Impulsive stabilization of two kinds of second-order linear delay differential equations

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J. Math. Anal. Appl. 91 (004) 70 81 www.elsevier.

1 J. Math. Anal. Appl. 91 (004) Impulsive stabilization of two kinds of second-order linear delay differential equations Xiang Li a, and Peixuan Weng b,1 a Department of Applied Mathematics, Nanjing Audit Institute, Nanjing 1009, PR China b Department of Mathematics, South China Normal University, Guangzhou , PR China Received 4 September 00 Submitted by S. Heikkilä Abstract In this paper, we consider the impulsive stabilization problems for two kinds of nd-order linear delay differential equations. By the method of Lyapunov functionals, criteria on the exponential stabilization by impulses and exponential stabilization by periodical impulses are gained. The control effects of impulses are especially stressed in the definitions and results. Some examples are also given to show the applications of our results. 003 Elsevier Inc. All rights reserved. 1. Introduction Recently, the problem of impulsive stabilization for differential equations has attracted many authors attentions and some results have been published. For example, see [1 11] and references therein. Among these investigations, one problem is very interesting, which assume that a given equation without impulses is stable, and it can be turned into asymptotically stable under proper impulsive controls ([3,5 9], etc.). Of course, the similar problems such as the impulsive stabilization of an equation which is originally unstable, are also very interesting and important. About this aspect, there are a few of articles. However, the presented referenceshere (see [1,,4,10,11]), dealt with mostly the first-order delay differential equations (abbreviated as DDEs, see [4]) and the second-order ordinary differential * Corresponding author. address: idealli@1cn.com (X. Li). 1 Supported by Natural Science Foundation of Guangdong Province (980018), China X/$ see front matter 003 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) equations (abbreviated as ODEs, see [1,]). The article which we find for n-order linear DDE with impulses is [10]. Therein the following n-order linear DDE with impulses was studied: m ẋ(t) + A i (t)x [ h i (t) ] = r(t), t 0, i=1 x(ξ) = ϕ(ξ), ξ < 0, x(τ j ) = B j x ( τj ), j = 1,,..., (I) with the following assumptions: (a1) 0 = τ 0 <τ 1 <τ < are fixed points, lim j τ j =+ ; (a) r L, columns of A i, i = 1,...,m,areinL,whereL is a Banach space of essentially bounded Lebesgue measurable functions y :[0, + ) R n, R n is the space of n-dimensional column vectors x = col(x 1,...,x n ) with the norm x = max 1 i n x i ; (a3) h i : [0, + ) R are Lebesgue measurable functions, h i (t) t, i = 1,...,m; (a4) ϕ : (, 0) R n is a Borel measurable bounded function; (a5) B = sup j B j ; (a6) τ j+1 τ j ρ>0. On the basis of Bohl Perron type theorem for impulsive DDE, the following sufficient stability conditions for (I) are given: Theorem [10, Theorem 5.]. Suppose the hypotheses (a1) (a6) hold and there exist positive constants δ,σ,ρ such that t h k (t) < δ, ρ τ j+1 τ j σ, j = 1,,..., B = sup j B j < 1, and m vrai sup Ak (t) ) ( σb k=1 t>0 ln B + σ < 1. (II) Then the impulsive equations (I) are exponentially stable. In [1], Feng proposes two concepts: exponential stabilization by impulses and exponential stabilization by periodical impulses for the following nd-order ODE: x (t) + p(t)x(t) = 0, t t 0, (III) and proves that if the coefficient p(t) is continuous on [t 0, + ), then there exist explicitly impulsive controls such that Eq. (III) under these controls will become exponentially stable. In [11], these definitions and related results are generalized to linear ordinary differential systems. Our present paper deals with the impulsive stabilization which includes exponential stabilization by impulses and exponential stabilization by periodical impulses for two

