Tohoku Math. Journ. 37 (1985), 489-511. STABILITY BY DECOMPOSITIONS FOR VOLTERRA EQUATIONS THEODORE ALLEN BURTON AND WADI ELIAS MAHFOUD (Received September 19, 1984) form Abstract. We consider a system of integrodifferential equations of the (1) x'=a(t)x+ çt0c(t, s)x(s)ds which we then write as A number of Lyapunov functionals are constructed for (2) yielding necessary sufficient conditions for stability of the zero solution of (1). 1, Introduction. We consider the system (1.1) x'=a(t)x+ çt0c(t, s)x(s)ds in which A(t) is an n ~n matrix continuous for 0 t<, C(t, s) is an n ~n matrix continuous for 0 s t<, n 1. We write (1.1) as discuss stability instability of the zero solution of (1.1) via the construction of Lyapunov functionals for the system (1.2). Evidently, (1.1) can be regarded as a special case of (1.2) therefore any stability result for (1.2) is also a stability result for (1.1). However, the most interesting stability results of this paper are those obtained by converting (1.1) to (1.2). It turns out that (1.1) can be reduced to (1.2) in several ways consequently a variety of stability results will be obtained. In most cases, we obtain simple practical results under mild conditions. The following terminology is used throughout this paper. For any t0 0 any continuous function ƒó:[0, t0] Rn, a solution of (1.2) hence of (1.1) is a continuous function x:[0, ) Rn, denoted by x(t, t0, ƒó) or x(t), which satisfies (1.2) for t t0 such that x(t)=ƒó(t) for 0 t t0. The solution x=0 is called the zero solution.
490 T.A. BURTON AND W.E. MAHFOUD be DEFINITION. The zero solution of (1.2) hence of (1.1) is said to 1. stable if for every ƒã>0 every t0 0, there exists a ƒâ= ƒâ (ƒã, t0)>0 such that ƒó(t) <ƒâ on [0, t0] implies x(t, t0,ƒó) <ƒã for t t0. 2. uniformly stable if it is stable the ƒâ in the definition of stability is independent of t0. 3. asymptotically stable if it is stable for each t0 0 there is a ƒà=ƒà(t0)>0 such that ƒó(t) <ƒà on [0, t0] implies x(t, t0, ƒó) 0 as t. 4. uniformly asymptotically stable if it is uniformly stable, the ƒà in the definition of asymptotic stability is independent of t0, for each ƒå >0 there is a T=T(ƒÅ)>0 such that ƒó(t) <ƒà on [0, t0] implies x(t, t0, ƒó)<ƒå for t t0+t. An n ~n matrix is said to be stable if all of its characteristic roots have negative real parts. Also, when a function is written without its argument, then it is understood that the argument is t. If D is a matrix or a vector, D means the sum of the absolute values of its elements. Most stability results for the system (1.1) assume that A(t)=A= constant. In this paper we allow A(t) to be a function of t which may be unbounded for n=1. For this reason we discuss the scalar the vector cases separately. 2. Scalar equations. Consider the scalar equation where L(t) is continuous for 0 t<, C1(t, s) H(t, s) are continuous for 0 s t<. Here L, C, H x are all scalars. Let (2.2) P(t)= çt0 C1,(t,s) ds, (2.3) J(t)= çt0 H(t,s) ds, (2.4) ƒ³(t, s)= ç t[(1+j(u))i C1(u,s) +( L(u) +P(u)) H(u,s) ]du assuming of course that ƒ³(t, s) exists for 0 s t<. Let
VOLTERRA EQUATIONS 491 where đ is an arbitrary constant. The functional V(t, x( E)) plays a central role in the derivation of most of the stability results in this section. Thus, if x(t)=(t, t0,ė) is a solution of (2.1), then the derivative V'(2.1,(t, x( E)) of V(t, x( E)) along x(t) satisfies Thus, Using the Schwarz inequality, we may write so that
