Finitetime stabilization and guaranteed cost control of linear autonomous delay systems with bounded controls

For the first time, the finitetime stabilization with guaranteed cost control for linear autonomous timevarying delay systems with bounded controls is studied in this paper. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel sufficient conditions for designing nonlinear feedback controllers that guarantee the robust finitetime stabilization of the closedloop system are established. The obtained stabilization condition is then adapted to solve the problem of guaranteed cost control. A numerical example is given to show the effectiveness of the proposed results

Finite-time stabilization and guaranteed cost control of linear autonomous delay systems with

P NIAMSUPa, V N PHATb, ∗

aDepartment of Mathematics Chiang Mai University, Chiang Mai 50200, Thailand

bInstitute of Mathematics, VAST

18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

∗ Corresponding author: vnphat@math.ac.vn

Abstract

For the first time, the finite-time stabilization with guaranteed cost control for linear autonomous time-varying delay systems with bounded controls is studied in this paper Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel sufficient conditions for designing nonlinear feedback controllers that guarantee the robust finite-time stabilization of the closed-loop system are established The obtained stabilization condition is then adapted to solve the problem of guaranteed cost control.

A numerical example is given to show the effectiveness of the proposed results.

Key words. Finite-time stabilization, Bounded control, Guaranteed cost control, Time-varying delay, Lyapunov function, Linear matrix inequality

1 Introduction

In the last decade, we have witnessed an increasing interest to the problem of finite-time stability and control for linear time-delay systems [1-4] The finite-time stability (FTS) in-troduced in [5] means that the state of a system does not exceed some bound during a fixed interval time The finite-time stabilization concerns with the design of a feedback controller which ensures the FTS of the closed-loop system and the problem of guaranteed cost control (GCC) is to find a feedback controller to finite-time stabilize the system guaranteeing an ad-equate cost level of performance Based on linear matrix inequality techniques, some results have been obtained for FTS and GCC for a class of linear time-delay systems, for instance [6-10] However, according to the author’s knowledge, there is no result available yet on FTS and GCC for linear time-delay systems with bounded controls

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1 This work was completed when the second author was visiting the Vietnam Institute for Advance Study in Mathematics (VIASM) He would like to thank the VIASM for financial support and hospitality

The study of stabilizations of systems with control constraints is not only a natural math-ematical problem, but also arises often in many applied areas [11-14] It is clear that control constraints on the structure of the feedbacks and the neglect of geometric constraints on the control are hardly in accord with present-day requirements for control systems More-over, finite-time stability analysis for linear time-varying delay systems is more difficult, because time-varying delay systems have more complicated dynamic behaviors than the sys-tems without delay or with constant delays Furthermore, it is difficult to find an suitable Lyapunov-Krasovskii functional satisfying the derivative conditions of the FTS and GCC as shown in [15]

Traditionally, Lyapunov function theory has served as a powerful tool for stability and control analysis The idea of a Lyapunov function was extended in [16] in the context of control design to yield control Lyapunov functions (CLFs) For continuous linear control systems, there exist a well known method to construct CLFs, which essentially involves finding a positive definite solution of Riccati equations or linear matrix inequalities (LMIs) [14, 17-19] However, the procedures are derived under the assumption of unconstrained control action For linear bounded control systems without delays, the system matrix satisfies some appropriate spectral and controllability properties, papers [20-22] proposed a nonlinear feedback control to Lyapunov stabilizes the system without delays It is worth noting that the approach in these works cannot be applied readily to the systems with time-varying delays The main difficulty is that the investigation of the spectrum of the time-varying delay system matrices is still complicated and there are no appropriate properties available

as in the un-delayed case Consequently, the problem of the finite-time control of linear time-delay systems with bounded controls is of interest in its own right

