singularly perturbed systems of impulsive differential equations. We successively determine all terms of the asymptotic expansion by means of pseudoinverse matrices and orthoprojections.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

Asymptotic analogs of the Floquet-Lyapunov theorem for some classes of periodic systems of ordinary differential equations

We prove asymptotic analogs of the Floquet-Lyapunov theorem and some reducibility theorems for various classes of linear and quasilinear systems of ordinary differential equations with periodic matrices with large and small amplitudes. We study such problems with the help of new versions of the splitting method in the theory of regular and singular perturbations, which complements the known results. We also adduce several examples.     

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.     

Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations

We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of nth-order quasilinear differential equations.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

On the continuous dependence of solutions of linear systems of differential-algebraic equations on a parameter

We consider linear systems of ordinary differential equations with identically degenerate matrix multiplying the derivative of the unknown vector function. The matrices specifying the system are assumed to depend on a parameter. We obtain criteria for the continuous dependence of the solutions of the system on the parameter and the asymptotic equivalence of solutions of the original and perturbed systems.     

Method of freezing in systems with impulse action

Linear and weakly linear systems of differential equations with variable matrices, subjected to impulse action at fixed moments of time, are investigated. Sufficient conditions for the asymptotic stability of solutions of the systems being considered, expressed in terms of the eigenvalues of the variable matrix, are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 848–853, June, 1991.     

Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

     

On the Explicit Solution of Certain First Order Systems of Partial Differential Equations

We consider two classes of systems of partial differential equations of first order. One consists of generalized Stokes-Beltrami equations $ Aw_z = w^*_z $ , $ \lambda Bw_{\bar z} -w^*_{\bar z} $ with square matrices A and B and a scalar factor u . The other may be written in matrix notation as $ v_{\bar z} = c{\bar v} $ where c denotes a square matrix. This system is known as a Pascali system. Both systems are in close connections to certain systems of second order for which the solutions can be represented using particular differential operators. On the basis of these relations we give the solutions of the first order systems explicitly.     

A numerical algorithm for analyzing the asymptotic stability of solutions to linear systems with periodic coefficients

Linear systems of ordinary differential equations with periodic coefficients are considered and some numerical algorithm is proposed for analyzing the asymptotic stability of the zero solution that is based on the Runge-Kutta method of the fourth order of accuracy.     

Weakly non-linear high frequency waves for a first order quasi-linear system involving source-like terms

Making use of the method of asymptotic expansion of multiple scales, a study of weakly non-linear, high frequency waves, through “homogeneous” media characterized by dissipative or dispersive hyperbolic systems of partial differential equations, is proposed. Within the present theoretical framework, asymptotic waves in a heat-conducting fluid are considered.     

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

Asymptotic Stability for a Class of Equations of Neutral Type

Under study is the problem of asymptotic but exponential stability for a class of linear autonomous neutral functional differential equations. We demonstrate that the asymptotic stability of an equation of the class takes place for all integrable initial functions if the roots of the characteristic equation lie on the left of and approach the imaginary axis.

     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.