Time-Delay Systems

This chapter serves as a brief introduction to the theory of the retarded type time-delay system. It starts with a discussion of such basic notions as solutions, initial conditions, and the state of a time-delay system. Then some results on the existence and uniqueness of an initial value problem are presented. Continuity properties of the solutions are discussed as well. The main part of the chapter is devoted to stability analysis. Here we define concepts of stability, asymptotic stability, and exponential stability of the trivial solution of a time-delay system. Classical stability results, obtained using the Lyapunov–Krasovskii approach, are given in the form of necessary and sufficient conditions. A short section with historical comments concludes the chapter.

In this chapter we consider the class of retarded type linear systems with one delay. There are several reasons for restricting our attention to this class before proceeding to more general ones. First, from a methodological point of view, it seems that dealing with single-delay systems simplifies the understanding of basic concepts and creates a firm basis for developing the concepts for more general cases. Second, for the case of single delay we often obtain more complete results than in a more general setting. Finally, results for the single-delay case are not as cumbersome as those for the more general classes of time-delay systems.

In this chapter we address the case of retarded type linear time-delay systems with multiple delays. The fundamental matrix of such a system is defined. Then this matrix is used to derive an explicit expression for the solution of an initial value problem. Applying the scheme presented in the previous chapter, a general form of quadratic functionals with prescribed time derivatives along the solutions of the time-delay systems is obtained. It is shown that the functionals are defined by special matrix valued functions known as Lyapunov matrices for the system. A special system of matrix equations that defines Lyapunov matrices is derived. It is shown that the system admits a unique solution if and only if the spectrum of the time-delay system does not contain points arranged symmetrically with respect to the origin of the complex plane. This spectrum property is known as a Lyapunov condition. Two numerical schemes for the computation of Lyapunov matrices are presented. The first one is applicable to the case where time delays are multiple to a basic one. The other one makes it possible to compute approximate Lyapunov matrices in the case of general time delays. A measure that allows one to estimate the quality of the approximation is provided as well. Quadratic functionals of the complete type are defined, and several important applications of the functionals are presented in the final part of the chapter.

The last chapter of the first part of the book is dedicated to the retarded type linear time-delay systems with distributed delay. First, quadratic functionals with a prescribed time derivative along the solutions of such a system are introduced and Lyapunov matrices that define the functional are derived. Then, such issues as the existence and uniqueness of Lyapunov matrices are studied. A numerical procedure for the computation of Lyapunov matrices is proposed. It is shown that in some cases the computation of the matrices is reduced to the solution of a boundary value problem for an auxiliary system of linear matrix ordinary differential equations. Complete type quadratic functionals are defined, and various application of the functionals are discussed.

This chapter starts the second part of the book, where neutral type time-delay systems are studied. Issues related to the existence, uniqueness, and continuation of the solutions of an initial value problem for such systems are discussed. In addition, stability concepts and basic stability results obtained with the use of the Lyapunov–Krasovskii approach, mainly in the form of necessary and sufficient conditions, are presented.

In this chapter we consider the class of neutral type linear systems with one delay. We define the fundamental matrix of such a system and present the Cauchy formula for the solution of an initial value problem. This formula is used to compute a quadratic functional with a given time derivative along the solutions of the time-delay system. It is demonstrated that this functional is defined by a special matrix valued function, which is called a Lyapunov matrix for a time-delay system. A thorough analysis of the basic properties of the matrix is included. Complete type functionals are introduced, and various applications of the functionals are discussed.

This chapter is dedicated to the case of neutral type linear systems with distributed delay. The structure of quadratic functionals that have prescribed time derivatives along the solutions of such a system is defined, and the corresponding Lyapunov matrices are introduced. A system of matrix equations that defines Lyapunov matrices is given. It is proven that under some conditions this system admits a unique solution. A general class of system with distributed delay for which Lyapunov matrices are solutions of special standard boundary value problems for an auxiliary system of linear matrix ordinary differential equations is presented. Complete type functionals are defined. It is shown that these functionals can be presented in a special form that is more convenient for the computation of lower and upper bounds for the functionals.