492 T.A. BURTON AND W.E. MAHFOUD Thus, Using (2.4), we obtain (2.6) THEOREM 1. Let P, J, ƒ³ be defined by (2.2)-(2.4). If L(t)<0 (i) supt 0 J(t)<1 (ii) J(t) L(t) +P(t)+ƒ³(t,t) 2 L(t), then the zero solution of (2.1) is stable. PROOF. Taking ƒë=1 in (2.5) observing that (2.7) we obtain from (2.6) (ii) that V'(2.1)(t,x( E)) 0. Since V(t,x( E)) is not positive definite, we still need to prove stability. Let t0 0 >0 be given. We must find ƒâ>0 so that if ƒó: [0, t0] R is continuous with ƒó(t) <ƒâ, then x(t, t0,ƒó) <ƒã for all t t0. As V'(t, x( E)) 0 for t t0, we have
VOLTERRA EQUATIONS 493 On the other h, Thus, x(t) ƒân+ çt0 H(t,s)) x(s)ds. As long as x(t) <ƒã, we have x(t) ƒân+ƒãsupt 0J(t)def=ƒÂN+ƒÃƒÀ<ƒÃ for all t t0 provided that ƒâ<ƒã(1-ƒà)/n. Thus, the solution x=0 is stable. THEOREM 2. Let P, J, ƒ³ be defined by (2.2)-(2.4) suppose there is a continuous function h: [0, ) [0, ) such that H(t, s) h(t-s) with h(u) 0 as u. If L(t)>0 there is a positive constant ƒ such that (2.8) J(t) L(t) +P(t)+ƒ³(t, t)+ƒ 2 L(t), then the zero solution of (2.1) is unstable. PROOF. Taking ƒë=-1 in (2.5) using (2.6)-(2.8) we obtain V'(2,1)(t, x( E)) ƒ x2 for all t t0. Now, for any t0 0 any ƒâ>0 we can find a continuous function ƒó: [0, t0] R with ƒó(t) <ƒâ V(t0, ƒâ( E))>0 so that if x(t)=x(t, to, ƒó) is a solution of (2.1), then We will show that x(t) is unbounded. Suppose x(t) is bounded; then as J(t)<2 by (2.8), we have çt0 H(t, s)x(s)ds bounded hence, by (2.9), x2(t) is in L1[0, ). By the Schwarz inequality we have The last integral is the convolution of an L1-function with a function tending to zero. Thus, the integral tends to zero as t hence H(t, s)x(s)ds 0 as t. Now, by (2.9), we have x(t)- çt0h(t, s)x(s)ds [V(t0,ƒÓ( E))]1/2 so that for sufficiently large T it follows that x(t) ƒá for some ƒá>0 all t T. This contradicts x2(t) being in L1. Thus, x(t) is unbounded
494 T.A. BURTON AND W.E. MAHFOUD so the zero solution of (2.1) is unstable. This completes the proof. Now, consider the scalar equation (2.10) x'=a(t)x+ çt0c(t, s)x(s)ds, where A(t) C(t, s) are continuous for 0 t< 0 s t<, respectively. Suppose that C(t,s)=C1(t,s)+C2(t,s), where Ci(t,s),i=1, 2, are continuous. Select a continuous function H(t, s) such that (2.11) ( Ý/ Ýt)H(t,s)=C2(t,s). For example, if ç tc2(u,s)du exists, then H(t,s) may be defined by H(t,s)=- ç tc2(u,s)du. If we let (2.12) L(t)=A(t)-H(t,t), then (2.10) takes the form (2.1). Thus, Theorems 1 2 combined yield the following necessary sufficient condition for stability of (2.10). THEOREM 3. Let P, J ƒ³ be defined by (2.2)-(2.4), where H L are defined by (2.11) (2.12). Suppose there is a continuous function h: [0, ) [0, ) with H(t, s) h(t-s) h(u) 0 as u, suppose there is a positive constant a such that (i) supt 0 J(t)<1 (ii) J(t) L(t) +P(t)+ƒ³(t,t)+ƒ 2 L(t). Then the zero solution of (2.10) is stable if only if L(t)<0. corollaries. Regarding (2.10) as a special case of (2.1), we obtain the following COROLLARY 1. Suppose that for some ƒ >0, Then the zero solution of (2.10) is stable if only if A(t)<0. PROOF. Let C2(t, s) ß0 in Theorem 3. Then C1(t, s)=c(t, s), J(t) ß0,
