Preservation of exponential stability for linear non ... - BGU Math
Automatica 46 (2010) 2077–2081
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Preservation of exponential stability for linear non-autonomous functional differential systems✩ Leonid Berezansky a , Elena Braverman b,∗ a
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
b
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB T2N 1N4, Canada
article
info
Article history: Received 24 September 2009 Received in revised form 11 May 2010 Accepted 27 July 2010 Available online 20 October 2010
abstract We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Systems with time delays Distributed delay Exponential stability Perturbations
1. Introduction Equations with a distributed delay provide a more realistic description for real world delay models in mathematical biology, mechanical engineering and other applications (Kuang, 1993). Here we consider a type of delay which involves, as special cases, equations with concentrated delays and integrodifferential equations. To the best of our knowledge the first systematic study of equations with a distributed delay can be found in the monograph of Myshkis (1972), the results obtained by 1993 are summarized in the book by Kuang (1993). In most publications on distributed delays (see, for example, Bernard, Bélair, & Mackey, 2001) integrodifferential equations are studied, however sometimes models incorporate both integral terms and concentrated delays; such equations are considered in the present paper. Here we study the following question: assuming that the original equation is exponentially stable, when can we state that the perturbed equation (generally, the perturbation has a distributed delay) preserves this property? As corollaries, we obtain some known results (see, for example, Fu, Olbrot, & Polis, 1989;
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Weiss under the direction of Editor Miroslav Krstic. L. Berezansky was partially supported by the Israeli Ministry of Absorption. E. Braverman was partially supported by NSERC. ∗ Corresponding author. E-mail addresses: [email protected] (L. Berezansky), [email protected] (E. Braverman).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.09.007
Halanay, 1966; Louisell, 1992; Stokes, 1974; Su & Huang, 1992; Tsypkin & Fu, 1993) for equations with concentrated delays x˙ (t ) +
m −
Ak (t )x(hk (t )) = 0,
(1)
k=1
where the equation is perturbed by adding new delay terms: x˙ (t ) +
m −
Ak (t )x(hk (t )) +
k=1
l −
Bk (t )x(gk (t )) = 0.
(2)
k=1
Delay perturbations were studied in Driver (1977): the equation x˙ (t ) = A0 (t )x(t ) +
m −
Ak (t )x(t − τk )
(3)
k=1
was considered as a perturbation of the equation x˙ (t ) =
m −
Ak (t )x(t )
(4)
k=0
without delays. Suppose that the fundamental matrix of (4) has exponential estimation ‖X0 (t , s)‖ ≤ K e−λ(t −s) for K > 0, λ > 0. Then (Driver, 1977) Eq. (3) is asymptotically stable as far as
max τk k
sup
m −
t ≥0 j=1
‖Ak (t )‖
m −
sup ‖Ak (t )‖
k=0 t ≥0
0, where the Lebesgue–Stiltjes matrix measure produced by ds R(t , s) is multiplied by the column vector x(·) for any t (x is not under the sign of the differential). Thus, α(t ) defined in (9) has the form
α(t ) =
t
∫
h(t )
‖ds R(t , s)‖ =
m −
‖Ak (t )‖
k=1
for (1) at points t where hk (t ) ̸= hj (t ), k ̸= j, and obviously satisfies
α(t ) ≤
t
∫
h(t )
‖K (t , s)‖ ds
for the integro-differential equation x˙ (t ) +
t
∫
h(t )
K (t , s)x(s)ds = 0,
t ≥ t0 .
(10)
Definition. An absolutely continuous on [t0 , ∞) function x : R → Rn will be called a solution of the problem (7), (8) if it satisfies Eq. (7) for almost all t ∈ [t0 , ∞) and conditions (8) for t ≤ t0 . In addition to (7) we consider the non-homogeneous equation t
∫
2. Preliminaries
x˙ (t ) +
We consider a system of linear differential equations with a distributed delay
where f (t ) is a Lebesgue measurable locally essentially bounded vector function.
x˙ (t ) +
t
∫
h(t )
ds R(t , s) x(s) +
t
∫
g (t )
ds Q (t , s) x(s) = 0
(6)
for t ≥ t0 ≥ 0, which will be treated as a perturbation of the equation with one integral term x˙ (t ) +
h(t )
ds R(t , s) x(s) = 0,
t ≥ t0 .
