FIXED POINTS AND CONTROLLABILITY IN DELAY SYSTEMS

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1 FIXED POINTS AND CONTROLLABILITY IN DELAY SYSTEMS HANG GAO AND BO ZHANG Received 9 December 24; Revised 28 une 25; Accepted 6 uly 25 Schaefer s fixed point theorem is used to study the controllability in an infinite delay system x (t) = G(t,x t )+(Bu)(t). A compact map or homotopy is constructed enabling us to show that if there is an aprioribound on all possible solutions of the companion control system x (t) = λ[g(t,x t )+(Bu)(t)], <λ<1, then there exists a solution for λ = 1. The aprioribound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach. Copyright 26 H. Gao and B. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction This paper is concerned with the problem of controllability in an infinity delay system x (t) = G ( t,x t ) +(Bu)(t), t = [,b], (1.1) where x(t) R n, x t (θ) = x(t + θ) for <θ, u(t), the control, is a real m-vector valued function on, B is a bounded linear operator acting on u, andg is defined on C with C being the Banach space of bounded continuous functions φ :(,] R n with the supremum norm C. The problem of controllability in delay systems has been the subject of extensive investigations by many scientists and researchers for over half of a century. A large number of applications have appeared in biology, medicine, economics, engineering, and information technology. Many actual systems have the property of after-effect, that is, the future state depends not only on the present, but also on the past history. It is well-known that such a delay factor, when properly controlled, can essentially improve system s qualitative and quantitative characteristics in many aspects. For historical background and discussion of applications, we refer the reader to the work of Balachandran and Sakthivel [1], Chukwu [4], Górniewicz and Nistri [8], and references therein. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 26, Article ID 4148, Pages 1 14 DOI /FPTA/26/4148

2 2 Fixed points and controllability in delay systems Equation (1.1) describes the state of a system (physical, chemical, economic, etc.) whose evolution in time t is governed by G(t,x t )+(Bu)(t). In general, we view a solution of (1.1) asafunctionofu(t) so that the behavior of the system depends on (or is controlled by) the choice of u within a set U given in advance. Assume that two states of the system are given, one to be considered as an initial state φ, and the other as a final state γ. The problem of controllability is to determine whether there are available controls which can transfer the state x from φ to γ along a solution (1.1), that is, whether there exists a u U such that ẋ(t) = G(t,x t )+Bu (t) has a solution joining φ and γ. When this is possible for arbitrary choice of φ and γ, we can say, roughly speaking, that system (1.1) is controllable by means of U. The main method of proving controllability has been to write (1.1)asanintegralequation t [ ( ) ] x(t) = x + G s,xs + Bu(s) ds (1.2) viewing the right-hand side as a mapping Px on an appropriate space, when u is properly chosen in terms of x. Then, apply a fixed point theorem, say Schauder s, to the mapping P when P : K K is compact for a closed, convex subset K of a Banach space. However, P, in general, does not satisfy this condition unless the growth of G(t,x t )inx is restricted. This presents a significant challenge to investigators. A modern approach to such a problem is to use topological degree or transversality method (see Górniewicz and Nistri [8]). Here we will use a fixed point theorem of Schaefer [12] which is a variant of the nonlinear alternative of Leray-Schauder degree theory, but much easier to use. The paper is organized as follows. In Section 2, we prove our main result on controllability of (1.1). Applications to specific systems are given in Section 3 which also contains some general results and remarks concerning the approach. 2. Controllability We start this section with some descriptions of spaces associated with our discussion. Let R = (, ), R = (,], and R + = [, ), respectively, and denote the Euclidean norm in R n.forann n matrix A = (a ij ) n n,define A =sup x 1 Ax. We denote by C(X,Y) the set of bounded continuous functions φ : X Y for normed spaces X and Y. For given in (1.1), we define C ={φ C(,R n ):φ() = } with the supremum norm C,andsetC (μ) ={φ C : φ C μ}. LetL 2 () denote the Banach space of square Lebesgue integrable functions x : R n with the norm x 2 = ( x(s) 2 ds) 1/2 with the understanding x = L 2 ()ifandonlyifx(t) = a.e.on. We denote by U the space of admissible controls and choose U as a complete subspace of L 2 ()withanorm U.The symbol is reserved for the norm of a linear operator. We assume that B : U L 2 () is a bounded linear operator. For a given u U,wesay that x :(,b] R n is a solution x = x(t,φ) of(1.1) on with initial function φ if x is absolutely continuous on and satisfies x (t) = G ( t,x t ) +(Bu)(t), a.e. t = [,b] (2.1) with x = φ, that is, x(s) = φ(s)foralls R.