VOLTERRA EQUATIONS 495 L(t)=A(t), (ii) of Theorem 3 reduces to P(t)+ƒ³(t, t)+ƒ 2 A(t) with ƒ³(t, t)= ç t C(u, t) du. If we choose h(t) ß0, then all the conditions of Theorem 3 are satisfied the proof is complete. Letting H(t, s) L(t) satisfy (2.13) ( Ý/ Ýt)H(t,s)=C(t,s) (2.14) L(t)=A(t)-H(t,t), we have: COROLLARY 2. Let H L satisfy (2.13)-(2.14). Suppose there is a continuous function h: [0, ) [0, ) with H(t, s) h(t-s) h(u) 0 as u, suppose there is a positive constant ƒ such that (i) supt 0 çt0 H(t,s) ds<1 (ii) Then the zero solution of (2.10) is stable if only if L(t)<0. PROOF. Let C1(t, s) ß0 in Theorem 3. Then C2(t, s)=c(t, s), P(t) ß0, (ii) of Theorem 3 reduces to J(t) L(t) +ƒ³(t,t)+ƒ 2 L(t) with ƒ³(t, t)= ç t L(u) H(u,t) du. Thus all the conditions of Theorem 3 are satisf ed this completes the proof. REMARK 1. Corollary 1 is [3, Theorem 3]. Also, Corollary 2 improves [3, Theorem 4] by relaxing the boundedness condition on L(t). Now, if L(t) is bounded, say Q1 L(t) Q2 for some positive constants Q1 Q2, then, for ƒ =2Q2-RQ1 with 0<R<2, Corollary 2 reduces to [3, Theorem 4]. We give below two illustrative examples in which both A(t) L(t) are unbounded. EXAMPLE 1. Consider the equation x'=-(t+2)x- çt0(s+1)(t-s+1)-2x(s)ds. As A(t)=-(t+2)<0, çt0 C(t, s) ds=t+2-(t+2)(t+1)-1-ln(t+1),
496 T.A. BURTON AND W.E. MAHFOUD ç t C(u,t) du=t+1. Then the conditions of Corollary 1 are satisfied hence the zero solution is stable. However, Corollary 2 does not apply as H(t,s)=(s+1)(t-s+1)+g(s) does not satisfy condition (i) for every choice of g. EXAMPLE 2. Consider the equation x'=[(2t+1)-3/3-t-1/10]x- çt0(t+s+1)-4x(s)ds. As A(t) changes sign, Corollary 1 does not apply. However, by letting H(t, s)=(t+s+1)-3/3, we have L(t)=A(t)-H(t,t)=-t-1/10, H(t,s) ds=[(t+1)-2-(2t+1)-2]/6 1/6, L(u) H(u,t) du=(30t+11)(2t+1)-2/60. Thus, All conditions of Corollary 2 are satisfied hence the zero solution is stable. REMARK 2, while Corollary 1 requires a sign condition on A(t), Corollary 2 requires a strong integral condition on C(t, s). On the other h, both results can be considered as extreme cases of Theorem 3. Consequently, Theorem 3 does not only extend unify Theorems 3 4 of [3] but also yields a new stability result which is more refined than either of these theorems. This is illustrated by the following example. EXAMPLE 3. Consider the equation x'=[(sint)/2-1/4]x+[(t+s+1)-2/9-(t-s+1)-3 sin s]x(s)ds, where neither Corollary 1 nor Corollary 2 apply. However, by letting
VOLTERRA EQUATIONS 497 C1(t,s)=(t+s+1)-2/9, C2(t,s)=-(t-s+1)-3 sin s, H(t,s)= C2(u, s)du, we have Thus, J(t) L(t) +P(t)+ƒ³(t,t) 9/20<2 L(t) hence, by Theorem 3, the zero solution is stable. We now apply Theorem 3 to the convolution equation (2.15) x'=ax+ çt0c(t-s)x(s)ds in which A is constant C(t) is continuous for 0 t<. Let C(t)=C1(t)+C2(t), where Ci(t), i=1, 2, are continuous for 0 t<. We assume that C1(t) ç t C2(v)dv are L1-functions let H(t)= C2(v)dv. Then P(t), J(t), L(t) defined by (2.2), (2.3), (2.12) reduce to P(t)= çt0 C1(v) dv, J(t)= çt0 ç uc2(v)dv du, L(t)=A+ C2(v)dv. As L(t) is constant, then, without loss of generality, we may replace P(t) J(t) in ƒ³(t, t) in (ii) of Theorem 3 by P( ) J( ). Thus, letting (2.16) P=C1(v) dv, (2.17) J=0 t C2(v)dv dt,