Motivated by the above discussions, we study the problem of finite-time control for linear autonomous time-varying delay systems with bounded controls Our main propose is to design a nonlinear feedback controller which guarantees the closed-loop system finite-time stable and guarantees an upper bound on the quadratic cost performance First, we show how to obtain sufficient conditions for robust finite-time stabilization of linear autonomous delay systems with bounded control by using Lyapunov function method and LMI techniques Then, we will demonstrate how the obtained stabilization result can be applied to solve the GCC problem for the system The conditions are obtained in terms of LMIs, which can be determined by utilizing MATLABs LMI Control Toolbox [23] Finally, an example is given

to show the effectiveness of the proposed results

The structure of the paper is as follows Section 2 gives the necessary background on linear time-varying delay systems with bounded controls and some technical lemmas In Section 3, the state feedback controller design for robust finite-time stabilization and GCC

is presented together with an illustrative example Section 4 gives some conclusions

2 Preliminaries

In this section, we introduce some notations and lemmas R+ denotes the set of all real

non-negative numbers; R n denotes the n −dimensional space with the scalar product x ⊤ y;

M n ×r denotes the space of all matrices of (n × r)−dimensions A ⊤ denotes the transpose

of matrix A; A is symmetric if A = A ⊤ ; I denotes the identity matrix; λ(A) denotes the

set of all eigenvalues of A; λmax(A) = max {Reλ; λ ∈ λ(A)} x t := {x(t + s) : s ∈ [−h, 0]}, ∥

x t ∥= sup s ∈[−h,0] ∥ x(t + s) ∥; C1([0, t], R n ) denotes the set of all R n −valued continuously

differentiable functions on [0, t] with the norm: ∥φ∥ C1 = max{∥φ∥ C , ∥ ˙φ∥ C } L2([0, t], R m)

denotes the set of all the R m −valued square integrable functions on [0, t]; Matrix A is called semi-positive definite (A ≥ 0) if x T Ax ≥ 0, for all x ∈ R n ; A is positive definite (A > 0)

if x ⊤ Ax > 0 for all x ̸= 0; A > B means A − B > 0 The notation diag{ .} stands for

a block-diagonal matrix Matrix (M ij)n ×m denotes the matrix of M i,j , i = 1, 2, , n, j =

1, 2, , m, M ij = M ji , i ̸= j The symmetric term in a matrix is denoted by ∗.

Consider the following linear autonomous time-varying delay systems with bounded

˙

x(t) = Ax(t) + Dx(t − h(t)) + Bu(t) + B1w(t)) t ≥ 0,

where x(t) ∈ R n , u(t) ∈ R m , w ∈ R r are, respectively, the state, the control, the disturbance

vector, A, D ∈ R n ×n , B ∈ R n ×m , B

1 ∈ R n ×pare given constant matrices The initial function

φ(t) ∈ C1([−h2, 0], R n ) The delay function h(t) is continuous and satisfying

The control u ∈ L2([0, T ], R m) satisfies

∃r > 0 : ||u(t)|| ≤ r, ∀t ≥ 0. (2.3)

The disturbance w(t) ∈ L2([0, T ], R p) satisfies

∃d > 0 :

∫ T

0

The performance index associated with the system (2.1) is the following function

J (u) =

∫ T

0

[x ⊤ (t)Q

1x(t) + x ⊤ (t − h(t))Q2x(t − h(t)) + u ⊤ (t)Q3u(t)]dt, (2.5)

where Q1, Q2 ∈ R n×n , Q

3 ∈ R m×m , are given symmetric positive definite matrices The

objective of this paper is to design a feedback controller u(t) = Kx(t) satisfying (2.3) and a positive number J ∗ such that the resulting closed-loop system

˙

x(t) = (A + BK)x(t) + Dx(t − h(t)) + B1w(t), (2.6)

is finite-time stable for all disturbance w(t) satisfying (2.2) and the value of the cost function (2.4) is bounded by J ∗ .

