INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int J Robust Nonlinear Control (2012) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/rnc.2885 New criteria for exponential stability of nonlinear time-varying differential systems Pham Huu Anh Ngoc* ,† International University, Vietnam National University-HCMC, Thu Duc, Ho Chi Minh City, Vietnam SUMMARY General nonlinear time-varying differential systems are considered An explicit criterion for exponential stability is presented Furthermore, an explicit robust stability bound for systems subjected to nonlinear time-varying perturbations is given In particular, it is shown that the generalized Aizerman conjecture holds for positive linear systems Some examples are given to illustrate obtained results Copyright © 2012 John Wiley & Sons, Ltd Received 25 April 2012; Revised July 2012; Accepted 10 July 2012 KEY WORDS: nonlinear differential system; time-varying; exponential stability; perturbation INTRODUCTION AND PRELIMINARIES Stability analysis of time-varying differential systems is always a central issue of control theory of dynamical systems Problems of stability and robust stability of time-varying differential systems have attracted much attention from researchers and have been studied intensively during the past decades (see, e.g [1–20] and references therein) In this paper, we investigate exponential stability of general nonlinear time-varying differential systems of the form x.t P / D f t , x.t //, t> > (1) Stability analysis of the nonlinear time-varying differential system (1) is, in general, hard Several approaches have been proposed in the literature, and most of them are based on the classical Lyapunov method and its variants (see, e.g [1, 2, 10–15]) In the present paper, we propose a new approach to problems of stability and robust stability of the nonlinear time-varying differential system (1) Our approach is based on the celebrated Perron–Frobenius theorem and ideas of the comparison principle We first present a new explicit criterion for exponential stability of the nonlinear time-varying differential system (1) Then, we give an explicit robust stability bound for (1) subjected to nonlinear time-varying perturbations In particular, we show that the generalized Aizerman conjecture holds for positive linear systems Let N be the set of all natural numbers For given m N, let us denote m WD ¹1, 2, : : : , mº Let K D C or R where C and R denote the sets of all complex and all real numbers, respectively For integers l, q > 1, Kl denotes the l-dimensional vector space over K, and Kl q stands for the set of all l q-matrices with entries in K Inequalities between real matrices or vectors will be understood componentwise; that is, for two real matrices A D aij / and B D bij / in Rł q , we write A > B if aij > bij for i D 1, , l, j D 1, , q In particular, if aij > bij for i D 1, , l, j D 1, , q, then we write A B instead of A > B We denote by RlC q the set of all nonnegative matrices *Correspondence to: Pham Huu Anh Ngoc, International University, Vietnam National University-HCMC, Thu Duc, Ho Chi Minh City, Vietnam † E-mail: phangoc@hcmiu.edu.vn Copyright © 2012 John Wiley & Sons, Ltd P H A NGOC A > Similar notations are adopted for vectors For x Kn and P Kl q , we define jxj D jxi j/ and jP j D jpij j A norm k k on Kn is said to be monotonic if kxk kyk whenever x, y Kn , jxj jyj Every p-norm on Kn (kxkp D jx1 jp C jx2 jp C C jxn jp / p , p < and kxk1 D maxi D1,2,:::,n jxi j), is monotonic Throughout the paper, if otherwise not stated, the norm of vectors on Kn is monotonic, and the norm of a matrix P Kl q is understood as its operator norm associated with a given pair of monotonic vector norms on Kl and Kq , that is kP k D max¹kP yk W kyk D 1º Note that P Kl q , Q RlC q , jP j Q ) kP k k jP j k kQk, (2) see, e.g [21] In particular, if Kn is endowed with k k1 or k k1 , then kAk D kjAjk for any A D aij / Kn n More precisely, one has kAk1 D kjAjk1 D max 16j 6n n X jaij jI kAk1 D kjAjk1 D max 16i 6n i D1 n X jaij j j D1 Let Br WD ¹x Rn W kxk rº for given r > For any matrix M C n n , the spectral abscissa of M is denoted by M / D max¹< W M /º, where M / WD ¹´ C W det.´In M / D 0º is the spectrum of M A matrix A Rn n is called Hurwitz stable if A/ < A matrix M Rn n is called a Metzler matrix if all off-diagonal elements of M are nonnegative We now summarize in the following theorem some properties of Metzler matrices Theorem 1.1 ([21]) Suppose that M Rn n is a Metzler matrix Then (i) (Perron–Frobenius) M / is an eigenvalue of M , and there exists a nonnegative eigenvector x ¤ such that M x D M /x (ii) Given ˛ R, there exists a nonzero vector x > such that M x > ˛x if and only if M / > ˛ (iii) tIn M / exists and is nonnegative if and only if t > M / (iv) Given B RnC n , C C n n Then jC j B H) M C C / M C B/ The following is immediate from Theorem 1.1 and is used in what follows Theorem 1.2 Let M Rn (i) (ii) (iii) (iv) (v) n be a Metzler matrix