3 H. Gao and B. Zhang 3 Our result rests on a fixed point theorem of Schaefer [12]. Its relation to Leray- Schauder degree theorem is explained in Smart [13, page 29]. Schaefer s theorem has been used in a variety of areas in differential equations and control theory (see Burton [2], Burton and Zhang [3],Balachandran and Sakthivel [1]). Theorem 2.1 (Schaefer). Let V beanormedspace,f a continuous mapping of V into V which is compact on each bounded subset of V. Then either (i) the equation x = λf(x) has a solution for λ = 1,or (ii) thesetofallsuchsolutions x, for <λ<1, is unbounded. If we view U λ (x) = λf(x) in Schaefer s theorem as a homotopy, then it can be restated in the form of Leray-Schauder Principle (cf. Zeidler [14, page 245]), which is often used in application. Definition 2.2. System (1.1) is said to be controllable on the interval if for each φ C and γ R n, there exists a control u U such that the solution x(t) = x(t,φ) of(1.1) satisfies x(b) = γ. Throughout this paper, we let φ C and γ R n be arbitrary, but fixed. For each y C,wedefine y(t)+φ() t, ȳ(t) = (2.2) φ(t) t R. This implies that ȳ = φ; that is, ȳ(s) = φ(s)foralls R. For each continuous function z :(,b] R with z = φ,wedefine [z] = γ φ() G ( ) s,z s ds. (2.3) We now introduce a companion to (1.1) x (t) = λ [ G ( t,x t ) +(Bu)(t) ], a.e.t = [,b] (1.1λ ) for λ [,1] and make the following assumptions. (H 1 ) The linear operator T : U R n defined by T(u) = Bu(s)ds (2.4) is invertible; that is, for each α R n, there exists a unique u α U such that α = (Bu α )(s)ds. (H 2 )Foreachy C, G(s, ȳ s )islebesguemeasurableinson, andforeachμ>, there exists an integrable function M μ : R + such that G(s, ȳ s ) M μ (s) foralls whenever y C (μ). (H 3 )Foranyε>andy C, there exists a δ>suchthat[x C, x y C <δ], imply G ( ) ( ) s, x s G s, ȳs ds<ε. (2.5)

4 4 Fixed points and controllability in delay systems (H 4 )Foreachφ C and γ R n, there exists a constant L = L(φ,γ) > suchthat x λ (t) L for all t whenever x λ (t) = x(t,φ) is a solution of (1.1 λ )withu(t) = u α (t) for α = [x λ ]andλ (,1). Theorem 2.3. If (H 1 ) (H 4 ) are satisfied, then (1.1)iscontrollableon. Proof. Let φ C and γ R n be fixed. We define a function F : C C by (Fy)(t) = t [ G ( s, ȳs ) + Bu[ȳ] (s) ] ds (2.6) for each y C and t. It is clear that F is well-defined. For the linear operator T defined in (H 1 ), we have T(u) = Bu(s)ds Bu(s) ds b 1/2 Bu 2 b 1/2 B u U. (2.7) Thus, T : U R is bounded, and hence T 1 is also bounded, say T 1 M 1 (see Friedman [6, page 143]). We now show that F is continuous on C.Foreachε>andy C,by(H 3 )there exists a δ>suchthat[x C, x y C <δ]imply Observe Bu [ x] (s) Bu [ȳ] (s) ds G ( ) ( ) s, x s G s, ȳs ε ds < ( ). 1+b 1/2 B T 1 (2.8) b 1/2 Bu [ x] Bu 2 [ȳ] b 1/2 B u [ x] u U [ȳ] = b 1/2 B T 1 [ x] T 1 [ȳ] U b 1/2 B T 1 [ x] [ȳ] b 1/2 B T 1 [ ( ) ( )] G s, xs G s, ȳs ds, (2.9) where [ x] = γ φ() G(s, x s )ds and [ȳ] = γ φ() G(s, ȳ s )ds. Thus, we obtain F(x)(t) F(y)(t) G ( ) ( ) s, x s G s, ȳs ds+ Bu [ x] (s) Bu [ȳ] (s) ds ( 1+b 1/2 B T 1 ) G ( ) ( ) s, x s G s, ȳs ds<ε. (2.1) This implies that F(x) F(y) C <εwhenever x y C <δ, and hence F is continuous on C. Next, we show that for each μ>, the set {F(y):y C (μ)} is uniformly bounded and equicontinuous. By the definition of F, we have F(y)(t) t = G ( ) t ( ) s, ȳ s ds+ Bu[ȳ] (s)ds M μ (s)ds+ b 1/2 B u U [ȳ]. (2.11)