Definition 2.1 For given positive numbers T, c1, c2, c2 > c1, and symmetric positive definite matrix R, the unforced control system (2.1) is robustly finite-time stable w.r.t (c1, c2, T, R) if

sup

−h2≤s≤0

{

φ ⊤ (s)Rφ(s)}

≤ c1 =⇒ x ⊤ (t)Rx(t) < c

2, ∀t ∈ [0, T ], for all disturbance w(t) satisfying (2.4).

Definition 2.2. If there exist a feedback control law u ∗ (t) = Kx(t) satisfying (2.3) and a

positive number J ∗ such that the closed-loop system (2.6) is robustly finite-time stable and

the cost function (2.5) satisfies J (u ∗)≤ J ∗ , then the value J ∗ is a guaranteed cost value and

the designed control u ∗ (t) is said to be a guaranteed cost controller.

We introduce the following technical propositions, which will be used in the proof of our results

Proposition 2.1 (Schur complement lemma [24]) Given constant matrices X, Y, Z with

appropriate dimensions satisfying Y = Y ⊤ > 0, X = X ⊤ Then X + Z ⊤ Y −1 Z < 0 if and

X Z ⊤

Z −Y

)

< 0.

Proposition 2.2 (Generalized Jensen inequality [25]) For a given symmetric matrix R > 0,

any differentiable function φ : [a, b] → R n , then the following inequality holds

∫ b

a

˙

φ ⊤ (u)R ˙ φ(u)du ≥ 1

b − a (φ(b) − φ(a)) ⊤ R(φ(b) − φ(a)) +

12

b − aΩ⊤ RΩ,

where Ω = φ(b) + φ(a)

b − a

∫ b

a

φ(u)du.

3 Main result

In this section, Lyapunov function approach is applied in order to design guaranteed cost controllers for the time-delay system (2.1) The following lemma is necessary for the proof

of main theorem

Lemma Let f (x) = −r BB ⊤ P x

1+∥B T P x ∥ , b = ||B|| Then (i) f (x) is global Lipschitz in R n

(ii)||f(x)|| ≤ 3rb2||P x||, ∀x ∈ R n

Proof Let x1, x2 ∈ R n and y1= B ⊤ P x1, y2 = B ⊤ P x2 We have

∥f(x1)− f(x2)∥ = r [ y2

1 +∥y2∥ −

y1

1 +∥y1∥

≤ rb y2

1 +∥y2∥ −

y1

1 +∥y1∥

≤ rb ∥y2− y1∥ + ∥y1∥y2− ∥y2∥y1

(1 +∥y1∥)(1 + ∥y2∥) .

Since

y2∥y1∥ − y1∥y2∥ = y2(∥y1∥ − ∥y2∥) + ∥y2∥(y2− y1)

≤ ∥y2∥(∥y1− y2∥) + ∥y2∥(∥y1− y2∥)

= 2∥y2∥(∥y1− y2∥).

and

∥y1− y2∥

(1 +∥y1∥)(1 + ∥y2∥) ≤ ∥y1− y2∥, ∥y2∥

(1 +∥y1∥)(1 + ∥y2∥) ≤ 1,

we have

||f(x1)− f(x2)|| ≤ 3rb||y1− y2|| ≤ 3rpb2||x1− x2||, (3.1)

where p = ||P ||.

(ii) From (3.1) , we have

||f(x1)− f(x2)|| ≤ 3rb||y1− y2|| = 3rb||B ⊤ P x1− B ⊤ P x2||.

Taking x2 = 0, we have

||f(x)|| ≤ 3rb||B ⊤ P x || ≤ 3rb2||P x||, ∀x ∈ R n

Theorem 3.1 If there exist symmetric positive definite matrices P, S i , i = 1, 2, 3, X1, X2, a positive number η > 0 satisfying the following conditions

α2c1+ 2βd

then

1 +||B ⊤ P x(t) || B ⊤ P x(t), t ≥ 0, (3.4)

is a guaranteed cost controller for the system (2.1) and the guaranteed cost value is given by