(7)
Here x(t ) is a column n-vector, R and Q are n × n matrix functions, ‖x‖ is the Euclidean norm of the vector x ∈ Rn ,
‖A‖ = sup ‖Ax‖. ‖x‖=1
We consider Eqs. (6) and (7) for a fixed t0 ≥ 0 with the initial condition x(t ) = ϕ(t ),
t ≤ t0
R(t , h(t )) = Q (t , g (t )) = 0, R(t , s), Q (t , s) are constant for s > t and coincide with the right limits R(t , t + ), Q (t , t + ); the functions
α(t ) =
t h(t )
‖ds R(t , s)‖,
β(t ) =
∫
t g (t )
‖ds Q (t , s)‖
(9)
are Lebesgue measurable and bounded on [0, ∞); (a2) h, g : [0, ∞) → R are Lebesgue measurable functions, h(t ) ≤ t , g (t ) ≤ t, lim supt →∞ [t −h(t )] < ∞, lim supt →∞ [t −g (t )] < ∞. t The Lebesgue–Stiltjes integral h(t ) ds R(t , s) x(s) is understood as
∫
t+
h(t )
[ds R(t , s)] x(s), i.e.,
t h(t )
ds R(t , s) x(s) =
∫
t +ε
h(t )
[ds R(t , s)] x(s)
(11)
Definition. For each s ≥ t0 and t ≥ s the solution X (t , s) of the problem x˙ (t ) +
t
∫
h(t )
dτ R(t , τ ) x(τ ) = 0, t < s,
t ≥ s,
x(s) = I ,
(12)
where I is the identity matrix, is called the fundamental matrix of Eq. (7). Here X (t , s) = 0, 0 ≤ t < s. Lemma 1 (Azbelev, Berezansky, & Rahmatullina, 1977; Hale & Verduyn Lunel, 1993). The solution of the initial value problem (11), (8) has the following representation x(t ) = X (t , t0 )ϕ(t0 ) −
∫
t
X ( t , s)
[∫
t0
(8)
under the following assumptions: (a1) R(t , ·) and Q (t , ·) are left continuous matrix functions of bounded variation for any t; R(·, s) and Q (·, s) are locally integrable for any s,
∫
ds R(t , s)x(s) = f (t ),
x( t ) = 0 ,
t
∫
h(t )
∫
s h(s)
]
dτ R(s, τ ) ϕ(τ ) ds
t
X (t , s)f (s)ds,
+
where ϕ(t ) = 0, t > t0 .
(13)
t0
Definition. Eq. (7) is (uniformly) exponentially stable, if there exist K > 0, λ > 0, such that the fundamental matrix X (t , s) defined by (12) has the estimate
‖X (t , s)‖ ≤ K e−λ(t −s) ,
t ≥ s ≥ 0.
(14)
Let us introduce some functional spaces on a halfline. Denote by L∞ [t0 , ∞) the space of all essentially bounded functions y : [t0 , ∞) → Rn with the essential supremum norm ‖y‖L∞ = ess supt ≥t0 ‖y(t )‖, by C[t0 , ∞) the space of all continuous bounded functions on [t0 , ∞) with the sup-norm. We will use the following Bohl–Perron type result presented in Azbelev and Simonov (2003) and Halanay (1966). Lemma 2. Suppose that for any f ∈ L∞ [t0 , ∞) the solution of (11) with the zero initial conditions belongs to C[t0 , ∞). Then Eq. (7) is exponentially stable.
L. Berezansky, E. Braverman / Automatica 46 (2010) 2077–2081
3. Perturbation by adding new terms
≤ K e−λt sup
t ≥ t0
K e−λ(t −s)
s
∫
g (s)
t0
] ‖dτ Q (s, τ )‖ ds ≤ µ.
(15)
Then Eq. (6) is also exponentially stable. Proof. Let us demonstrate that the absolutely continuous solution of the non-homogeneous equation x˙ (t ) +
t
∫
h(t )
ds R(t , s) x(s) +
∫
t
ds Q (t , s) x(s) = f (t ),
g (t )
(16)
x(t ) +
t
X (t , s)
s
∫
g (s)
t0
∫
dτ Q (s, τ ) x(τ )ds =
By Theorem 1 Eq. (6) is exponentially stable.
t ≥ t0
t0
(1) lim supt →∞
(3)
s
]
‖dτ Q (s, τ )‖ ds < q0
(17)
q0 =
−1
K e2λh
∫
t
.
x˙ (t ) +
e−λ(t −s) ‖p(s)‖ds [t ]+1
≤K
−λ(t −s)
e
‖p(s)‖ ds
[t0 ]−1
= K e−λt
[∫
m −
[ t0 ]
eλs ‖p(s)‖ ds + · · · +
[t0 ]−1
≤ K e−λt sup t ≥ t0
‖p(s)‖ ds t
∫
[t ]+1
]
eλs ‖p(s)‖ ds
[t ] t +1
∫
eλ[t0 ] + · · · + eλ([t ]+1)
Ak (t )x(gk (t )) = 0.