498 T.A. BURTON AND W.E. MAHFOUD (2.18) L=A+ ç 0 C2(v)dv, we have ƒ³(t, t)=(1+j)p+( L +P)J so that conditions (i) (ii) of Theorem 3 reduce to P(J+1)+(J-1) L <0. THEOREM 4. Let P, J L be defined by (2.16)-(2.18). (i) If P(J+1)+(J-1) L <0, then the zero solution of (2.15) is stable if only if L<0. (ii) If, in addition, C(t) is in Lj[0, ), j=1 or j=2, then the zero solution of (2.15) is asymptotically stable if only if L<0. (iii) If, furthermore, J ç t C(v) dv is in Lq[0, ), 0<q<2,then the zero solution of (2.15) is uniformly asymptotically stable if only if L<0. PROOF. If we let h(t)=h(t), Part (i) follows from Theorem 3. To prove Part (ii), we use (2.15) to obtain If L<0, then the Lyapunov functional V(t, x( E)) defined by (2.5) satisfies, as in the proof of Theorem 1, V'(t, x( E)) -axe, ƒ >0. Thus, x2(t) is in L1[0, ). By the above inequality the hypothesis in (ii), it follows that x'(t) is bounded. Since x2(t) is in L1 [x2(t)]'=2x(t)x'(t) is bounded, then x(t) 0 as t. Thus x=0 is asymptotically stable. To prove (iii) we consider separately the two cases J=0 J 0. Suppose that J 0. Then for any t0 0 any continuous function ƒó:[0, t0] R we write (2.15) as x't=ax(t)+ çt00c(t-s)ƒó(s)ds+ çtt0c(t-s)x(s)ds. If we let y(t)=x(t+t0), we obtain y' t)=ay(t)+ çt0c(t-s)y(s)ds+ çt00c(t+t0-s)ƒó(s)ds. Let Z(t) be the n ~n matrix (here, n=1) satisfying Z'(t)=AZ(t)+ çt0c(t-s)z(s)ds Z(0)=I. Then by the variation of parameters formula, we have y(t)=z(t)ƒó(t0)+ çt0z(t-u) çt00c(u+t0-s)ƒó(s)dsdu
VOLTERRA EQUATIONS 499 =Z(t)ƒÓ(t0)+ ç0tz(t-u) çu+t0uc(v)ƒó(u+t0-v)dvdu. Let K=max0 t t0 ƒó(t). Then y(t) K Z(t) +K çt0 Z(t-u) ç u C(v) dvdu. Letting G(t)= ç t C(v) dv, we have By the Schwarz inequality, we have y(t) K Z(t) +K( çt0[g(t-u)]qdu çt0[g(t-u)]2-q Z(u) 2du)1/2 As x=0 is asymptotically stable, then Z(t) 0 as t. By the hypothesis in (iii), G(t) is an Lq-function furthermore G(t) 0 as t, On the other h, Z 2 is an L1-function the last integral is the convolution of an L1-function with a function tending to zero. Thus, the right h side in the above inequality tends to zero hence x(t+t0) 0 as t uniformly in t0. Consequently, the zero solution is uniformly asymptotically stable. If J=0, then C2(t) ß0 C1(t)=C(t). The condition in (i) reduces to P< A hence the conditions in (ii) (iii) are satisfied. If we assume A<0 consider the functional W(t,x( E))= x + çt0 ç t C(u-s) du x(s) ds, then, for some ƒ >0, Thus, x(t) is in L1[0, ) this is equivalent to uniform asymptotic stability of the zero solution of (2.15); see [6]. COROLLARY 3. Suppose that C(v) dv< A.
500 T.A. BURTON AND W.E. MAHFOUD Then the zero solution of (2.15) is uniformly asymptotically stable if only if A<0. COROLLARY 4. (i) If ç 0 ç tc(v)dv dt<1, then the zero solution of (2.15) is stable if only if A+ ç 0C(v)dv 0. (ii) If, in addition, C(t) Lj[0, ), j=1 or j=2, then the zero solution of (2.15) is asymptotically stable if only if A+ ç 0C(v)dv<0. (iii) If, furthermore, ç t C(v)) dv Lq[0, ), 0<q<2, then the zero solution of (2.15) is uniformly asymptotically stable if only if A+ C(v)dv<0. PROOF. We need only to show that A+ ç 0C(v)dv=0 implies stability in (i). In fact, by letting C1(t) ß0 in Theorem 4, we have C2(t)=C(t), P=0, (i) of Theorem 4 reduces to (J-1) L <0. If L 0, the result follows from Theorem 4. If L=0, the result also follows from [3, Theorem 5]. In [4] Grossman Miller gave a