Systems with Distributed Delay
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 Lyapu
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Systems with Distributed Delay
In this chapter a linear retarded type system with distributed delays is studied. First, we introduce quadratic functionals and Lyapunov matrices for the system. Then we present the existence and uniqueness conditions for the matrices and provide some numerical schemes for the computation of the matrices. In the last part of the chapter functionals of the complete type are introduced, and some applications of the functionals are discussed.
4.1 System Description We start with the following retarded type time-delay system: d x(t) = A0 x(t) + A1x(t − h) + dt
0
G(θ )x(t + θ )dθ ,
t ≥ 0.
(4.1)
−h
Here A0 and A1 are given real n × n matrices, delay h > 0, and G(θ ) is a continuous matrix defined for θ ∈ [−h, 0].
4.2 Quadratic Functionals Given a symmetric matrix W , we are looking for a quadratic functional v0 : PC([−h, 0], Rn ) → R
V.L. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering, DOI 10.1007/978-0-8176-8367-2 4, © Springer Science+Business Media, LLC 2013
133
134
4 Systems with Distributed Delay
such that along the solutions of system (4.1) the following equality holds: d v0 (xt ) = −xT (t)W x(t), dt
t ≥ 0.
(4.2)
Definition 4.1. The matrix U(τ ) is said to be a Lyapunov matrix of system (4.1) associated with a symmetric matrix W if it satisfies the following properties: 1. Dynamic property: d U(τ ) = U(τ )A0 + U(τ − h)A1 + dτ
0
U(τ + θ )G(θ )dθ ,
τ ≥ 0;
(4.3)
−h
2. Symmetry property: U(−τ ) = U T (τ ),
τ ≥ 0;
(4.4)
3. Algebraic property: − W = U(0)A0 + U(−h)A1 + + AT1 U(h) +
0
0
U(θ )G(θ )dθ + AT0 U(0)
−h
GT (θ )U(−θ )dθ .
(4.5)
−h
Remark 4.1. The algebraic property can also be written as U (+0) − U (−0) = −W.
(4.6)
For a given matrix U(τ ) we define on PC([−h, 0], Rn ) a functional of the form
v0 (ϕ ) = ϕ (0)U(0)ϕ (0) + 2ϕ (0) T
+
0
T
⎛
ϕ T (θ1 )AT1 ⎝
−h
+ 2ϕ T (0)
0
−h
⎛ ⎝
−h
U(−h − θ )A1ϕ (θ )dθ
−h
0
−h
θ
0
⎞
U(θ1 − θ2 )A1 ϕ (θ2 )dθ2 ⎠ dθ1 ⎞
U(ξ − θ )G(ξ )dξ ⎠ ϕ (θ )dθ
4.2 Quadratic Functionals
+2
0
135
⎛
ϕ T (θ1 )AT1 ⎝
−h
⎡
0 θ2
⎣
−h
⎤
⎞
U(h + θ1 − θ2 + ξ )G(ξ )dξ ⎦ ϕ (θ2 )dθ2 ⎠ dθ1
−h
⎧ ⎡ ⎞ ⎤ ⎛ 0 θ2 ⎨0 θ1 ⎣ GT (ξ1 ) ⎝ U(θ1 − θ2 − ξ1 + ξ2 )G(ξ2 )dξ2 ⎠ dξ1 ⎦ + ϕ T (θ1 ) ⎩ −h
−h
× ϕ (θ2 )dθ2
⎫ ⎬ ⎭
−h
−h
dθ1 .
(4.7)
We can now prove the theorem. Theorem 4.1. Let U(τ ) be a Lyapunov matrix of system (4.1) associated with W . Then the time derivative of functional (4.7) along the solutions of the system satisfies equality (4.2). Proof. Let x(t), t ≥ 0, be a solution of system (4.1); then v0 (xt ) = x (t)U(0)x(t) + 2x (t) T
+
T
0
⎛ xT (t + θ1 )AT1 ⎝
−h
0 θ
−h
+2
0
−h
⎣
U(−h − θ )A1x(t + θ )dθ
−h
0
⎞
U(θ1 − θ2 )A1 x(t + θ2 )dθ2 ⎠ dθ1
−h
⎡
+ 2xT (t)
0
⎤
U(ξ − θ )G(ξ )dξ ⎦ x(t + θ )dθ
−h
⎡
xT (t + θ1 )AT1 ⎣ ⎤
0
⎛ ⎝
−h
θ2
⎞ U(h + θ1 − θ2 + ξ2)G(ξ2 )dξ2 ⎠
−h
× x(t + θ2 )dθ2 ⎦ dθ1
+
0 −h
⎛ xT (t + θ1 )⎝
⎡
⎛
⎣
GT (ξ1 )⎝
0 θ1
−h
⎞
−h
θ2
⎞
⎤
U(θ1 − θ2 − ξ1 + ξ2 )G(ξ2 )dξ2 ⎠dξ1 ⎦
−h
× x(t + θ2 )dθ
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