J ∗ = α

3∥φ∥2+ 2βd, where

¯

P = R −1/2 P R −1/2 , ¯ S

i = R −1/2 S

i R −1/2 , i = 1, 2, 3, γ = r2λmax(Q3), α1 = λmin( ¯P ),

β = λmax(B ⊤

1P B1) + 2λmax(B ⊤

1S4B1), ∥φ∥ = sup

−h2≤s≤0

{

φ ⊤ (s)R(s)φ(s), ˙ φ ⊤ (s)R ˙ φ(s)}

,

α2= λmax( ¯P ) +

2

∑

i=1

h i λmax( ¯S i ) + 0.5.(h2− h1)2(h2+ h1)λmax( ¯S3) +

2

∑

i=1

0.5h3i λmax( ¯X i ),

α3 = λmax(P ) +

2

∑

i=1

h i λmax(S i ) + 0.5.(h2− h1)2(h2+ h1)λmax(S3) +

2

∑

i=1

0.5h3i λmax(X i ),

M11= A T P + P A − 4

2

∑

i=1

X i+

2

∑

i=1

S i + P + Q1, M22=−S1− 4S3− 4X1,

M33=−S2− 4S3− 4X2, M44=−8S3, M55= (h2− h1)2S3+ h21X1+ h22X2− 2S4,

M66=−12X1, M77=−12X2, M88= M99=−12S3, M 10,10=− 1

1 + 27r2b4I,

M 11,11=−1

γ , M 12,12=−0.5I, M12=−2X1, M13=−2X2, M14= P D, M15= A ⊤ S

4,

M16= 6X1, M17= 6X2, M18= M19= 0, M 1,10 = P, M 1,11 = P B, M 1,12= 0

M23= 0, M24=−2S3, M25= 0, M26= 6X1, M27= 0, M28= 6S3,

M29= M 2,10 = M 2,11 = M 2,12 = 0, M34=−2S3, M35= M36= 0, M37= 6X2,

M38= 0, M39= 6S3, M 3,10 = M 3,11 = M 3,12 = 0, M45= D T S4, M46= M47= 0,

M48= M49= 6S3, M 4,10 = M 4,11 = M 4,12 = 0,

M56= M57= M58= M59= M 5,10 = M 5,11 = 0, M 5,12 = S4,

M67= M68= M69= M 6,10 = M 6,11 = M 6,12 = 0,

M78= M79= M 7,10 = M 7,11 = M 7,12 = 0, M89= M 8,10 = M 8,11 = M 8,12 = 0,

M 9,10 = M 9,11 = M 9,12 = M 10,11 = M 10,12 = 0, M 11,12 = 0.

Proof Let us consider the bounded feedback control (3.4) By Lemma (i), the function

f (x) = −r BB T P x

1+∥B T P x ∥ is global Lipschitz, hence, the closed-loop system

˙

x(t) = Ax(t) + A1x(t − h(t)) + B1w(t) + f (x(t)), t ∈ R+, (3.5) has an unique solution Consider the following Lyapunov-Krasovskii functionals for system (3.5):

V (t, x t) =

4

∑

i=1

V i (t, x t ),

where

V1(t, x t ) =e ηt x ⊤ (t)P x(t), V

2(t, x t) =

2

∑

i=1

e ηt

∫ t

t −h i

x ⊤ (s)S

i x(s)ds,

V3(t, x t ) =(h2− h1)e ηt

∫ −h1

−h2

∫ t

t+s

˙

x ⊤ (τ )S

3x(τ )dτ ds˙

V4(t, x t) =

2

∑

i=1

h i e ηt

∫ 0

−h i

∫ t

t+s

˙

x(τ ) ⊤ X

i x(τ )dτ ds,˙

We prove that

α1x ⊤ (t)Rx(t) ≤ V (t, x t ), ∀t : 0 ≤ t ≤ T. (3.6) Since

V1(.) = e ηt x ⊤ P x = e ηt x ⊤ R 1/2 R −1/2 P R −1/2 R 1/2 x = e ηt x T R 1/2 P R¯ 1/2 x,

where ¯P = R −1/2 P R −1/2 , we have

V1(.) = e ηt x ⊤ R 1/2 P R¯ 1/2 x ≥ x ⊤ R 1/2 P R¯ 1/2 x ≥ λmin( ¯P )x ⊤ Rx,

and α1 = λmin( ¯P ) Similarly, we can verify the following estimations

V (0, x0)≤ α2 sup

−h2≤s≤0

{

φ ⊤ (s)R(s)φ(s), ˙ φ ⊤ (s)R ˙ φ(s)}

≤ α2c1, (3.7)