3 7 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) kinds of nd-order linear DDEs. It is organized as follows. In Section, we propose the definitions of exponential stabilization by impulses and exponential stabilization by periodical impulses. In Section 3, by using Lyapunov methods, we obtain sufficient conditions of impulsive stabilization. It is worth noticing that these results actually deal with the global stabilization, i.e., under the conditions, all solutions of the equation exponentially convergent to zero. At the end of this paper, by using Mathematica software, we give some examples to show that impulses can make some unstable nd-order DDEs into exponentially stable. From this point, we know that the impulses are very important for the stabilization of the equations.. Preliminaries Let R be the real axis, and N be the set of all positive integers. We consider the following two equations: { x (t) + a(t)x(t τ)= 0, t t 0, x(t) = ϕ(t), t 0 τ t t 0, x (1) (t 0 ) = y 0, and { x (t) + b(t u)x(u) du = 0, t t 0, x(t) = ϕ(t), t 0 τ t t 0, x () (t 0 ) = y 0. Meanwhile we also consider the corresponding equations with impulses: x (t) + a(t)x(t τ)= 0, t t 0,t t k, x(t) = ϕ(t), t 0 τ t t 0, x (t 0 ) = y 0, (1 ) x(t k ) = I k (x(tk )), x (t k ) = J k (x (tk )), and x (t) + b(t u)x(u) du = 0, t t 0,t t k, x(t) = ϕ(t), t 0 τ t t 0, x (t 0 ) = y 0, ( ) x(t k ) = I k (x(tk )), x (t k ) = J k (x (tk )), with the following assumptions: (H 1 ) x(t): [t 0 τ,+ ) R; (H ) a(t) is continuous on [t 0, + ), b(t) is continuous on [0,τ]; (H 3 ) t 0 <t 1 < <t k <, k N, lim k t k =+ ; (H 4 ) I k, J k : R R are both continuous, and I k (0) = J k (0) 0, k N; (H 5 ) For each continuous function z(t),wedenotebyz (t) the right derivative of z(t),i.e., z (t) lim h 0+ z(t + h) z(t), h all the symbols x (t), x (t) [x (t)] in the equations satisfy the above definition; (H 6 ) ϕ : [t 0 τ,t 0 ] R, has at most finite discontinuity points of the first kind and is right continuous at these points.

4 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) Definition 1. A function x : [t 0 τ,t 0 + α) R, α>0issaidtobeasolutionofthe impulsive equation (1 ) (or ( )), starting from (t 0,ϕ,y 0 ),if (i) x(t) and x (t) are both continuous on [t 0,t 0 + α)\{t k,k N}; (ii) x(t) = ϕ(t), t [t 0 τ,t 0 ], x (t 0 ) = y 0 ; (iii) x(t) satisfies the first equality of (1 ) (or ( ))on[t 0,t 0 + α)\{t k,k N}; (iv) x(t) and x (t) both have two-side limits and right continuous at points t k, k N, and x(t k ), x (t k ) satisfy the 3rd equality of (1 ) (or ( )). We denote by x(t; t 0,ϕ,y 0 ) the solution of (1 ) (or ( )), starting from (t 0,ϕ,y 0 ). Remark 1. The global existence and uniqueness of solutions for (1 ) (or ( )) can be found in [3,1]. In the following, we always assume that the solutions of (1 ) (or ( ))existin [t 0, + ). Next we give definitions about the impulsive stabilization of Eq. (1 ) (or ( )), i.e., the definitions for exponential stabilization by impulses and exponential stabilization by periodical impulses. Definition. Equation (1) (or ()) is said to be exponentially stabilized by impulses, if there exist α>0, sequence {t k }, and function sequences on R: {I k }, {J k }, satisfying (H 3 ), (H 4 ), such that for all ε>0, there exists δ>0 and once the solution of (1 ) (or ( )) x(t; t 0,ϕ,y 0 ) meets ϕ + y0 δ; then x (t) + x (t) ε exp [ α(t t 0 ) ], t t 0, where ϕ sup t0 τ s t 0 ϕ(s). Definition 3. Equation (1) (or ()) is said to be exponentially stabilized by periodical impulses, ifthereexistα>0, sequence {t k } satisfying (H 3 ) and t k t k 1 = c (here c is a positive constant), and function sequences on R: {I k }, {J k } satisfying (H 4 ) and I 1 (u) = =I k (u) =, J 1 (u) = =J k (u) =, k= 1,,..., u R, such that for all ε>0, there exists δ>0and once the solution of (1 ) (or ( )) x(t; t 0,ϕ,y 0 ) meets ϕ + y 0 δ; then x (t) + x (t) ε exp [ α(t t 0 ) ], t t 0. In the next section, we give two criteria for the impulsive stabilization of Eqs. (1) and ().

5 74 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) Main results First we consider Eq. (1). Theorem 1. If a(t) is continuous and bounded on [t 0, + ), with a(t) A (here A 0), and if Aτ <exp { (1 + A)τ }, (3) then Eq. (1) can be exponentially stabilized by impulses. Proof. We have from the inequality (3) that we could find α>0, l τ such that Aτ exp [ α(l + τ) ] exp [ (1 + A)l ]. (4) Let α, l be as in (4). For any sequence {t k } such that t 0 <t 1 < <t k <, lim k t k = +, with τ t k t k 1 l, wedefinei k (u) = J k (u) = d k u, k = 1,,...,where pk Aτ d k =, p k = exp [ α(t k+1 t k + τ) ] exp [ (1 + A)(t k+1 t k ) ]. By (4) it is easy to verify that d k is a non-negative real number. For all ε>0, let δ be as follows: ε δ = exp [ α(t 1 t 0 ) ] [ exp 1 ] 1 + Aτ (1 + A)(t 1 t 0 ). We shall prove that for each solution x(t; t 0,ϕ,y 0 ) of Eq. (1 ), the inequality ϕ + y0 δ implies x (t) + x (t) ε exp [ α(t t 0 ) ], t t 0. (1 0 ) If t [t 0,t 1 ),letv(t) [x (t) + x (t)]+ a(s + τ) x (s) ds; thenv(t) has the following properties: (i) x (t) + x (t) V(t); (ii) V(t) [ x (t) + x (t) ] + Aτ x t (1 + Aτ) x t + x (t) (1 + Aτ) [ x t + x (t) ], where x t sup s t x(s) ; (iii) when t (t 0,t 1 ), if we denote by V(t) the right upper derivative of V(t) along the solution of (1 ),thenwehave V(t)= x(t)x (t) + x (t)x (t) + a(t + τ) x (t) a(t) x (t τ) = x(t)x (t) a(t)x (t)x(t τ)+ a(t + τ) x (t) a(t) x (t τ) [ x (t) + x (t) ] + [ a(t) x (t τ)+ x (t) ] + a(t + τ) x (t) a(t) x (t τ) (1 + A) [ x (t) + x (t) ] (1 + A)V (t),

6 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) Thus that is V(t) V(t 0 ) exp [ (1 + A)(t t 0 ) ] V(t 0 ) exp [ (1 + A)(t 1 t 0 ) ]. x (t) + x (t) V(t 0 ) exp [ (1 + A)(t 1 t 0 ) ] (1 + Aτ) [ x t0 + x (t 0 ) ] exp [ (1 + A)(t 1 t 0 ) ] (1 + Aτ)δ exp [ (1 + A)(t 1 t 0 ) ] = ε exp [ α(t 1 t 0 ) ] ε exp [ α(t t 0 ) ], therefore, we obtain on (t 0,t 1 ) that x (t) + x (t) ε exp [ α(t t 0 ) ]. By the right continuity of x(t) and x (t), the inequality above also holds on [t 0,t 1 ).Especially, we have sup t 1 τ t t 1 [ x (t) + x (t) ] ε exp [ α(t 1 t 0 τ) ]. ( 0 ) If t (t 1,t ),wealsoletv(t) [x (t)+x (t)]+ a(s + τ) x (s) ds, similarly to (1 0 ),wehave V(t) (1 + A)V (t), that is x (t) + x (t) V(t) V ( t + ) [ 1 exp (1 + A)(t t 1 ) ] { [x = ( t + ) 1 + x ( t 1 } t + )] 1 + a(s + τ) x (s) ds exp [ (1 + A)(t t 1 ) ] = { [x (t 1 ) + x (t 1 ) ] + t 1 τ 1 t 1 τ exp [ (1 + A)(t t 1 ) ] { = d1[ x ( t1 ) + x ( t1 exp [ (1 + A)(t t 1 ) ] } a(s + τ) x (s) ds t 1 )] + t 1 τ } a(s + τ) x (s) ds

7 76 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) d 1 [ sup x (t) + x (t) ] exp [ (1 + A)(t t 1 ) ] t 1 τ t t 1 + Aτ sup t 1 τ t t 1 [ x (t) ] exp [ (1 + A)(t t 1 ) ] ( d1 + Aτ) [ sup x (t) + x (t) ] exp [ (1 + A)(t t 1 ) ] t 1 τ t t 1 ε ( d1 + Aτ) exp [ α(t 1 t 0 τ) ] exp [ (1 + A)(t t 1 ) ]. By the definition of d 1,wederivethat ( ) x (t) + x (t) ε p1 Aτ + Aτ exp [ α(t 1 t 0 τ) ] exp [ (1 + A)(t t 1 ) ]. Because of Aτ p 1, we obtain x (t) + x (t) ε p 1 exp [ α(t 1 t 0 τ) ] exp [ (1 + A)(t t 1 ) ] = ε exp [ α(t t 1 + τ) ] exp [ (1 + A)(t t 1 ) ] exp [ α(t 1 t 0 τ) ] exp [ (1 + A)(t t 1 ) ] = ε exp [ α(t t 0 ) ] ε exp [ α(t t 0 ) ]. Therefore, we have on (t 1,t ) that x (t) + x (t) < ε exp [ α(t t 0 ) ]. Similarly, by the right continuity of x(t) and x (t), the above inequality holds on [t 1,t ), and thus it leads to [ sup x (t) + x (t) ] ε exp [ α(t t 0 τ) ]. t τ t t (3 0 ) Similar to (1 0 ) and ( 0 ), we can show by induction that for all k N, whent [t k 1,t k ),wehave x (t) + x (t) < ε exp [ α(t t 0 ) ]. Note that the above inequality holds for all t t 0. Thus we complete the proof. Remark. Under the conditions in Theorem 1, if t k,i k,j k also satisfy t k t k 1 = l, I k (u) = J k (u) = du, k = 1,,..., where p Aτ d =, p= exp [ α(l + τ) ] exp [ (1 + A)l ], then according to the proof of Theorem 1, (1) can also be exponentially stabilized by periodical impulses.

8 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) Next we consider Eq. (). Theorem. If b(t) is continuous and bounded on [0,τ], with b(t) B (B 0), and if Bτ < exp { (1 + Bτ)τ }, (5) then Eq. () can be exponentially stabilized by impulses. Proof. First by (5), we can find α>0, l τ such that Bτ exp [ α(l + τ) ] exp [ (1 + Bτ)l ]. (6) Let α, l be as in (6). For any sequence {t k } such that t 0 <t 1 < <t k <, lim k t k = +, with τ t k t k 1 l, wedefinei k (u) = J k (u) = d k u, k = 1,,...,where p k Bτ d k =, p k = exp [ α(t k+1 t k + τ) ] exp [ (1 + Bτ)(t k+1 t k ) ]. By (6) it is easy to verify that d k is a non-negative real number. For any ε>0, let δ be as follows: ε δ = exp[ α(t 1 t 0 ) ] [ exp 1 ] 1 + Bτ (1 + Bτ)(t 1 t 0 ). Define a Lyapunov functional V(t) [ x (t) + x (t) ] + then V(t)satisfies [ t ] b(u s + τ) x (s) ds du; u (i) x (t) + x (t) V(t); (ii) V(t) [ x (t) + x (t) ] + Bτ x t ( 1 + Bτ )[ x t + x (t) ] ; (iii) when t (t k 1,t k ), k N, if we denote by V(t) the right upper derivative of V(t) along the solution of ( ),then V(t)satisfies V(t)= x(t)x (t) + x (t)x (t) + b(u t + τ) x (t) du = x(t)x (t) b(t s) x (s) ds + b(t s)x(s)x (t) ds b(t s) x (s) ds b(u t + τ) x (t) du

9 78 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) [ x (t) + x (t) ] [ = x (t) 1 + b(t s) x (t) ds b(t s) x (s) ds b(u t + τ) x (t) du [ + x (t) 1 + ] b(s t + τ) ds ] b(t s) ds (1 + Bτ) [[ x (t) + x (t) ]] (1 + Bτ)V(t). b(t s) x (s) ds Similarly to the proof of Theorem 1 and by the inductive method, we can prove that for each solution x(t; t 0,ϕ,y 0 ) of Eq. ( ), the inequality ϕ + y0 δ implies x (t) + x (t) ε exp [ α(t t 0 ) ], t t 0. We omit the details and thus complete the proof of Theorem. Remark 3. Under the conditions in Theorem, if t k,i k,j k also satisfy t k t k 1 = l, I k (u) = J k (u) = du, k = 1,,..., where p Bτ d =, p= exp [ α(l + τ) ] exp [ (1 + Bτ)l ], then according to the proof of Theorem, () can also be exponentially stabilized by periodical impulses. Remark 4. According to the proof of Theorem 1, we have that ε = δ 1 + Aτ exp [ α(t 1 t 0 ) ] [ ] 1 exp (1 + A)(t 1 t 0 ). Thus, for each solution of Eq. (1 ), starting from (t 0,ϕ,y 0 ),wederivethat x (t) + x (t) δcexp [ α(t t 0 ) ], t t 0, (7)

10 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) where C = 1 + Aτ exp [ α(t 1 t 0 ) ] [ ] 1 exp (1 + A)(t 1 t 0 ). Since C is a constant and δ is determined only by ϕ and y 0, (7) implies that under the conditions of Theorem 1, all solutions of Eq. (1) can exponentially convergent to zero under impulsive controls. Similarly, we can see that under the conditions of Theorem, all solutions of Eq. () can exponentially convergent to zero under impulsive controls. Thus, our theorems actually deal with the global stabilization for (1) and (). 4. Examples In this section, we give some examples to illustrate the applications of our results. Example 1. Consider the following equation: x (t) = x(t 1), t 0, t k, k N, x(t) = ϕ(t), 1 t 0, x (0) = y 0, x(k) = dx(k ), x (k) = dx (k ), where d = (exp( ) )/. It is known that the non-impulsive equation x (t) = x(t 1) is unstable. However, if we let A = , l = τ = 1, α = 1, it is easy to verify manually (or by Mathematica software) that Aτ = < exp( ) = exp [ α(l + τ) ] exp [ (1 + A)τ ] exp [ (1 + A)τ ], which meets the assumptions in Theorem 1, and d satisfies the conditions in Remark. Thus, the zero solution of (8) is exponentially stable, i.e., an unstable equation x (t) = x(t 1) can be exponentially stabilized by periodical impulses. Example. Consider the following equation: x (t) = x(t 0.38), t t 0,t t k, x(t) = ϕ(t), t t t 0, x (t 0 ) = y 0, (9) x(t k ) = dx(tk ), x (t k ) = dx (tk ), where d = (exp( 1.48) 0.38)/. Obviously, equation x (t) = x(t 0.38) is also unstable. But if we let A = 1, l = τ = 0.38, α = 1, then we have Aτ <exp [ α(l + τ) ] exp [ (1 + A)τ ] exp [ (1 + A)τ ]. Thus, x (t) = x(t 0.38) can also be exponentially stabilized by periodical impulses. (8) Example 3. Consider the following equation: x (t) = t 1 x(u)du, t 0, t k, k N, x(t) = ϕ(t), 1 t 0, x (0) = y 0, x(k) = dx(k ), x (k) = dx (k ), (10)