(18)
k=1
Theorem 2. Suppose that Eq. (1) is exponentially stable, its fundamental matrix has the exponential estimate (14), and in addition there exist t0 ≥ 0 and µ ∈ (0, 1) such that for any t ≥ t0 t
−λ(t −s)
∫ m gk (s) − ‖Ak (s)‖ ‖Ak (τ )‖dτ ds ≤ µ. h ( s ) k k=1 k=1
m −
Proof. Similarly to the proof of Theorem 1, we demonstrate that the solution of the equation x˙ (t ) +
m −
Ak (t )x(gk (t )) = f (t ),
t ≥ t0 ,
(19)
k=1
with the zero initial conditions is bounded on [t0 , ∞) for any f ∈ L∞ [t0 , ∞). Eq. (19) can be rewritten as x˙ (t ) +
m −
Ak (t )x(hk (t )) +
k=1
x˙ (t ) +
m −
m −
Ak (t )
k=1
∫
gk (t ) hk (t )
x˙ (s) ds = f (t ),
Ak (t )x(hk (t ))
k=1 m − k=1
t0
∫
‖dτ Q (s, τ )‖ ds = 0 for some h > 0;
4. Delays perturbation
−
For simpler notations, we assume h = 1, the proof for an arbitrary h s is similar. Denote p(s) = g (s) ‖dτ Q (s, τ )‖. We have for t ≥ t0 ≥ 0 K
g (s)
hence after substituting x˙ from (19) we have
Proof. Suppose that for the Eq. (7) the fundamental matrix has the exponential estimate (14). Let us prove that the statement of the corollary holds for e
t t g (t )
Then Eq. (18) is also exponentially stable.
implies that Eq. (6) is exponentially stable.
λh
0
Ke
Corollary 1. Suppose that Eq. (7) is exponentially stable. Then for every h > 0 there exists q0 > 0 such that the inequality
g (s)
t +h s
∞ s
t0
thus the norm of H in L∞ [t0 , ∞) does not exceed µ < 1, consequently, the inverse (I + H )−1 = I − H + H 2 − H 3 + H 4 · · · is a bounded operator and the function x = (I + H )−1 r is bounded. By Lemma 2 Eq. (6) is exponentially stable.
t
< 1.
‖ds Q (t , s)‖ = 0. ‖dτ Q (s, τ )‖ ds < ∞. g (s)
(2) lim supt →∞
∫
g (s)
t0
t →∞
e
eλ − 1
Consider the following equation as a perturbation of Eq. (1)
∫ t [ ] ∫ s X (t , s) dτ Q (s, τ ) x(τ ) ds ‖(Hx)(t )‖ = t0 g (s) ∫ t ∫ s ≤ K e−λ(t −s) ds ‖dτ Q (s, τ )‖ ‖x‖L∞ [t0 ,∞)
lim sup
t
2λ
−1
Corollary 2. Suppose that (7) is exponentially stable and at least one of the following conditions holds:
where
[∫
‖p(s)‖ ds
≤ K sup
X (t , s)f (s) ds,
x + Hx = r ,
t +h
e
λ([t ]−[t0 ]+2)
t
t +1
∫
t
where X (t , s) is the fundamental matrix of Eq. (7) which has the t estimate (14), thus the right hand side is r (t ) := t X (t , s)f (s) ds ∈ 0 L∞ [t0 , ∞). Eq. (16) has the form
∫
Then Eq. (6) is also exponentially stable.
for t ≥ t0 with the zero initial conditions is bounded on [t0 , ∞) for any f ∈ L∞ [t0 , ∞). This solution also satisfies the following integral equation
∫
e
λ[t0 ]
−1
t ≥ t0
∫ t[
‖p(s)‖ ds
eλ
Theorem 1. Suppose that (7) is exponentially stable, its fundamental matrix satisfies the exponential estimate (14) and there exist t0 ≥ 0 and µ ∈ (0, 1) such that sup
2079
t +1
∫
Ak (t )
∫
gk (t ) hk (t )
m −
Ak (s)x(gk (s))ds = F (t ),
k=1
g
(t )
k where F (t ) = f (t ) − k=1 Ak (t ) hk (t ) f (s) ds. Evidently F ∈ L∞ [t0 , ∞). Hence a solution of Eq. (19) is also a solution of the t operator equation x + Hx = r, where r (t ) := t X (t , s)F (s) ds ∈ 0 L∞ [t0 , ∞) and
∑m
‖(Hx)(t )‖ ∫ ∫ gk (s) − m m t − = X (t , s) Ak (s) Ak (τ )x(gk (τ )) dτ ds t0 h ( s ) k k =1 k=1 ∫ ∫ t m m gk (s) − − K e−λ(t −s) ‖Ak (s)‖ ‖Ak (τ )‖dτ ds ≤ t0 h ( s ) k k=1 k=1 × ‖x‖L∞ [t0 ,∞) ≤ µ‖x‖L∞ [t0 ,∞) .