characterization of the uniform asymptotic stability of the zero solution of the convolution system (2.19) x'=ax+ çt0c(t-s)x(s)ds in terms of the location of the zeros of det(s-a-c(s)) in the complex plane; C(s) denotes the Laplace transform of C(t). THEOREM (Grossman-Miller). Suppose C L1[0, ). Then the zero solution of (2.19) is uniformly asymptotically stable if only if det[si-a-c(s)] 0 for Re s 0. Letting F(s)=s-A-C(s), we have F(0)=-A- ç 0C(t)dt. Thus, in the case n=1, if A+ ç 0C(t)dt 0, then F(0) 0. Since F(s) as s along the real axis, then F(s) has zeros in the right halfplane. By the above theorem, the zero solution of (2.19) is not uniformly asymptotically stable. In other words, A+ ç 0C(t)dt<0 is a necessary condition for uniform asymptotic stability. But, when C(t) 0, it follows from Corollary 3 that A+ ç 0C(v)dv<0 is a sufficient condition for uniform asymptotic stability. Thus, when C(t) 0, Grossman-Miller's result is equivalent to the condition A+ ç 0C(v)dv<0. This result was also obtained by Brauer [2] under stronger conditions on C(t). In general,
VOLTERRA EQUATIONS 501 such a result is much easier to apply than Grossman-Miller's result. In fact, apart from kernels such as C(t)=ke-at, ƒ >0, the location of the zeros of F(s) is not an easy task, especially when A>0. When C(t) 0 or C(t) changes sign, the condition A+ ç 0C(v)dv<0 is no longer sufficient for uniform asymptotic stability. In this case, we may apply Corollary 3 if A<0 A sufficiently large or Corollary 4 if A+ ç 0C(v)dv<0 ç 0 ç tc(v)dv dt<1. The last condition is mild if A+ ç 0C(v)dv is small; see Example 2 of [3]. Here is another example. EXAMPLE 4. Consider the equation x'= çt0c(t-s)x(s)ds, where with b>0. As A+ ç 0C(v)dv=0 ç 0 ç tc(v)dv dt=2ƒîb, we conclude, by Corollary 4, that the zero solution is stable if b<1/(2ƒî). However, for b=ƒ (ƒ 2+1)/(1-e-2ƒÎƒ ), ƒ >0, the function x(t)=exp(ƒ t) is a solution on [2ƒÎ, ). Since b 1/(2ƒÎ) as ƒ 0, b as ƒ, then the zero solution is unstable for every b>1/(2ƒî). If A+ ç 0C(v)dv is large, then the condition ç 0 ç tc(v)dv dt<1 may be severe. In this case, Theorem 3 provides us with a stability criterion under mild donditions. EXAMPLE 5. Consider the equation x'=-x-(1/a) çt0e-(t-s)/ax(s)ds, a>0. Here, A+ ç 0G(v)dv=-2, J=a, A+ ç 0 C(v) dv=0. Thus, Corollary 3 does not apply Corollary 4 yields uniform asymptotic stability only if a<1. However, the zero solution is uniformly asymptotically stable for all large values of a. If we choose C2(t)= -[exp(-2t)]/a C1(t)=[-exp(-t/a)+exp(-2t)]/a, then L=-1-1/(2a), J=1/(4a), P=1-1/(2a), P(J+1)+(J-1) L =-1/(2a). By Theorem 3, the zero solution is uniformly asymptotically stable.
502 T.A. BURTON AND W.E. MAHFOUD REMARK 3. Example 5 can easily be hled by Grossman-Miller's result. The purpose here is to show that, by an appropriate decomposition of the kernel C(t), Theorem 3 can yield a well refined criterion for uniform asymptotic stability. This criterion will be quite useful when the kernel is of the form C(t)=-k(ƒ t+1)-p as Grossman-Miller's result is very hard to apply. Below we give an effective decomposition of kernels of the form C(t)=(at-b)(ct+d)-p, where a 0, b>0, c>0, d>0. We ask that p>3/2 if a=0 p>5/2 if a 0. If a=0, then C(t)=-k(ƒ t+1)-p for some k>0 ƒ >0. In this case, we let C2(t)={ C(t) if 0 t ƒà, ƒà 0 -ƒá(ƒ t+1)-q if t ƒà, where ƒá=k(ƒ ƒà+1)q-p q>max(2, p). We choose ƒà q so that J<1, L=A+ ç C2(v)dv<0, P(J+1)+(J-1) L <0. If a 0, then C(t) changes sign. We may then write C(t) as C(t)=k1/(ƒ t+1)p-1-k2/(ƒ t+1)p, 0<k1<k2 let C1(t)=k1/(ƒ t+1)p-1 C2(t)=k2/(ƒ t+1)p. EXAMPLE 6. Consider the equation x'=(1/5)x-(1/3) çt0(t-s+1)-2x(s)ds. Here, C(t)=-(1/3)(t+1)-2. Thus, by choosing ƒà=6 q=4, we have A+ ç 0C2(v)dv=-32/315, P=2/63, J=(42In7-17)/126<1. Thus, all conditions of Theorem 4 are satisfied the zero solution is uniformly asymptotically stable. EXAMPLE 7. Consider the equation x'=ax+ çt0[(3t-3s+1)-2-ƒá(3t-3s+1)-3]x(s)ds, 0<ƒÁ<18. Letting C1(t)=(3t+1)-2 C2(t)=-ƒÁ(3t+1)-3, we have J=ƒÁ/18, P=1/3, A+ ç 0C2(v)dv=A-ƒÁ/6. If A<(1/3)[ƒÁ/2+(ƒÁ+18)/(ƒÁ-18)], then all conditions of Theorem 4 are satisfied the zero solution is uniformly asymptotically stable. Observe that the right-h side of the above inequality is an upper bounded for A which is maximum for ƒá= 18-6 ã2. Thus, for this choice of ƒá, we can take A=1/2. 3. Vector equations. We consider the system
VOLTERRA EQUATIONS 503(3.1) where L is a constant n ~n matrix, L1(t) is an n ~n matrix continuous for 0 t<, C1(t, s) H(t, s) are n ~n matrices continuous for 0 s t<, n 1. If B is any positive definite n ~n matrix, then there is a positive constant k such that (3.2) k x 2 xtbx for all x Rn. Let D be an n ~n symmetric matrix satisfying (3.3) LTD+DL=-I. For a detailed discussion of the existence of such a matrix D, see [1] [3]. Let (3.4) P=supt 0 çt0 C1(t,s) ds, (3.5) J=supt 0 çt0 H(t,s) ds, (3.6) m=supt 0 L1(t), (3.7)du assuming that ƒ³(t, s) exists for 0 s t<. THEOREM 5. Let D, P, J, m ƒ³ be defined by (3.3)-(3.7) suppose that for some constant ƒ, (1) J<1 (ii) D [P+2m+J(m+ L )+ƒ³(t,t)] ƒ <1. In addition, suppose there is a continuous function h: [0, ) [0, ) such that H(t, s) h(t-s) h(u) 0 as u. Then the zero solution of (3.1) is stable if only if D is positive definite. PROOF. Let
504 T.A. BURTON AND W.E. MAHFOUD The derivative of V(t, x( E)) along a solution x(t)=x(t,t0,ė) of (3.1) is given by Using (3.3), we may write Applying the Schwarz inequality as in the proof of Theorem 1, we obtain By (3.4)-(3.7), we have Suppose that D is positive definite let ċ>0 t0>0 be given. We must find ĉ>0, ĉ<ċ, so that if ė: [0, t0] Rn is continuous with ė(t) <ĉ,
VOLTERRA EQUATIONS 505 then x(t,t0,ƒó) <ƒã for all t t0. As V'3,1)(t,x( E)) 0, then Using (3.2), we may write Thus, x(t) (ƒân/r)+ çt0 H(t,s) x(s) ds. As long as x(t) <ƒã, we have x(t) (ƒâ/r)+ƒãj<for all t t0 provided that ƒâ<ƒâr(1-j)/n. Thus, the zero solution of (3.1) is stable. Now, suppose that x=0 is stable but D is not positive definite. Then there is an x0 0 such that xt0dx0 0. If k is any non-zero constant, then x1=kx0 0 xt1dx1 0. Let ƒã=1 t0=0. Since x=0 is stable, then there is a ƒâ>0 x1 0 such that x1 <ƒâ x(t,0,x1)<ƒã for all t 0. Let x(t)=x(t,0,x1). Then along the solution x(t), we have V(0, x1)=xt1dx1 0 V'(t,x( E)) -ƒáx2 so that for some t1>0, V(t1,x( E))<0 V(t,x( E)) V(t1,x( E))-ƒÁ çtt1x(s)2ds=-ƒå-ƒá çtt1x(s)2ds, where =-V(t1,x( E))>0 t t1. Thus, Since x(t)<1 for all t 0 ƒå+ƒátt1x(s) 2ds D[x(t)+H(t,s)x(s)ds D(1+J)2 4D then x(t)2 is in L1[0, ). By the Schwarz inequality, H(t,s)x(s)ds]2 H(t,s) ds çh(t,s)x(s)2ds J çh(t-s)x(s)2ds.