5 Notice that [ȳ] = Bu [ȳ] (s)ds = T(u [ȳ] )sothatu [ȳ] = T 1 [ȳ]. Therefore, H. Gao and B. Zhang 5 u U [ȳ] T 1 [ȳ] M1 [ γ + φ() + M 1 [ γ + φ() ] + M μ (s)ds =: M 2. G ( ] ) s, ȳ s ds (2.12) Combine (2.11) and(2.12) toobtain F(y) C M 3 for some M 3 > andforally C (μ). Thus, {F(y):y C (μ)} is uniformly bounded. Now let t 1,t 2 with t 1 <t 2. Then for y C (μ), we have F(y) ( t 2 ) F(y) ( t1 ) t2 t2 t 1 M μ (s)ds+ t2 t 1 Bu [ȳ] (s) ds t 1 M μ (s)ds+ ( t 2 t 1 ) 1/2 ( t2 t 1 Bu [ȳ] (s) 2 ds t2 M μ (s)ds+ ( ) 1/2 B t 2 t 1 u U [ȳ] t 1 t2 M μ (s)ds+ ( ) 1/2 B M2 t 2 t 1 t 1 ) 1/2 (2.13) by (2.12). Thus, {F(y):y C (μ)} is uniformly bounded and equicontinuous on,and hence F is compact by Ascoli-Arzela s theorem. Finally, suppose that x = λf(x)forsomex C and <λ<1; that is, [ t x(t) = λ G ( ) t ] s, x s ds+ Bu [ x] (s)ds (2.14) for t.differentiate (2.14)with respect to t to obtain x (t) = x (t) = λ [ G ( t, x t ) + Bu(t) ], a.e.t (2.15) with u = u [ x].from(2.15), we see that x is a solution of (1.1 λ )with x = φ. Moreover, x C L by (H 4 ). This implies that x C L + φ(). Therefore, alternative (i) of Theorem 2.1 must hold, and there exists y C such that y = F(y). Following the argument in (2.14)and(2.15), we see that ȳ is a solution of (1.1)withȳ = φ and b ȳ(b) ȳ() = G ( ) b s, ȳ s ds+ = b b Bu [ȳ] (s)ds = G ( ) s, ȳ s ds+[ȳ] G ( (2.16) ) b s, ȳ s ds+ γ φ() G ( ) s, ȳ s ds. Since ȳ() = φ(), we obtain ȳ(b) = γ, andso(1.1) is controllable on the interval with u U. This completes the proof.

6 6 Fixed points and controllability in delay systems 3. Examples and remarks In this section, we give several examples to illustrate how to apply Theorem 2.3 to some specific equations and systems. Our emphasis will be on obtaining a priori bounds. The examples are shown in simple forms for illustrative purposes, and they can easily be generalized. Example 3.1. Consider the control problem t x (t) = Ax(t)+ E ( t,s,x(s) ) ds+ u(t), a.e. t, (3.1) where A = (a ij ) n n is an n n matrix, E : Ω R n R n is measurable with Ω ={(t,s) R 2 : t s},andu(t)isanarbitrarycontrol(tobedeterminedlater). It is well known that e At = ω 1 (t)i + ω 2 (t)a + + ω n (t)a n 1, (3.2) where ω i : R R (i = 1,2,...,n) are continuous and linearly independent (see Godunov [7, page 32]). Denote the space of linear span of functions w 1, w 2,..., w n by Let c R n be fixed. We define U as span { ω 1,ω 2,...,ω n }. (3.3) U = { u = u c u span { ω 1,ω 2,...,ω n }}, (3.4) and view (U, U ), U = 2, as a complete subspace of (L 2 (), 2 ). Introduce a transformation y(t) = e At x(t)towrite(3.1)as y (t) = G ( t, y t ) + e At u(t), a.e. t, (3.5) where G ( ) t t, y t = e At E ( t,s,e As y(s) ) ds+ e At E ( t,s,φ(s) ) ds. (3.6) We assume that E ( t,s,φ(s) ) ds is integrable on with respect to t (3.7) for each φ C. It is clear that (3.1)iscontrollableifandonlyif(3.5) is controllable. We now write (Bu)(t) = e At u(t) (3.8) and show that the linear operator T : U R n defined by T(u) = (Bu)(s)ds = e As u(s)ds (3.9)

7 H. Gao and B. Zhang 7 is invertible. To this end, we assume span { c,ac,a 2 c,...,a n 1 c } = R n. (3.1) Let α R n. We will find a unique u U such that T(u) = α. By(3.1), there exists a unique a = (a 1,a 2,...,a n ) T R n satisfying Since ω 1, ω 2,..., ω n are linearly independent, we have α = a 1 c + a 2 Ac + + a n A n 1 c. (3.11) =:det [( ω i,ω j )n n], (3.12) where ω i,ω j = w i (s)w j (s)ds is the inner product in L 2 (). Thus, the system of equations ω1,ω j k1 + ω 2,ω j k2 + + ω n,ω j kn = a j (3.13) (j = 1,2,...,n) has a unique solution (k 1,k 2,...,k n ). We now define u (t) = ω 1 (t)k 1 + ω 2 (t)k ω n (t)k n. (3.14) Multiply (3.14)bye As c and integrate on to obtain [ n ][ e As u(s)ds = e As cu (s)ds = ω j (s)a j 1 n c ω i (s)k i ]ds j=1 i=1 n n = ωi,ω 1 ki c + + ωi,ω n ki A n 1 c i=1 i=1 = a 1 c + a 2 Ac + a 3 A 2 c + + a n A n 1 c = α. (3.15) This implies that T is invertible. By Cramer s rule, we write k j in (3.13) ask j = j (a)/ where j (a) isthen n determinant obtained by replacing the jth column of by a = (a 1,a 2,...,a n ) T defined in (3.11). Thus Since u (t) n j=1 j (a) w j (t). (3.16) j (a) = a 1 C 1j + a 2 C 2j + + a n C nj, (3.17)

8 8 Fixed points and controllability in delay systems where C ij is the cofactor of w i,w j in the matrix ( w i,w j ) n n,andl 1 α a l 2 α for some positive constants l 1 and l 2,wehave b j (a) n a C ij 2 l2 α n C ij 2, (3.18) i=1 e At u(t) n l dt 2 c n C ij 2 j=1 i=1 b i=1 e A t w j (t) dt α =: K α. (3.19) This implies that e At u α (t) dt K α. (3.2) Theorem 3.2. Suppose that (3.7), (3.1), and the following conditions hold. (i) There exist a positive constant M and a measurable function q : Ω R +, Ω = {(t,s) R n : s t b} such that E(t,s,x) q(t,s) ( x + M ). (3.21) (ii) For any μ>, there exists k μ : Ω R + with t k μ (t,s)dsdt < such that E(t,s,x) E(t,s, y) kμ (t,s) x y (3.22) for all t,s Ω, x μ,and y μ. (iii) (K +1) where K is defined in (3.19). Then (3.1) iscontrollable. b t e A (t+s) q(t,s)dsdt =: r<1, (3.23) Proof. It suffices to show that (3.5) is controllable; that is, for any φ C and γ 1 R n,there exists a control u U such that the solution y(t) = y(t,φ)of(3.5) satisfies y(b) = γ 1.We have shown that (H 1 )holds.forφ C and y C (μ), it is clear that G(t, ȳ t )ismeasurable in t.forφ C and y C (μ), we also have G ( ) t t, ȳ t = e At E ( t,s,e As( y(s)+φ() )) ds+ e At E ( t,s,φ(s) ) ds t e A t q(t,s) [ e A s( y(s) + φ() ) + M ] ds+ e A b E ( t,s,φ(s) ) ds ( μ + φ() + M ) t e A (t+s) q(t,s)ds+ e A b E ( t,s,φ(s) ) ds =: M μ (t). (3.24)