2080
L. Berezansky, E. Braverman / Automatica 46 (2010) 2077–2081
Thus the norm of H in L∞ [t0 , ∞) does not exceed µ < 1. Consequently, the inverse (I + H )−1 = I − H + H 2 − H 3 + H 4 · · · is a bounded operator and the function x = (I + H )−1 r is bounded. By Lemma 2 Eq. (18) is exponentially stable.
x˙ (t ) + x(t ) + α(1 + cos t )x(t − τ (t )) = 0.
Next, consider the equation with a distributed delay t
∫
x˙ (t ) +
g (t )
x(s)ds R(t , s) = 0,
t ≥ t0 ,
Theorem 1 yields that (22) is exponentially stable if for some µ ∈ ∑ (0, 1), ak (t ) ≥ 0, k = 1, . . . , m, m k=1 ak (t ) < µ < 1, which is a delay-independent stability condition similar to Zevin and Pinsky (2006). In particular, for the equation
(20)
Theorem 1 implies exponential stability of (24) for any α < 2/(2 + √ 2) ≈ 0.8284 since
∫
t
as a perturbation of Eq. (7). Denote H (t ) = min{h(t ), g (t )},
t
sup t ≥ t0
K e−λ(t −s)
G(t )
∫
H (t )
t0
G(s)
lim sup t →∞
H (s)
t
]
‖dτ R(s, τ )‖ ds ≤ q0
implies exponential stability of Eq. (20). Corollary 4. Suppose that Eq. (7) is exponentially stable, and at least one of the following conditions holds: (1) lim supt →∞
t +δ G(s) H (s)
t
dτ ‖R(s, τ )‖ ds = 0 for some δ > 0;
2
2
Hence Theorem 1 implies that for α < 0.8284, while (23) yields
We also mention that the equation considered in the present paper is more general than most of the previously considered models. Finally, let us formulate some open problems. (1) Formulate sufficient conditions for (6) which for the case (2) generalizes inequality (5). (2) Most results of the present paper are based on the explicit estimates for the fundamental functions of equations with a distributed delay. Develop techniques to obtain such estimates. (3) Based on the results of the present paper, deduce explicit exponential stability results for Eq. (7), as well as for its particular cases (1) and (10).
∞ G(t )
‖ d R ( s , τ )‖ ds < ∞; τ 0 H (t ) G(t ) (3) lim supt →∞ H (t ) ‖ds R(t , s)‖ = 0. (2)
2
t ≥ t0
Corollary 3. Suppose that Eq. (7) is exponentially stable. For every δ > 0 there exists q0 > 0 such that the inequality
[∫
e−(t −s) α(1 + cos s) ds
0
thus Theorem 2 outperforms Theorem 1 (and thus Zevin & Pinsky, 2006) for √small τ , where τ (t ) ≤ τ , but is obviously worse than the 2/(2 + 2) bound for τ > 0.23.
The proof is similar to the proof of Theorem 2.
t +δ
t
α(1 + 2α) sup[τ (t )(1 + cos(t ))] < 1,
‖dτ R(s, τ )‖ ds ≤ µ.
Then Eq. (20) is also exponentially stable.
∫
∫
√ ] [ 1 α(2 + 2) cos t + sin t − e− t ≤ = µ < 1. =α 1+
Theorem 3. Suppose that (7) is exponentially stable, its fundamental matrix satisfies (14) and there exist t0 ≥ 0 and µ ∈ (0, 1) such that
∫
e−(t −s) α(1 + cos s) ds ≤
t0
G(t ) = max{h(t ), g (t )}.