506 T.A. BURTON AND W.E. MAHFOUD The last integral tends to zero as it is the convolution of an L1-function with a function tending to zero. Thus, çt0 (H(t, s) x(s) ds 0 as t hence, by (3.8), xt(t)dx(t) -ƒå/2 for all sufficiently large t. This implies that x(t) 2 ƒê for some ƒê>0 all sufficiently large t, a contradiction to x(t) 2 being L1. Thus, the assumption that D is not positive definite is false the proof is now complete. Let us apply Theorem 5 to the system (1.1) assuming that A(t)= A=constant the matrix D satisfies (3.9) ATD+DA=-I. Taking L1(t)=H(t,s)=0 in Theorem 5, we obtain the following, which is [3, Theorem 8]: COROLLARY 5. Suppose that A(t)=A=constant, D satisfies (3.9), there is a constant ƒ such that D [ çt0 C(t,s) ds+ ç t C(u,t) du] ƒ <1. Then the zero solution of (1.1) is stable if only if D is positive definite. Now, suppose that A(t) is not constant ç t A(v)dv exists in L1[0, ). Letting C1(t, s)=c(t,s)-a(t-s) H(t)=- ç ta(v)dv, Equation (1.1) takes the form x'=-h(0)x+a(t)x+ çt0 C1(t,s)x(s)ds+d/dt ç0th(t-s)x(s)ds with (3.10) P=supt 0 çt0 C(t,s)-A(t-s) ds, (3.11) J= ç 0 ç ta(v)dv dt, (3.12) m=supt 0 A(t), (3.13) [AT(t)D+DA(t)]dt=-I. THEOREM 6. Let P, J, m D be defined by (3.10)-(3.13) suppose there is a constant a such that
VOLTERRA EQUATIONS 507 Then the zero solution of (1.1) is stable if only if D is positive definite. PROOF. As H(t,s)=H(t-s), then ç t H(u,t)du= H(v) dv=j. Also, L=-H(0)= ç 0A(v)dv (ii) of Theorem 5 reduces to (3.14). But, (3.14) implies that 2 D JA ç 0(t)dt<1 (3.13) implies that 2 D ç 0A(t)dt n. Thus, J<(1/n) 1 hence (i) of Theorem 5 is satisfied. Taking h(t)=h(t), we have all the conditions of Theorem 5 satisfied. This completes the proof. In the special case where C(t,s)=C(t-s), Equation (1.1) reduces to (3.15) x'=a(t)x+ çt0c(t-s)x(s)ds P of (3.10) reduces to (3.16) P= ç C(v)-A(v) dv. THEOREM 7. Let J, m, D P be defined by (3.11)-(3.13) (3.16) respectively, suppose that (3.17) (m+p)(j+1)+j ç 0A(t)dt <1/(2 D ). Then the zero solution of (3.15) is asymptotically stable if only if D is positive definite. PROOF. As C(t,s)=C(t-s), then C(u,C)-A(u-t) du= ç 0 C(v)-A(v) dv=p. Thus, (3.17) implies (3.14) the stability or instability of (3.15) follows from Theorem 6. To show asymptotic stability, we observe from (3.17) that m P are bounded. Hence, by (3.16) the assumption on A(t), it follows that C(t) is in L1[0, ). As x=0 is stable, then by (3.15), x'(t) is bounded. Using the functional V(t,x( E)) in the proof of Theorem 5, we see that along a solution x(t)=x(t,t0,ƒó) of (3.15), V'(t,x( E)) -ƒâ x2, ƒâ>0. Thus, x(t) 2 is in L1[0, ) since ( x(t) 2)'= (XT)'x+XTx' is bounded, then x(t) 0 as t. This completes the proof.
508 T.A. BURTON AND W.E. MAHFOUD COROLLARY 6. Let J, m P be defined by (3.11), (3.12), (3.16) respectively, suppose that n=1 (3.18) (m+p)(j+1)+(j-1) ç 0A(t)dt <0. Then the zero solution of (3.15) is asymptotically stable if only if A(t)dt<0. PROOF. For n=1 we have by (3.13), 2 D ç 0A(t)dt =1. Thus, (3.17) reduces to (3.18) this completes the proof. EXAMPLE 6. Consider the equation x'=-(1/2)ƒ 2e-ƒ tx-(1/2)ƒ 2 çt0e-ƒ (t-s)x(s)ds, 0<ƒ <1/3. Here, C(t)=A(t)=-ƒ 2te-t/2. Thus, P=0, m=ƒ 2/2, J=1/2, A(t)dt=-ƒ /2<0. Hence (m+p)(j+1)+(j-1) ç 0A(t)dt =ƒ (3ƒ -1)/4<0,, by Corollary 6, the zero solution is asymptotically stable. Now, we present another interesting application of Theorem 5. Consider the scalar equation (3.19) x'=ax+ çt0[k(t-s)+c(t,s)]x(s)ds in which A is a constant, C(t, s) is continuous for 0 s t<, k: [0, ) (-, ) is differentiable with k' in L'[0, ). We differentiate (3.19) to obtain x"=ax'+k(0)x+ çt0k'(t-s)x(s)ds+d/dt çt0c(t,s)x(s)ds. Let x'=y. Thus, y'=k(0)x+ay+ çt0k'(t-s)x(s)ds+d/dt çt0c(t,s)x(s)ds. Then we have z'=lz+ çt0c1(t-s)z(s)ds+d/dt çt0h(t,s)z(s)ds, where
