General Theory
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
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General Theory
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.
1.1 Preliminaries We begin with a class of retarded type time-delay systems of the form dx(t) = g(t, x(t), x(t − h)), dt
(1.1)
where x ∈ Rn and the time delay h > 0. Let the vector-valued function g(t, x, y) be defined for t ≥ 0, x ∈ Rn , and y ∈ Rn . We assume that this function is continuous in the variables.
1.1.1 Initial Value Problem It is well known that a particular solution of a delay-free system, x˙ = G(t, x), is defined by its initial conditions, which include an initial time instant t0 and an initial
V.L. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering, DOI 10.1007/978-0-8176-8367-2 1, © Springer Science+Business Media, LLC 2013
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1 General Theory
state x0 ∈ Rn . This is not the case when dealing with a solution of system (1.1). Here the knowledge of t0 and x0 is not sufficient even to define the value of the time derivative of x(t) at the initial time instant t0 . To define a solution of system (1.1), one needs to select an initial time instant t0 ≥ 0 and an initial function ϕ : [−h, 0] → Rn . The initial value problem for system (1.1) is formulated as follows. Given an initial time instant t0 ≥ 0 and an initial function ϕ , find a solution of the system that satisfies the condition x(t0 + θ ) = ϕ (θ ),
θ ∈ [−h, 0].
(1.2)
The initial function ϕ belongs to a certain functional space. It may be the space of continuous functions, C ([−h, 0], Rn ), the space of piecewise continuous functions, PC ([−h, 0], Rn ), or some other functional space. The choice of the space is dictated by a specific problem under investigation. In our case we assume that initial functions belong to the space PC ([−h, 0], Rn). Recall that the function ϕ belongs to the space if it admits at most a finite number of discontinuity points and for each continuity interval (α , β ) ∈ [−h, 0] the function has a finite right-hand-side limit at θ = α , ϕ (α + 0) = limε →0 ϕ (α + |ε |), and a finite left-hand-side limit at θ = β , ϕ (β − 0) = limε →0 ϕ (β − |ε |). The Euclidean norm is used for vectors and the corresponding induced norm for matrices. The space PC ([−h, 0], Rn ) is supplied with the standard uniform norm [24, 65, 66], ϕ h = sup ϕ (θ ) . θ ∈[−h,0]
On the one hand, the fact that initial function
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