(24)
6. Conclusion
Then Eq. (20) is exponentially stable. 5. Discussion, example and open problems For equations with a distributed delay, exponential stability of (7) and the condition lim supt →∞ |h(t ) − g (t )| = 0 in general do not imply exponential stability of Eq. (20). A perturbation of the value of a pointwise delay may be large even for small perturbations of the lower bound. Theorem 2 improves and extends stability condition (5) obtained in Driver (1977) for Eq. (3) with constant delays. Indeed, as a corollary of Theorem 2 the following inequality implies asymptotic stability of (3) with variable delays τk (t ):
sup
m −
‖Ak (t )‖τk (t )
m −
t ≥0 k=1
sup ‖Ak (t )‖
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Preservation of exponential stability for linear non-autonomous functional differential systems✩ Leonid Berezansky a , Elena Braverman b,∗ a
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
b
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB T2N 1N4, Canada
article
info
Article history: Received 24 September 2009 Received in revised form 11 May 2010 Accepted 27 July 2010 Available online 20 October 2010
abstract We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Systems with time delays Distributed delay Exponential stability Perturbations
1. Introduction Equations with a distributed delay provide a more realistic description for real world delay models in mathematical biology, mechanical engineering and other applications (Kuang, 1993). Here we consider a type of delay which involves, as special cases, equations with concentrated delays and integrodifferential equations. To the best of our knowledge the first systematic study of equations with a distributed delay can be found in the monograph of Myshkis (1972), the results obtained by 1993 are summarized in the book by Kuang (1993). In most publications on distributed delays (see, for example, Bernard, Bélair, & Mackey, 2001) integrodifferential equations are studied, however sometimes models incorporate both integral terms and concentrated delays; such equations are considered in the present paper. Here we study the following question: assuming that the original equation is exponentially stable, when can we state that the perturbed equation (generally, the perturbation has a distributed delay) preserves this property? As corollaries, we obtain some known results (see, for example, Fu, Olbrot, & Polis, 1989;
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Weiss under the direction of Editor Miroslav Krstic. L. Berezansky was partially supported by the Israeli Ministry of Absorption. E. Braverman was partially supported by NSERC. ∗ Corresponding author. E-mail addresses: [email protected] (L. Berezansky), [email protected] (E. Braverman).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.09.007
Halanay, 1966; Louisell, 1992; Stokes, 1974; Su & Huang, 1992; Tsypkin & Fu, 1993) for equations with concentrated delays x˙ (t ) +
m −
Ak (t )x(hk (t )) = 0,
(1)
k=1
where the equation is perturbed by adding new delay terms: x˙ (t ) +
m −
Ak (t )x(hk (t )) +
k=1
l −
Bk (t )x(gk (t )) = 0.
(2)
k=1
Delay perturbations were studied in Driver (1977): the equation x˙ (t ) = A0 (t )x(t ) +
m −
Ak (t )x(t − τk )
(3)
k=1
was considered as a perturbation of the equation x˙ (t ) =
m −
Ak (t )x(t )
(4)
k=0
without delays. Suppose that the fundamental matrix of (4) has exponential estimation ‖X0 (t , s)‖ ≤ K e−λ(t −s) for K > 0, λ > 0. Then (Driver, 1977) Eq. (3) is asymptotically stable as far as
max τk k
sup
m −
t ≥0 j=1
‖Ak (t )‖
m −
sup ‖Ak (t )‖
k=0 t ≥0
0, where the Lebesgue–Stiltjes matrix measure produced by ds R(t , s) is multiplied by the column vector x(·) for any t (x is not under the sign of the differential). Thus, α(t ) defined in (9) has the form
α(t ) =
t
∫
h(t )
‖ds R(t , s)‖ =
m −
‖Ak (t )‖
k=1
for (1) at points t where hk (t ) ̸= hj (t ), k ̸= j, and obviously satisfies
α(t ) ≤
t
∫
h(t )
‖K (t , s)‖ ds
for the integro-differential equation x˙ (t ) +
t
∫
h(t )
K (t , s)x(s)ds = 0,
t ≥ t0 .
(10)
Definition. An absolutely continuous on [t0 , ∞) function x : R → Rn will be called a solution of the problem (7), (8) if it satisfies Eq. (7) for almost all t ∈ [t0 , ∞) and conditions (8) for t ≤ t0 . In addition to (7) we consider the non-homogeneous equation t
∫
2. Preliminaries
x˙ (t ) +
We consider a system of linear differential equations with a distributed delay
where f (t ) is a Lebesgue measurable locally essentially bounded vector function.
x˙ (t ) +
t
∫
h(t )
ds R(t , s) x(s) +
t
∫
g (t )
ds Q (t , s) x(s) = 0
(6)
for t ≥ t0 ≥ 0, which will be treated as a perturbation of the equation with one integral term x˙ (t ) +
h(t )
ds R(t , s) x(s) = 0,
t ≥ t0 .
