Linear Functional Differential Systems Research Articles

ON THE CONTROLLABILITY OF LINEAR EVOLUTIONARY FUNCTIONAL-DIFFERENTIAL SYSTEMS

New theorem on the controllability of linear functional-differential system of evolutionary type and algorithm for practical verification of controllability are obtained, which can be applied even in the case when the coefficients of system are not continuously differentiable over the time interval under consideration. Special cases of this system are nonstationary systems with distributed and concentrated delay, integro-differential systems with a Voltaire integral and ordinary differential systems. The main results obtained are formulated in the form of 12 theorems and 3 corollaries. On their basis an algorithm for practical verification of the controllability of the system under consideration using a computer is built. Examples illustrating the operability of the obtained theorems and the algorithm are given. The algorithm is implemented in the Maple 17 package for examples of second-order differential systems with delay.

Some Economic Dynamics Problems for Hybrid Models with Aftereffect

In this paper, we consider a class of economic dynamics models in the form of linear functional differential systems with continuous and discrete times (hybrid models) that covers many kinds of dynamic models with aftereffect. The focus of attention is periodic boundary value problems with deviating argument, control problems with respect to general on-target vector-functional and questions of stability to solutions. For boundary value problems, some sharp sufficient conditions of the unique solvability are obtained. The attainability of on-target values is under study as applied to control problems with polyhedral constraints with respect to control, some estimates of the attainability set as well as estimates to a number of switch-points of programming control are presented. For a class of hybrid systems, a description of asymptotic properties of solutions is given.

On a class of linear continuous-discrete systems with discrete memory

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.

ДОСТИЖИМЫЕ ЗНАЧЕНИЯ ЦЕЛЕВЫХ ФУНКЦИОНАЛОВ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОЙ СИСТЕМЫ С ИМПУЛЬСНЫМ ВОЗДЕЙСТВИЕМ

For a linear functional differential system with aftereffect and impulses, a description of the attainability set is given. The attainability is considered in the term of a given system of on-target functionals in the case of polyhedral constraints with respect to control and impulses.

Exponential Stability of Functional Differential Systems

We present a novel approach to exponential stability of functional differential systems. Our approach is relied upon the theory of positive linear functional differential systems and a comparison principle. Consequently, we get some comparison tests and explicit criteria for exponential stability of functional differential systems. Two examples are given to illustrate the obtained results.

Erratum to: Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II (in Russian)

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. I

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II

In this paper we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for the solutions of the considered systems in a critical case. In the second part of the work we establish the existence of a critical manifold for the considered class of systems and study its main properties. We also investigate the asymptotic integration problem for a reduced system. We illustrate the proposed method with an example of constructing the asymptotics for the solutions of a certain scalar delay differential equation.

Control of functional differential system in conditions of impulse disturbances

We consider the control problem for a linear functional differential system with time delay in a general form. The purpose of controlling is prescribed with the use of a finite set of linear functionals. The number of these functionals is independent of the dimension of the system. The system is subject to impulse disturbances which result in trajectory jumps with unknown previously time moments and values. To solve the control problem, we construct a control that contains both programand jumps-positional components. We establish solvability conditions for the stated control problem.

Non-oscillatory Solutions of Odd Order Linear Functional Differential System of Neutral Type with Distributed Delay

In the present paper we establish explicit and easy computable sufficient conditions for existing of several types of non-oscillatory solutions of linear delayed system of neutral type with distributed delay. The results are proved by numerical range technique, and generally they are applicable in the case of non-monotone measures too.

Asymptotic integration of functional differential systems with oscillatory decreasing coefficients

We use the method of averaging and the extension of the Levinson fundamental theorem to study the problem of asymptotic integration of a class of linear functional differential systems that contain oscillatory decreasing coefficients. Moreover, we construct the asymptotics for solutions of the second order delay differential equation that is close, in some sense, to harmonic oscillator.

Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

The classical Wazewski theorem claims that the condition p ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities , x i (0) ≥ 0, i =1, ..., n. Although this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients p ij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Green's matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Green's matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Green's matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the n th component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered. Mathematics Subject Classification 2000: 34K06; 34K10.

Slowly growing solutions of singular linear functional differential systems

AbstractWe obtain new conditions under which a system of linear functional differential equations with singular coefficients has a unique solution possessing a given initial value and satisfying a certain growth restriction.

Stability of linear functional differential systems with a multivalued feedback with respect to impulse disturbances

We consider a linear controlled functional differential system with a linear feedback channel. The system is assumed to be exponentially stable in the closed state. The feedback is assumed to be nonideal, which consists in an uncertain delay either distributed or not. The only assumption is that this delay is sufficiently small. Such a nonideal system is described by a functional differential inclusion of special type. A generalized derivative of a function of locally bounded variation is admitted to the input of the system as a free term. We obtain exponential estimates for solutions of the resulting system.

Differential Inequalities for One Component of Solution Vector for Systems of Linear Functional Differential Equations

The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green's matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for the th component of the solution vector and then to use assertions about positivity of its Green's functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval , where the system is considered.

On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new.

An optimal condition for the uniqueness of a periodic solution for linear functional differential systems

Unimprovable effective efficient conditions are estab- lished for the unique solvability of the periodic problem u '(t) = i+1 X j=2 li,j(uj)(t) + qi(t) for 1 ≤ i ≤ n − 1, u ' (t) = n X j=1 ln,j(uj)(t) + qn(t),

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem $$ u^{\prime }_{i} (t) = {\ell }_{i} (u_{{i + 1}} )(t) + q_{i} (t)\quad (i = \overline{{1,n - 1}} ), $$ $$ u^{\prime }_{n} (t) = {\ell }_{n} (u_{1} )(t) + q_{n} (t), $$ $$ u_{j} (0) = u_{j} (\omega )\quad (j = \overline{{1,n}} ), $$ where ω > 0, l i : C([0, ω])→ L([0,ω]) are linear bounded operators, and q i ∈L([0, ω]).