VOLTERRA EQUATIONS 509 (3.20) Thus, (3.21) P= ç 0 k'(v) dv (3.22) J=supt 0 çt0 C(t,s) ds. If A<0 k(0)<0, then L is a stable matrix there is a symmetric positive definite matrix D such that (3.23) LTD+DL=-I. THEOREM 8. Let L, P, J D be defined by (3.20)-(3.23). If A<0, k(0)<0, there is a constant a such that (i) J<1 (ii) then the zero solution of (3,19) is stable. PROOF. If we observe that the condition H(t,s) h(t-s) is needed only to prove the converse of Theorem 5, then Theorem 8 is an immediate consequence of Theorem 5. REMARK 4. There is no integrability condition on the kernel in (3.19). Thus, if we take k(t)=k=constant C(t,s)=C(t-s), then (3.19) reduces to (3.24) x'=ax+ çt0[k+c(t-s)]x(s)ds. In this case P=0, (3.25) J= ç 0 C(v) dv, Theorem 8 reduces to the following:
510 T.A. BURTON AND W.E. MAUFOUD COROLLARY 7. Let L, D J be defined by (3.20), (3.23), (3.25) respectively. If A<0, k<0, 2 L D J<1, then the zero solution of (3.24) is stable. EXAMPLE 7. Consider the equation x'=-x+ çt0[-ƒ +ƒà(t-s+1)-2]x(s)ds, where ƒ ƒà are positive constants. As a simple calculation yields with a=(ƒ +1+1/ƒ )/2, b=1/(2ƒ ) c=(1+1/ƒ )/2. Thus, by Corollary 7, the zero solution is stable if ƒà(ƒ 2+4ƒ +8+8/ƒ )<1. system The following result is an extension of Theorem 4 to the convolution (3.26) x'=ax+ çt0c(t-s)x(s)ds, where A is an n ~n constant matrix C(t) is an n ~n matrix continuous for 0 t<. Let C(t)=C1(t)+C2(t), where Ci(t), i=1, 2, are continuous for 0 t<. We assume that C1(t) ç tc2(v)dv are L1-functions let (3.27) P= ç 0 C1(v) dv, (3.28) J= ç 0 ç tc2(v)dv dt, (3.29) L=A+ ç 0C2(v)dv.
VOLTERRA EQUATIONS 511 Let D be an n ~n symmetric matrix satisfying (3.30) LTD+DL=-I. THEOREM 9. Let P, J, L D be defined by (3.27)-(3.30). (i) If P(J+1)+J L <1/(2 D ), then the zero solution of (3.26) is stable if only if D is positive definite. (ii) If, in addition, C(t) is in Lj[0, ), j=1 or j=2, then the solution of (3.26) is asymptotically stable if only if D is positive definite. (iii) If, furthermore, J 0 ç t C(v) dv is in Lq[0, ), 0<q<2, then the zero solution of (3.26) is uniformly asymptotically stable if only if D is positive definite. PROOF. Let H(t)=- ç tc2(v)dv. Then (3.26) may be written as x'=lx+ çt0c1(t-s)x(s)ds+d/dt çt0h(t-s)x(s)ds. This is (3.1) with L1(t) ß0, C1(t,s)=C1(t-s) H(t,s)=H(t-s). Thus m=0, ç C1(u,t) du= ç 0 C1(v)dv=P, ç t H(u,t)du= H(v) dv=j. Hence, (ii) of Theorem 5 reduces to P(J+1)+J L < 1/(2 D ). This condition with (3.30) implies that J<1. If we let h(t)= H(t), then Part (i) of the theorem follows from Theorem 5. Parts (ii) (iii) are proved exactly in the same way as the corresponding parts of Theorem 4. REFERENCES [1] E.A. BARBASHIN, The construction of Lyapunov functions, Differential Equation 4 (1968), 1097-1112. (This is the translation of Differentsial'nye Uravneniya 4 (1968), 2127-2158). [2] F. BRAUER, Asymptotic stability of a class of integrodifferential equations, J. Differential Equations 28 (1978), 180-188. [3] T.A. BURTON AND W.E. MAHFOUD, Stability criteria for Volterra equations, Trans. Amer. Math. Soc. 279 (1983), 143-174. [4] S.I. GROSSMAN AND R.K. MILLER, Nonlinear Volterra integrodifferential systems with L1-kernels, J. Differential Equations 13 (1973), 551-566. [5] G.S. JORDAN, Asymptotic stability of a class of integrodifferential systems, J. Differential Equations 31 (1979), 359-365. [6] R.K. MILLER, Asymptotic stability properties of linear Volterra integrodifferential equations, J. Differential Equations 10 (1971), 485-506. DEPARTMENT OF MATHEMATICS SOUTHERN ILLINOIS UNIVERSITY CARBONDALE, ILLINOIS 62901 AND DEPARTMENT OF MATHEMATICS MURRAY STATE UNIVERSITY MURRAY, KENTUCKY 42071