(7)
Here x(t ) is a column n-vector, R and Q are n × n matrix functions, ‖x‖ is the Euclidean norm of the vector x ∈ Rn ,
‖A‖ = sup ‖Ax‖. ‖x‖=1
We consider Eqs. (6) and (7) for a fixed t0 ≥ 0 with the initial condition x(t ) = ϕ(t ),
t ≤ t0
R(t , h(t )) = Q (t , g (t )) = 0, R(t , s), Q (t , s) are constant for s > t and coincide with the right limits R(t , t + ), Q (t , t + ); the functions
α(t ) =
t h(t )
‖ds R(t , s)‖,
β(t ) =
∫
t g (t )
‖ds Q (t , s)‖
(9)
are Lebesgue measurable and bounded on [0, ∞); (a2) h, g : [0, ∞) → R are Lebesgue measurable functions, h(t ) ≤ t , g (t ) ≤ t, lim supt →∞ [t −h(t )] < ∞, lim supt →∞ [t −g (t )] < ∞. t The Lebesgue–Stiltjes integral h(t ) ds R(t , s) x(s) is understood as
∫
t+
h(t )
[ds R(t , s)] x(s), i.e.,
t h(t )
ds R(t , s) x(s) =
∫
t +ε
h(t )
[ds R(t , s)] x(s)
(11)
Definition. For each s ≥ t0 and t ≥ s the solution X (t , s) of the problem x˙ (t ) +
t
∫
h(t )
dτ R(t , τ ) x(τ ) = 0, t < s,
t ≥ s,
x(s) = I ,
(12)
where I is the identity matrix, is called the fundamental matrix of Eq. (7). Here X (t , s) = 0, 0 ≤ t < s. Lemma 1 (Azbelev, Berezansky, & Rahmatullina, 1977; Hale & Verduyn Lunel, 1993). The solution of the initial value problem (11), (8) has the following representation x(t ) = X (t , t0 )ϕ(t0 ) −
∫
t
X ( t , s)
[∫
t0
(8)
under the following assumptions: (a1) R(t , ·) and Q (t , ·) are left continuous matrix functions of bounded variation for any t; R(·, s) and Q (·, s) are locally integrable for any s,
∫
ds R(t , s)x(s) = f (t ),
x( t ) = 0 ,
t
∫
h(t )
∫
s h(s)
]
dτ R(s, τ ) ϕ(τ ) ds
t
X (t , s)f (s)ds,
+
where ϕ(t ) = 0, t > t0 .
(13)
t0
Definition. Eq. (7) is (uniformly) exponentially stable, if there exist K > 0, λ > 0, such that the fundamental matrix X (t , s) defined by (12) has the estimate
‖X (t , s)‖ ≤ K e−λ(t −s) ,
t ≥ s ≥ 0.
(14)
Let us introduce some functional spaces on a halfline. Denote by L∞ [t0 , ∞) the space of all essentially bounded functions y : [t0 , ∞) → Rn with the essential supremum norm ‖y‖L∞ = ess supt ≥t0 ‖y(t )‖, by C[t0 , ∞) the space of all continuous bounded functions on [t0 , ∞) with the sup-norm. We will use the following Bohl–Perron type result presented in Azbelev and Simonov (2003) and Halanay (1966). Lemma 2. Suppose that for any f ∈ L∞ [t0 , ∞) the solution of (11) with the zero initial conditions belongs to C[t0 , ∞). Then Eq. (7) is exponentially stable.
L. Berezansky, E. Braverman / Automatica 46 (2010) 2077–2081
3. Perturbation by adding new terms
≤ K e−λt sup
t ≥ t0
K e−λ(t −s)
s
∫
g (s)
t0
] ‖dτ Q (s, τ )‖ ds ≤ µ.
(15)
Then Eq. (6) is also exponentially stable. Proof. Let us demonstrate that the absolutely continuous solution of the non-homogeneous equation x˙ (t ) +
t
∫
h(t )
ds R(t , s) x(s) +
∫
t
ds Q (t , s) x(s) = f (t ),
g (t )
(16)
x(t ) +
t
X (t , s)
s
∫
g (s)
t0
∫
dτ Q (s, τ ) x(τ )ds =
By Theorem 1 Eq. (6) is exponentially stable.
t ≥ t0
t0
(1) lim supt →∞
(3)
s
]
‖dτ Q (s, τ )‖ ds < q0
(17)
q0 =
−1
K e2λh
∫
t
.
x˙ (t ) +
e−λ(t −s) ‖p(s)‖ds [t ]+1
≤K
−λ(t −s)
e
‖p(s)‖ ds
[t0 ]−1
= K e−λt
[∫
m −
[ t0 ]
eλs ‖p(s)‖ ds + · · · +
[t0 ]−1
≤ K e−λt sup t ≥ t0
‖p(s)‖ ds t
∫
[t ]+1
]
eλs ‖p(s)‖ ds
[t ] t +1
∫
eλ[t0 ] + · · · + eλ([t ]+1)
Ak (t )x(gk (t )) = 0.