Taking the derivative of V1(.) we have

˙

V1 = ηe ηt 2y ⊤ (t)P x(t) + 2e ηt x ⊤ (t)P ˙ x(t)

= e ηt

[

y T (t)[A ⊤ P + P A]x(t) + 2x ⊤ (t)P Dx(t − h(t)) + 2x ⊤ (t)P f (x(t)) + 2x T (t)P B1w(t)

]

+ ηV1(.)

˙

V2 = e ηt

[

x ⊤ (t)(S

1+ S2)x(t) −

2

∑

i=1

x ⊤ (t − h i )S i x(t − h i)

]

+ ηV2(.)

˙

V3 = e ηt

[

(h2− h1)2x˙⊤(t)S3x(t)˙ − (h2− h1)

∫ t −h1

t −h2

˙

x ⊤ (s)S

3x(s)ds˙

]

+ ηV3(.)

= e ηt

[

(h2− h1)2x˙⊤ (t)S

3x(t)˙ − (h2− h1)

∫ t −h(t)

t −h2

˙

x ⊤ (s)S

3x(s)ds˙

− (h2− h1)

∫ t −h1

t −h(t) x˙

⊤ (s)S

3x(s)ds˙

]

+ ηV3(.)

Applying Proposition 2.2 gives

−(h2− h1)

∫ t −h(t)

t −h2

˙

x ⊤ (s)S

3x(s)ds˙ ≤ −(h2− h(t))

∫ t −h(t)

t −h2

˙

x ⊤ (s)S

3x(s)ds˙

≤ − (x(t − h(t)) − x(t − h2))⊤ S

3(x(t − h(t) − x(t − h2)))

− 12

(

x(t − h(t)) + x(t − h2)

h2− h(t)

∫ t −h(t)

t −h2

x(s)ds

)⊤

S3

(

x(t − h(t)) + x(t − h2)

h2− h(t)

∫ t −h(t)

t −h2

x(s)ds

)

≤ − 4x T (t − h(t))S3x(t − h(t)) − 4x ⊤ (t − h2)S3x(t − h2)− 4x ⊤ (t − h(t))Ux(t − h2) + 12

h2− h(t) x ⊤ (t − h(t))S3

∫ t −h(t)

t −h2

x(s)ds + 12

h2− h(t) x ⊤ (t − h2)S3

∫ t −h(t)

t −h2

x(s)ds

(h2− h(t))2

∫ t −h(t)

t −h2

x ⊤ (s)dsS

3

∫ t −h(t)

t −h2

x(s)ds.

and similarly we have

−(h2− h1)

∫ t −h1

t −h(t) x˙

T (s)S3x(s)ds˙ ≤ −4x ⊤ (t − h(t))Ux(t − h(t)) − 4y ⊤ (t − h1)S3x(t − h1)

− 4x ⊤ (t − h(t))Ux(t − h1) + 12

h(t) − h1

x ⊤ (t − h1)S3

∫ t −h1

t −h(t) x(s)ds

+ 12

h(t) − h1

x ⊤ (t − h(t))S3

∫ t −h1

t −h(t) x(s)ds − 12

(h(t) − h1)2

∫ t −h1

t −h(t) x

⊤ (s)dsS

3

∫ t −h1

t −h(t) x(s)ds.