9 H. Gao and B. Zhang 9 Thus, (H 2 ) holds. Next, let μ 1 > andx, y C (μ 1 ). By (ii), there exists k μ (t,s), μ = (μ 1 + φ() )e A b,suchthat E ( t,s,e As( x(s)+φ() )) E ( t,s,e As( y(s)+φ() )) e A b k μ (t,s) x(s) y(s) (3.25) for all (t,s) Ω. This yields b G ( ) ( ) t, x t G t, ȳt dt b t = e At E ( t,s,e As( x(s)+φ() )) t ds e At E ( t,s,e As( y(s)+φ() )) ds dt t e 2 A b k μ (t,s)dsdt x y C (3.26) for all x, y C (μ 1 ), and hence (H 3 ) is satisfied. We now show that (H 4 )holds.lety = y(t,φ) be a solution of y (t) = λ [ G ( t, y t ) + e At u(t) ], a.e.t (3.2 λ ) with λ (,1), y = φ,and e As u(s)ds = γ 1 φ() G ( ) s, y s ds. (3.27) Integrate (3.2 λ )fromtot to obtain Thus, t y(t) = φ() + λ G ( ) t τ, y τ dτ + λ e Aτ u(τ)dτ. (3.28) It follows from (3.2)that y(t) φ() + b b G ( ) τ, y τ ds+ e Aτ u(τ) dτ K [ γ 1 + φ() + Substituting (3.3)into(3.29), we arrive at y(t) K γ 1 +(1+K) φ() +(1+K) K γ 1 +(1+K) φ() +(1+K) +(1+K) b τ b b b b e Aτ u(τ) dτ. (3.29) G ( ) τ, y τ dτ ]. (3.3) G ( ) τ, y τ dτ e Aτ E ( τ,s,e As y(s) ) dsdτ e Aτ E ( τ,s,φ(s) ) ds dτ

10 1 Fixed points and controllability in delay systems K γ 1 +(1+K) φ() +(1+K) +(1+K) =:(1+K) b τ b τ r y C + M. b e A t q(τ,s) [ e A s y(s) + M ] dsdτ e A (t+s) q(τ,s) y(s) dsdτ + M e Aτ E ( τ,s,φ(s) ) ds dτ (3.31) This yields y C M /(1 r), and hence (H 4 )holds.bytheorem 2.3, system(3.5) is controllable. The proof is complete. Remark 3.3. A more general expression can be introduced in the control term in (3.1) such as B(t)u(t)whereB(t)isann m matrix function and u(t) R m.whene(t,s,x), (3.1)isreducedto x (t) = Ax + u (t)c. (3.32) A classical result states that (3.32) is controllable if and only if (3.1) holds (see Godunov [7, page 211] and Conti [5, page 98]). Example 3.4. Consider the scalar Volterra equation t x (t) = a(t s)q ( x(s) ) ds+ u(t), a.e. t, (3.33) where a : R R, q : R R are continuous, and u U.Forafixedξ : R + with ξ 2 = 1, we define U = { u L 2 ():u = kξ, k R } (3.34) with u U = u 2 = k ξ 2 = k. Therefore, U is a Banach space (dimension 1) with U. Observe that (3.33)can be written in the formof(1.1)with G ( t,x t ) = t a(t s)q ( x(s) ) ds (3.35) and B : U L 2 () being the identity operator ( B =1). Define T : U R by T(u) = Bu(s)ds = u(s)ds. (3.36) Notice that ξ(s)ds since ξ(s) and ξ 2 = 1. For each α R,ifthereareu 1,u 2 U with u 1 = k 1 ξ, u 2 = k 2 ξ such that T(u 1 ) = T(u 2 ) = α,thenk 1 ξ(s)ds = k 2 ξ(s)ds, which yields k 1 = k 2.Thus,T is invertible.

11 H. Gao and B. Zhang 11 Theorem 3.5. Suppose the following conditions hold. (ĩ ) a(t s)q(φ(s))ds is integrable on with respect to t for each φ C. (ĩi) a C(R,R)witha(t), a (t), a (t) forallt. ( iii) There are positive constants α and K such that Q(x)+K>and q(x) α Q(x)+K x R with lim Q(x) =, (3.37) x where Q(x) = x q(s)ds. ( iv) ( α 2 ) t a(s)dsdt =: η<1. (3.38) 2 Then (3.33) is controllable. Proof. We verify that conditions (H 2 ) (H 4 )hold.foreachφ C and x C (μ), we have G ( ) t, x t = a(t s)q ( φ(s) ) t ds+ a(t s)q ( x(s)+φ() ) ds a(t s)q ( φ(s) ) ds + a(t s)q ( φ(s) ) ds + t t a(t s) q ( x(s)+φ() ) ds a(s)ds q μ =: M μ (t), (3.39) where q μ = sup{ q(z) : z μ + φ() }. Thus, (H 2 )issatisfied. Now let μ>andx, y C (μ). Since q is uniformly continuous on [ (μ + φ() ),μ + φ() ], for any ε>, there exists a δ>suchthat x y C <δimplies for all s,andso G ( ) ( ) s, x s G s, ȳs ds = q ( x(s)+φ() ) q ( y(s)+φ() ) <ε (3.4) t t a(t s) [ q ( x(s)+φ() ) q ( y(s)+φ() )] ds dt a(s)dsdt ε 2ε α 2. (3.41) This shows (H 3 )holds. Let x(t) = x(t,φ) be a solution of [ t x (t) = λ a(t s)q ( x(s) ) ] ds+ u(t), a.e.t, (3.42) where λ (,1), x = φ,and τ u(s)ds = γ φ() + a(τ s)q ( x(s) ) dsdτ. (3.43)

12 12 Fixed points and controllability in delay systems We apply Liapunov s direct method to derive a priori bounds on x.define E(t) = Q ( x(t) ) + K + 1 [ t 2 λa(t) q ( x(s) ) ] 2 ds 1 t [ t 2 λ a (t s) q ( x(ν) ) 2 dν] ds (3.44) s for all t andsetv(t) = E(t). Then {[ ( V 1 t (t) = 2 q(x)λ a(t s)q ( x(s) ) )] ds+ u(t) E(t) + 1 [ t 2 λa (t) q ( x(s) ) 2 [ t ds] + λa(t) q ( x(s) ) ] ds q ( x(t) ) 1 t [ t 2 λ a (t s) q ( x(ν) ) 2 dν] ds s t [ t λ a (t s) q ( x(ν) ) ] dν ds q ( x(t) )}. s (3.45) Integrating by parts in the last term, we get t λ a (t s) = λa(t s) [ t s t s q ( x(ν) ) ] dν ds q ( x(ν) ) t dν t +λ t = λa(t) q ( x(ν) ) dν + λ t a(t s)q ( x(s) ) ds a(t s)q ( x(s) ) ds. (3.46) Substitute (3.46)into(3.45) and apply condition ( iii) to obtain V 1 (t) = {λq 2 ( x(t) ) u(t) λq ( x(t) ) a(t s)q ( φ(s) ) ds E(t) + 1 [ t 2 λa (t) q ( x(s) ) ] 2 ds 1 t [ t 2 λ a (t s) q ( x(ν) ) 2 } dν] ds { 1 2 q ( x(t) ) ( u(t) + a(ν)dν sup q ( φ(s) ) )} Q(x)+K t R α ( u(t) + a(ν)dν sup q ( φ(s) ) ) =: α 2 s R 2 u(t) + Γ 1. s (3.47)

13 H. Gao and B. Zhang 13 Integrating V (t)fromtot and using (3.43), we find V(t) V() + α b 2 u(s) ds+ bγ 1 = Q ( x() ) + K + α b 2 u(s)ds + bγ 1 = Q ( φ() ) + K + bγ 1 + α τ 2 γ φ() + a(τ s)q ( x(s) ) dsdτ α τ a(τ s) q ( x(s) ) dsdτ + Q ( φ() ) + K + bγ α ( γ + φ() + 2 a(τ s)q ( φ(s) ) ) ds dt =: α τ a(τ s) q ( x(s) ) dsdτ + Γ 2. 2 (3.48) Observe Q ( x(t) ) + K V(t) α τ a(τ s) q ( x(s) ) dsdτ + Γ 2 2 ) α 2 τ ( ηsup s sup Q ( x(s) ) + K s 2 Q ( x(s) ) + K + Γ 2. a(ν)dν + Γ 2 (3.49) This implies sup Q ( x(s) ) + K Γ 2 s (1 η). (3.5) Since Q(x) as x, there exists a positive constant L = L(Γ 2,η)suchthat x(t) < L for all t. Therefore, (H 4 )holds.bytheorem 2.3,weconcludethat(3.33)iscontrollable. Remark 3.6. Equations such as (3.33) have been the center of much interest for a long time in connection with a problem of reactor dynamics (Levin and Nohel [11]). The Liapunov function here, having its root in the work of Levin [1], continues to play an important role in the investigation of Volterra equations. It is also well-known that under xq(x) > forx withq(x) as x, condition (ĩi) with small modifications guarantees that the zero solution of the unperturbed equation t x (t) = a(t s)q ( x(s) ) ds (3.51) is asymptotically stable (Hale [9]). Acknowledgment This research was supported in part by NSF of China (147121).

14 14 Fixed points and controllability in delay systems References [1] K. Balachandran and R. Sakthivel, Controllability of functional semilinear integrodifferential systems in Banach spaces, ournal of Mathematical Analysis and Applications 255 (21), no. 2, [2] T. A. Burton, Periodic solutions of a forced Liénard equation, Annali di Matematica Pura ed Applicata. Serie Quarta 167 (1994), [3] T. A. Burton and B. Zhang, Periodicity in delay equations by direct fixed point mapping,differential Equations and Dynamical Systems. An International ournal for Theory and Applications 6 (1998), no. 4, [4] E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Mathematics in Science and Engineering, vol. 188, Academic Press, Massachusetts, [5] R. Conti,Linear DifferentialEquations and Control, Institutiones Mathematicae, vol. I, Academic Press, New York, [6] A.Friedman,Foundations of Modern Analysis, Dover, New York, [7] S. K. Godunov, Ordinary DifferentialEquations withconstantcoefficient, Translations of Mathematical Monographs, vol. 169, American Mathematical Society, Rhode Island, [8] L. Górniewicz and P. Nistri, Topological essentiality and nonlinear boundary value control problems, Topological Methods in Nonlinear Analysis 13 (1999), no. 1, [9]. K. Hale, Dynamical systems and stability, ournal of Mathematical Analysis and Applications 26 (1969), [1].. Levin, The asymptotic behavior of the solution of a Volterra equation, Proceedings of the American Mathematical Society 14 (1963), [11].. Levin and. A. Nohel, On a system of integro-differential equations occurring in reactor dynamics, ournal of Mathematics and Mechanics 9 (196), [12] H. Schaefer, Über die Methode der a priori-schranken, Mathematische Annalen 129 (1955), (German). [13] D. R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics, vol. 66, Cambridge University Press, Cambridge, 198. [14] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer, New York, Hang Gao: Department of Mathematics, Northeast Normal University, Changchun, ilin 1324, China address: hangg@nenu.edu.cn Bo Zhang: Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC , USA address: bzhang@uncfsu.edu