Then the following statements are equivalent M / < 0; 0I Mp for some p Rn , p M is invertible and M 0; For given b Rn , b 0, there exists x RnC , such that M x C b D n For any x RC n ¹0º, the row vector x T M has at least one negative entry CRITERIA FOR EXPONENTIAL STABILITY Consider a nonlinear time-varying differential system of the form (1), where f W RC Rn ! Rn is continuous and is locally Lipschitz in the second argument, uniformly in t on compact intervals of RC and f t , 0/ D 0, for all t RC It is well known that for a fixed > and a given x0 Rn , there exists a unique local solution of (1) satisfying the initial condition x / D x0 Copyright © 2012 John Wiley & Sons, Ltd (3) Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc STABILITY OF NONLINEAR TIME-VARYING SYSTEMS This solution is continuously differentiable on Œ , / for some > and satisfies (1) for every t Œ , / (see, e.g [6]) It is denoted by x I , x0 / Furthermore, if the interval Œ , / is the maximum interval of existence of x I , x0 /, then x I , x0 / is said to be noncontinuable The existence of a noncontinuable solution follows from Zorn’s lemma, and the maximum interval of existence must be open Definition 2.1 The zero solution of (1) is said to be exponentially stable if there exist positive numbers r, K, ˇ such that for each RC and each x0 Br , the solution x I , x0 / of (1) and (3) exists on Œ , 1/ and furthermore satisfies kx.t I , x0 /k Ke ˇ t / kx0 k, 8t > When the zero solution of (1) is exponentially stable, we also say that (1) is exponentially stable We are now in the position to state the main result of this note Theorem 2.2 Suppose that for each t RC , f t , / is continuously differentiable on Rn Let  à @fi J.t , x/ WD t , x/ Rn n , t RC , x Rn , @xj be the Jacobian matrix of f t , / at x Assume that there exists a Hurwitz stable Metzler matrix A WD aij / Rn n such that for any t > and any x Rn , ˇ ˇ ˇ @fi ˇ @fi ˇ t , x/ i , i nI ˇ t , x/ˇˇ aij , i 6D j , i, j n (4) @xi @xj Then, (1) is exponentially stable Proof Let x0 Rn be given and let x.t / WD x.t I , x0 /, t Œ , / be a noncontinuable solution of (1) and (3) We first show that there exists ˇ > such that for any > and any r > and any x0 Br , we have kx.t I , x0 /k Ke ˇ t / , 8t Œ , /, (5) where K R is independent of t , and x0 Because A is a Hurwitz stable Metzler matrix, there exists p WD ˛1 , ˛2 , : : : , ˛n /T , ˛i > 0, 8i n such that Ap 0, (6) by (ii) of Theorem 1.2 Furthermore, (6) implies that Ap ˇp D ˇ.˛1 , : : : , ˛n /T , (7) for some sufficiently small ˇ > Fix r > and choose K > such that jx0 j Kp for any x0 Br Define u.t / WD Ke ˇ t / p, t Œ , 1/ Set x.t / WD x.t I , x0 /, t Œ , / Note that jx /j D jx0 j u / D Kp We claim that jx.t /j u.t / for any t Œ , / Assume on the contrary that there exists t0 > such that jx.t0 /j 66 u.t0 / Set t1 WD inf¹t , / W jx.t /j 66 u.t /º By continuity, t1 > , and there is i0 n such that jx.t /j u.t /, 8t Œ , t1 /I jxi0 t1 /j D ui0 t1 /, jxi0 t /j > ui0 t /, 8t t1 , t1 C /, (8) for some > By the mean value theorem [22], we have for each t R and for each i n à n ÂZ X @fi xP i t / D fi t , x.t // D fi t , x/ fi t , 0// D t , sx.t //ds xj t / @xj j D1 Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc P H A NGOC Thus, à n ÂZ X d @fi t , sx.t //ds xj t / jxi t /j D sgn.xi t //xP i t / D sgn.xi t // dt @xj j D1 ÂZ à à ÂZ n X @fi @fi t , sx.t //ds jxi t /j C sgn.xi t // t , sx.t //ds xj t / D @xi @xj j D1,j 6Di for almost any t Œ , / Then, (4) implies d jxi t /j i jxi t /j C dt n X aij jxj t /j, (9) j D1,j 6Di for almost any t Œ , / It follows that for any t Œ , / D C jxi t /j WD lim sup h!0C jxi t C h/j h jxi t /j h!0C n X i jxi t /j C D lim sup h Z t t Ch d jxi s/jds ds aij jxj t /j, j D1,j 6Di where D C denotes the Dini upper-right derivative In particular, it follows from (7) and (8) that 8/ D C jxi0 t1 /j ai0 i0 Ke ˇ t1 / n X ˛i0 C ai0 j Ke ˇ t1 / ˛j D Ke ˇ t1 / < K ˇ/e ˇ t1 / ai0 j ˛j j D1 j D1,j 6Di0 7/ n X ˛i0 D D C ui0 t1 / However, this conflicts with (8) Hence, jx.t I , x0 /j u.t / D Ke ˇ t / p, > 0I 8x0 Br I 8t Œ , / By the monotonicity of vector norms, this yields kx.t I , x0 /k K1 e ˇ t / , > 0I 8x0 Br I 8t Œ , /, for some K1 > Finally, we show that D 1, and so (1) is exponentially stable Seeking a contradiction, we assume that < Then it follows from (5) that x I , x0 / is bounded on Œ , / Furthermore, this together with (1) implies that x P / is bounded on Œ , / Thus, x / is uniformly continuous on Œ , / Therefore, limt ! x.t / exists, and x / can be extended to a continuous function on Œ ,  Then, one can find a solution of (1) through , x // to the right of This contradicts the noncontinuability hypothesis on x / Thus, must be equal to 1, and this completes the proof The following is immediate from Theorem 2.2 Corollary 2.3 Suppose f W Rn ! Rn is continuously differentiable Then, the system x.t P / D f x.t //, t> > 0, is exponentially stable provided that there exists a Hurwitz stable Metzler matrix A WD aij / