General Theory
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, sta
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General Theory
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 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 here.
5.1 System Description We consider a neutral type time-delay system of the form d [x(t) − Dx(t − h)] = f (t, xt ). dt
(5.1)
Here the functional f (t, ϕ ) is defined for t ∈ [0, ∞) and ϕ ∈ PC1 ([−h, 0], Rn ), f : [0, ∞) × PC1 ([−h, 0], Rn ) −→ Rn , and is continuous in both arguments. The matrix D is a given n × n matrix, delay h > 0. The information needed to begin the computation of a particular solution of the system includes an initial time instant t0 ≥ 0 and an initial function ϕ : [−h, 0] → Rn , and it is assumed that x(t0 + θ ) = ϕ (θ ),
θ ∈ [−h, 0].
(5.2)
As usual, the state of the system at the time instant t ≥ t0 is defined as the restriction, xt : θ → x(t + θ ),
θ ∈ [−h, 0],
V.L. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering, DOI 10.1007/978-0-8176-8367-2 5, © Springer Science+Business Media, LLC 2013
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5 General Theory
of the solution x(t) on the segment [t − h,t]. If the initial condition (t0 , ϕ ) is indicated explicitly, then we use the notations x(t,t0 , ϕ ) and xt (t0 , ϕ ). In the case of timeinvariant systems we usually assume that t0 = 0 and omit the argument t0 in these notations. We will use initial functions from the space PC1 ([−h, 0], Rn) ⊂ PC ([−h, 0], Rn ). Here it is assumed that a function ϕ ∈ PC ([−h, 0], Rn ) belongs to PC1 ([−h, 0], Rn) if on each continuity interval (α , β ) ∈ [−h, 0] the function is continuously differentiable and the first derivative of the function, ϕ (θ ), 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 |). On the one hand, such a choice creates certain technical difficulties. But on the other hand, it provides several advantages in the formulations and proofs of some statements presented in the chapter. In particular, it follows from Theorem 5.1 that if ϕ ∈ PC1 ([−h, 0], Rn ), then xt (t0 , ϕ ) ∈ PC1 ([−h, 0], Rn) for t > t0 . Henceforth we assume that the following assumptions hold. Assumption 5.1. The difference x(t,t0 , ϕ ) − Dx(t − h,t0 , ϕ ) is continuous and differentiable for t ≥ t0 , except possibly for a countable number of points. This does not imply that x(t,t0 , ϕ ) is differentiable, or even continuous, for t ≥ t0 . Assumption 5.2. In Eq. (5.1) the right-hand-side derivative of the difference x(t,t0 , ϕ ) − Dx(t − h,t0 , ϕ ) is assumed at the point t = t0 . By default, such agreement remains valid in situations where only a one-sided variation of the independent variable is allowed. Let x(t) be a solution of the initial value problem
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