Then, we have

˙

V3≤ e ηt[

(h2− h1)2x˙⊤ (t)S

3x(t)˙ − 8y ⊤ (t − h(t))S3x(t − h(t)) − 4x ⊤ (t − h2)S3x(t − h2)

− 4x ⊤ (t − h1)S3x(t − h1)− 4y ⊤ (t − h(t))Ux(t − h2)

− 4x ⊤ (t − h(t))S3x(t − h1)− 12

(h2− h(t))2

∫ t −h(t)

t −h2

x ⊤ (s)dsS

3

∫ t −h(t)

t −h2

x(s)ds

(h(t) − h1)2

∫ t −h1

t −h(t) x

T (s)dsS3

∫ t −h1

t −h(t) x(s)ds + ηV3(.)

+ 12

h2− h(t) x T (t − h(t))S3

∫ t −h(t)

t −h2

x(s)ds + 12

h2− h(t) x ⊤ (t − h2)S3

∫ t −h(t)

t −h2

y(s)ds

+ 12

h(t) − h1

x ⊤ (t − h1)S3

∫ t −h1

t −h(t) x(s)ds +

12

h(t) − h1

x ⊤ (t − h(t))S3

∫ t −h1

t −h(t) x(s)ds

]

.

Using the same calculation as in ˙V3(t, x t ), we get

˙

V4(t, x t)≤ηV4+ e ηt

(

˙

x(t) ⊤ (h2

1X1+ h22X2) ˙x(t) + x(t) ⊤(−4X1− 4X2)x(t)

− 4x(t − h1)⊤ X

1x(t − h1)− 4x(t − h2)⊤ X

2x(t − h2)− 4x(t) ⊤ X

1x(t − h1)

− 4x(t) ⊤ X

2x(t − h2) +

2

∑

i=1

12

h i

x(t) ⊤ X

i

∫ t

t −h i x(s)ds

+

2

∑

i=1

12

h i

x(t − h i)⊤ X

i

∫ t

t −h i x(s)ds −

2

∑

i=1

12

h2

i

∫ t

t −h i x(s) ⊤ dsX

i

∫ t

t −h i x(s)ds

)

.

Thus, we obtain that

˙

V (.) ≤e ηt[

x ⊤ (t)[A T P + P A − 4(X1+ X2)]x(t) + 2x ⊤ (t)P Dx(t − h(t)) + 2x ⊤ (t)P f (x(t)) + 2x ⊤ (t)P B

1w(t) + x ⊤ (t)(S

1+ S2)x(t) −

2

∑

i=1

x ⊤ (t − h i )S i x(t − h i) + ˙x ⊤ (t)[(h

2− h1)2S3+ h21X1+ h22X2)] ˙x(t) − 8x ⊤ (t − h(t))S3x(t − h(t))

−

2

∑

i=1

4x ⊤ (t − h i )(S3+ X i )x(t − h i)−

2

∑

i=1

4x ⊤ (t − h(t))S3y(t − h i)

−

2

∑

i=1

4x ⊤ (t)X

i x(t − h i)− 12

(h2− h(t))2

∫ t −h(t)

t −h2

x ⊤ (s)dsS

3

∫ t −h(t)

t −h2

x(s)ds

+

2

∑

i=1

12

h i

x(t) ⊤ X

i

∫ t

t −h i

x(s)ds − 12

(h(t) − h1)2

∫ t −h1

t −h(t) x

T (s)dsS3

∫ t −h1

t −h(t) x(s)ds

+

2

∑

i=1

12

h i

x(t − h i)⊤ X

i

∫ t

t −h i x(s)ds −

2

∑

i=1

12

h2

i

∫ t

t −h i

x(s) ⊤ dsX

i

∫ t

t −h i x(s)ds

)

+ 12

h2− h(t) x ⊤ (t − h(t))S3

∫ t −h(t)

t −h2

x(s)ds + 12

h2− h(t) y ⊤ (t − h2)S3

∫ t −h(t)

t −h2

x(s)ds

+ 12

h(t) − h1

x ⊤ (t − h1)S3

∫ t −h1

t −h(t) x(s)ds +

12

h(t) − h1

x ⊤ (t − h(t))S3

∫ t −h1

t −h(t) x(s)ds

]