(18)
k=1
Theorem 2. Suppose that Eq. (1) is exponentially stable, its fundamental matrix has the exponential estimate (14), and in addition there exist t0 ≥ 0 and µ ∈ (0, 1) such that for any t ≥ t0 t
−λ(t −s)
∫ m gk (s) − ‖Ak (s)‖ ‖Ak (τ )‖dτ ds ≤ µ. h ( s ) k k=1 k=1
m −
Proof. Similarly to the proof of Theorem 1, we demonstrate that the solution of the equation x˙ (t ) +
m −
Ak (t )x(gk (t )) = f (t ),
t ≥ t0 ,
(19)
k=1
with the zero initial conditions is bounded on [t0 , ∞) for any f ∈ L∞ [t0 , ∞). Eq. (19) can be rewritten as x˙ (t ) +
m −
Ak (t )x(hk (t )) +
k=1
x˙ (t ) +
m −
m −
Ak (t )
k=1
∫
gk (t ) hk (t )
x˙ (s) ds = f (t ),
Ak (t )x(hk (t ))
k=1 m − k=1
t0
∫
‖dτ Q (s, τ )‖ ds = 0 for some h > 0;
4. Delays perturbation
−
For simpler notations, we assume h = 1, the proof for an arbitrary h s is similar. Denote p(s) = g (s) ‖dτ Q (s, τ )‖. We have for t ≥ t0 ≥ 0 K
g (s)
hence after substituting x˙ from (19) we have
Proof. Suppose that for the Eq. (7) the fundamental matrix has the exponential estimate (14). Let us prove that the statement of the corollary holds for e
t t g (t )
Then Eq. (18) is also exponentially stable.
implies that Eq. (6) is exponentially stable.
λh
0
Ke
Corollary 1. Suppose that Eq. (7) is exponentially stable. Then for every h > 0 there exists q0 > 0 such that the inequality
g (s)
t +h s
∞ s
t0
thus the norm of H in L∞ [t0 , ∞) does not exceed µ < 1, consequently, the inverse (I + H )−1 = I − H + H 2 − H 3 + H 4 · · · is a bounded operator and the function x = (I + H )−1 r is bounded. By Lemma 2 Eq. (6) is exponentially stable.
t
< 1.
‖ds Q (t , s)‖ = 0. ‖dτ Q (s, τ )‖ ds < ∞. g (s)
(2) lim supt →∞
∫
g (s)
t0
t →∞
e
eλ − 1
Consider the following equation as a perturbation of Eq. (1)
∫ t [ ] ∫ s X (t , s) dτ Q (s, τ ) x(τ ) ds ‖(Hx)(t )‖ = t0 g (s) ∫ t ∫ s ≤ K e−λ(t −s) ds ‖dτ Q (s, τ )‖ ‖x‖L∞ [t0 ,∞)
lim sup
t
2λ
−1
Corollary 2. Suppose that (7) is exponentially stable and at least one of the following conditions holds:
where
[∫
‖p(s)‖ ds
≤ K sup
X (t , s)f (s) ds,
x + Hx = r ,
t +h
e
λ([t ]−[t0 ]+2)
t
t +1
∫
t
where X (t , s) is the fundamental matrix of Eq. (7) which has the t estimate (14), thus the right hand side is r (t ) := t X (t , s)f (s) ds ∈ 0 L∞ [t0 , ∞). Eq. (16) has the form
∫
Then Eq. (6) is also exponentially stable.
for t ≥ t0 with the zero initial conditions is bounded on [t0 , ∞) for any f ∈ L∞ [t0 , ∞). This solution also satisfies the following integral equation
∫
e
λ[t0 ]
−1
t ≥ t0
∫ t[
‖p(s)‖ ds
eλ
Theorem 1. Suppose that (7) is exponentially stable, its fundamental matrix satisfies the exponential estimate (14) and there exist t0 ≥ 0 and µ ∈ (0, 1) such that sup
2079
t +1
∫
Ak (t )
∫
gk (t ) hk (t )
m −
Ak (s)x(gk (s))ds = F (t ),
k=1
g
(t )
k where F (t ) = f (t ) − k=1 Ak (t ) hk (t ) f (s) ds. Evidently F ∈ L∞ [t0 , ∞). Hence a solution of Eq. (19) is also a solution of the t operator equation x + Hx = r, where r (t ) := t X (t , s)F (s) ds ∈ 0 L∞ [t0 , ∞) and
∑m
‖(Hx)(t )‖ ∫ ∫ gk (s) − m m t − = X (t , s) Ak (s) Ak (τ )x(gk (τ )) dτ ds t0 h ( s ) k k =1 k=1 ∫ ∫ t m m gk (s) − − K e−λ(t −s) ‖Ak (s)‖ ‖Ak (τ )‖dτ ds ≤ t0 h ( s ) k k=1 k=1 × ‖x‖L∞ [t0 ,∞) ≤ µ‖x‖L∞ [t0 ,∞) .