11 80 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) where d = (exp( ) )/. It is also easy to verify that the coefficient and impulses satisfy the conditions in Theorem, which implies that the unstable equation x (t) = t 1 x(u)du can be exponentially stabilized by periodical impulses. Example 4. Consider the following equation: x (t) = t 0.35 x(u)du, t t 0,t t k, x(t) = ϕ(t), t t t 0, x (t 0 ) = y 0, (11) x(t k ) = dx(tk ), x (t k ) = dx (tk ), where d = (exp( 1.875) 0.15)/. Similarly we can verify by Theorem the unstable equation x (t) = t 0.35 x(u)du can be exponentially stabilized by periodical impulses. Example 5. Consider the following equation with oscillating coefficients: x (t) = [cos4πt + sin 4πt]x(t 1/), t 0, t t k, x(t) = ϕ(t), 1/ t 0, x (0) = y 0, (1) x(t k ) = dx(tk ), x (t k ) = dx (tk ), where d = (exp( ) )/. It is easy to verify that x(t) = (0.3707/ (0π ))e cos4πt is a solution of the equation x (t) = [cos4πt + sin 4πt](t 1/), and it does not convergent to zero. But if we let A = , l = τ = α = 1/, we have [ cos 4πt + sin 4πt ] A and Aτ <exp [ α(l + τ) ] exp [ (1 + A)τ ]. That is to say, the equation x (t) = [cos4πt + sin 4πt]x(t 1/) can be exponentially stabilized by periodical impulses. From the examples above, we can see that if A,τ (or B,τ) in Eq. (1) (or ()) are properly small, then the unstable equation can be made into exponentially stable by explicit impulsive controls. Remark 5. We mention here that although the equation studied in paper [10] is more general (delays are non-constant, and coefficients are not necessarily continuous) than the equations herein, there exist still many cases on which [10, Theorem 5.] cannot be applied to verify the stability. For example, the impulsive stabilization of Eq. (8) cannot be judged by it, which implies that our results have their own applications. Now, please see Example 1. First, we rewrite Eq. (8) to the following matrix form: [ ] [ ][ ] [ ][ ] x 1 (t) 0 0 x1 (t 1) 0 1 x1 (t) x (t) = +, t 0, x (t 1) 0 0 x (t) x 1 (t) = ϕ(t), 1 t 0, x (0) = y 0, x 1 (k) = dx 1 (k ), x (k) = dx (k ), k = 1,,...

12 X. Li, P. Weng / J. Math. Anal. Appl. 91 (004) Comparing with Eq. (I) and the conditions in [10, Theorem 5.], we can find that [ ] 0 0 A 1 (t) =, A1 (t) = (by the norm defined in [10]), [ ] 0 1 A (t) =, A (t) = 1, 0 0 exp( ) σ = 1, B = d = ; then the left side of (II) becomes ) ) ( d ln d ( d ln d + > 1, i.e., the criteria in Theorem 5. cannot be applied effectively in such nd-order linear DDE as (8). References [1] W. Feng, Impulsive stabilization of second order differential equations, J. South China Normal Univ. Natur. Sci. Ed. 1 (001) [] W. Feng, Y. Chen, The weak exponential asymptotic stability of impulsive differential system, Appl. Math. J. Chinese Univ. 1 (00) 1 6. [3] J. Shen, Z. Luo, X. Liu, Impulsive stabilization of functional differential equations via Liapunov functionals, J. Math. Anal. Appl. 40 (1999) 1 5. [4] L. Berezansky, E. Braverman, Impulsive stabilization of linear delay differential equations, Dynam. Systems Appl. 5 (1996) [5] X. Liu, Impulsive stabilization and applications to population growth models, Rocky Mountain J. Math. 5 (1995) [6] X. Liu, Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform. 10 (1993) [7] X. Liu, Practical stabilization of control systems with impulse effects, J. Math. Anal. Appl. 166 (199) [8] X. Liu, Stability results for impulsive differential systems with applications to population growth models, Dynam. Stability Systems 10 (1995) 5 1. [9] X. Liu, R. Pirapakaran, Asymptotic stability for impulsive systems, Appl. Math. Comput. 30 (1989) [10] A. Anokhin, L. Berezansky, E. Braverman, Exponential stability of linear delay impulsive differential equations, J. Math. Anal. Appl. 193 (1995) [11] X. Li, Impulsive stabilization of linear differential system, J. South China Normal Univ. Natur. Sci. Ed. 1 (00) [1] L. Wen, Y. Chen, Razumikhin type theorems for functional differential equations with impulses, Dynam. Contin. Discrete Impuls. Systems 6 (1999)

Impulsive Stabilization of certain Delay Differential Equations with Piecewise Constant Argument

International Journal of Difference Equations ISSN0973-6069, Volume3, Number 2, pp.267 276 2008 http://campus.mst.edu/ijde Impulsive Stabilization of certain Delay Differential Equations with Piecewise