+ ηV (t, x t)

(3.9)

Multiplying both sides of equation (3.5) with 4e ηt x˙⊤ (t)S

4, we obtain

e ηt[−4 ˙x ⊤ (t)S

4x(t) + 4 ˙˙ x ⊤ (t)S

4Ax(t) + 4 ˙ x ⊤ (t)S

4Dx(t − h(t))

+ 4 ˙x ⊤ (t)S

4B1w(t) + 4 ˙ x ⊤ (t)S

4f (x(t))] = 0. (3.10) Adding all the zero items of (3.10) and zero term e ηt f0(t, x(t), x(t −h(t)), u(t))− e ηt f0(t, x(t), x(t − h(t)), u(t)) = 0 into (3.9), respectively and using Lemma (ii) and the Cauchy matrix

inequality for the following estimations

4 ˙x ⊤ (t)S

4B1w(t) ≤ 2 ˙x ⊤ (t)S4x(t) + 2w˙ ⊤ (t)B T

1S4B1w(t);

4 ˙x ⊤ (t)S

4f (x(t)) ≤ 2 ˙x ⊤ (t)S

4S4x(t) + 2˙ ∥f(x(t))∥2

≤ 2 ˙x ⊤ (t)S

4S4x(t) + 18r˙ 2b4x ⊤ (t)P P x(t);

2x ⊤ (t)P f (x(t)) ≤ x ⊤ (t)P P x(t) + ∥f(x(t))∥2 ≤ x ⊤ (t)P P x(t) + 9r2b4x ⊤ (t)P P x(t);

2x T (t)P B1w(t) ≤ w ⊤ (t)B ⊤

1 P B1w(t) + x ⊤ (t)P x(t),

u ⊤ Q

3u(t) ≤ r2λmax(Q3)x ⊤ P BB T P x(t)

we obtain

˙

V (.) ≤ηV (t, x t ) + e ηt ζ ⊤ (t) Vζ(t) − e ηt f0(t, x(t), x(t − h(t)), u(t)) + e ηt βw ⊤ (t)w(t) (3.11)

where β = λmax(B ⊤

1P B1) + 2λmax(B ⊤

1S4B1) and

ζ(t) = [x(t), x(t − h1), x(t − h2), x(t − h(t)), ˙x(t), 1

h1

∫t

t −h1x ⊤ (s)ds, 1

h2

∫t

t −h2x ⊤ (s)ds

1

(h(t) −h1 )

∫t −h1

t −h(t) x(s) ⊤ ds, (h2−h(t))1

∫t −h(t)

t −h2 x(s) ⊤ ds],

V = (N ij)9×9 , γ = r2λmax(Q3),

N11= A ⊤ P + P A − 4

2

∑

i=1

X i+

2

∑

i=1

S i + P + γP BB ⊤ P + (1 + 27r2b4)P P + Q1,

N22=−S1− 4S3− 4X1, N33=−S2− 4S3− 4X2, N44=−8S3,

N55= (h2− h1)2S3+ h21X1+ h22X2− 2S4+ 2S4S4,

N66=−12X1, N77=−12X2, N88= N99=−12S3, N12=−2X1,

N13=−2X2, N14= P D, N15= A ⊤ S

4, N16= 6X1, N17= 6X2, N18= N19= 0,

N23= 0, N24=−2S3, N25= 0, N26= 6X1, N27= 0, N28= 6S3, N29= 0,

N34=−2S3, N35= N36= 0, N37= 6X2, N38= 0, N39= 6S3,

N45= D T S4, N46= N47= 0, N48= N49= 6S3,

N56= N57= N58= N59= 0, N67= N68= N69= 0, N78= N79= 0, N89= 0.