2080
L. Berezansky, E. Braverman / Automatica 46 (2010) 2077–2081
Thus the norm of H in L∞ [t0 , ∞) does not exceed µ < 1. Consequently, the inverse (I + H )−1 = I − H + H 2 − H 3 + H 4 · · · is a bounded operator and the function x = (I + H )−1 r is bounded. By Lemma 2 Eq. (18) is exponentially stable.
x˙ (t ) + x(t ) + α(1 + cos t )x(t − τ (t )) = 0.
Next, consider the equation with a distributed delay t
∫
x˙ (t ) +
g (t )
x(s)ds R(t , s) = 0,
t ≥ t0 ,
Theorem 1 yields that (22) is exponentially stable if for some µ ∈ ∑ (0, 1), ak (t ) ≥ 0, k = 1, . . . , m, m k=1 ak (t ) < µ < 1, which is a delay-independent stability condition similar to Zevin and Pinsky (2006). In particular, for the equation
(20)
Theorem 1 implies exponential stability of (24) for any α < 2/(2 + √ 2) ≈ 0.8284 since
∫
t
as a perturbation of Eq. (7). Denote H (t ) = min{h(t ), g (t )},
t
sup t ≥ t0
K e−λ(t −s)
G(t )
∫
H (t )
t0
G(s)
lim sup t →∞
H (s)
t
]
‖dτ R(s, τ )‖ ds ≤ q0
implies exponential stability of Eq. (20). Corollary 4. Suppose that Eq. (7) is exponentially stable, and at least one of the following conditions holds: (1) lim supt →∞
t +δ G(s) H (s)
t
dτ ‖R(s, τ )‖ ds = 0 for some δ > 0;
2
2
Hence Theorem 1 implies that for α < 0.8284, while (23) yields
We also mention that the equation considered in the present paper is more general than most of the previously considered models. Finally, let us formulate some open problems. (1) Formulate sufficient conditions for (6) which for the case (2) generalizes inequality (5). (2) Most results of the present paper are based on the explicit estimates for the fundamental functions of equations with a distributed delay. Develop techniques to obtain such estimates. (3) Based on the results of the present paper, deduce explicit exponential stability results for Eq. (7), as well as for its particular cases (1) and (10).
∞ G(t )
‖ d R ( s , τ )‖ ds < ∞; τ 0 H (t ) G(t ) (3) lim supt →∞ H (t ) ‖ds R(t , s)‖ = 0. (2)
2
t ≥ t0
Corollary 3. Suppose that Eq. (7) is exponentially stable. For every δ > 0 there exists q0 > 0 such that the inequality
[∫
e−(t −s) α(1 + cos s) ds
0
thus Theorem 2 outperforms Theorem 1 (and thus Zevin & Pinsky, 2006) for √small τ , where τ (t ) ≤ τ , but is obviously worse than the 2/(2 + 2) bound for τ > 0.23.
The proof is similar to the proof of Theorem 2.
t +δ
t
α(1 + 2α) sup[τ (t )(1 + cos(t ))] < 1,
‖dτ R(s, τ )‖ ds ≤ µ.
Then Eq. (20) is also exponentially stable.
∫
∫
√ ] [ 1 α(2 + 2) cos t + sin t − e− t ≤ = µ < 1. =α 1+
Theorem 3. Suppose that (7) is exponentially stable, its fundamental matrix satisfies (14) and there exist t0 ≥ 0 and µ ∈ (0, 1) such that
∫
e−(t −s) α(1 + cos s) ds ≤
t0
G(t ) = max{h(t ), g (t )}.
(24)
6. Conclusion
Then Eq. (20) is exponentially stable. 5. Discussion, example and open problems For equations with a distributed delay, exponential stability of (7) and the condition lim supt →∞ |h(t ) − g (t )| = 0 in general do not imply exponential stability of Eq. (20). A perturbation of the value of a pointwise delay may be large even for small perturbations of the lower bound. Theorem 2 improves and extends stability condition (5) obtained in Driver (1977) for Eq. (3) with constant delays. Indeed, as a corollary of Theorem 2 the following inequality implies asymptotic stability of (3) with variable delays τk (t ):
sup
m −
‖Ak (t )‖τk (t )
m −
t ≥0 k=1
sup ‖Ak (t )‖