By Proposition 2.2, the conditions V < 0 is equivalent to the condition (2.2) Therefore,

we obtain from (3.11) that

˙

V (t, x t ) < ηV (t, x t ) + 2e ηt βw ⊤ (t)w(t), ∀t ∈ [0, T ]. (3.12)

Multiplying both sides of (3.12) with e −ηt we have

e −ηt V (t, x˙ t)− ηe −ηt V (t, x

t ) < 2βw ⊤ (t)w(t), ∀t ∈ [0, T ]. (3.13)

Integrating both sides of (3.13) from 0 to t, we obtain

e −ηt V (t, x

t ) < V (0, x0) + 2β

∫ t

0

w ⊤ (s)w(s)ds, ∀t ∈ [0, T ].

Therefore, from (3.6), (3.7) it follows that

α1e −ηt x(t) ⊤ Rx(t) < e −ηt V (t, x

t)≤ α2c1+ 2βd,

and hence

x(t) T Rx(t) < α2c1+ 2βd

α1 e

ηt ≤ c2, ∀t ∈ [0, T ].

which implies that the closed-loop systems is robustly finite-time stable w.r.t (c1, c2, T, R).

To find the value of the cost function (2.4), we derive from (3.13) that

e −ηt V (t, x˙

t)− ηe −ηt V (t, x t)≤ −f0(t, x(t), x(t − h(t)), u(t)) + 2βw ⊤ (t)w(t), t ≥ 0 (3.14) Integrating both sides of (3.14) from 0 to T leads to

∫ T

0

f0(t, x(t), x(t − h(t)), u(t))dt ≤V (0, x0)− e −ηt V (t, x t ) + 2β

∫ T

0

w ⊤ (t)w(t)dt

≤V (0, x0) + 2βd, due to V (t, x t)≥ 0 Hence, from (3.8), (3.14) it follows that

J ≤ V (0, x0) + 2βd ≤ α3∥ϕ∥2+ 2βd = J ∗ .

This completes the proof of the theorem

Remark 3.1 We note that the condition (3.3) is not LMI with respect to η Since η does not

include in (3.2), we can first find the solutions P, S i , X i from LMI (3.2) and then determine

η from (3.3).

Example 3.1 Consider the system (2.1) where

A =

[

−2 −0.2 0.5 −3

]

, D =

[

−0.5 −0.5

−0.2 0.4

]

, B =

[

−0.2

−0.5

]

, B1 =

[

−1

−3

]

with

h(t) =

{

0.1 + 0.4 cos t if t ∈ I = ∪ k≥0 [2kπ, (2k + 1)π]

0.1 if t ∈ R+\ I.

Note that the functions h(t) is non-differentiable Given

R = I, Q1 =

[

0.02 0

0 0.01

]

, Q2 =

[

0.01 0

0 0.05

]

, Q3 =[

0.5]

.

By using the LMI Toolbox in MATLAB [22], the conditions (3.2) and (3.3) are satisfied with

h1 = 0.1, h2 = 0.5, r = 0.4, r = 0.4, d = 0.01, c1= 1, c2 = 2.5, T = 10 and η = 0.01,

P =

[

1.1372 −0.0259

−0.0259 1.5739

]

, S1=

[

0.4505 −0.0518

−0.0518 0.8791

]

, S2 =

[

0.4013 −0.0512

−0.0512 0.9069

]

S3=

[

0.5447 −0.0097

−0.0097 0.5691

]

, S4 =

[

0.2029 −0.0011

−0.0011 0.1913

]

, X1=

[

1.6168 0.0173 0.0173 1.4009

]

,

X2 =

[

0.2733 −0.0054

−0.0054 0.2335

]

Thus the system is robustly finite-time stable w.r.t (1, 2.5, 10, I) by feedback controller

u(t) = 0.4(0.2145x1+ 0.7817x2)

1 + (0.2145x1+ 0.7817x2). The guaranteed cost valued of the closed-loop system is as follows:

J ∗ = 2.1654 ∥ϕ∥2+ 0.3797.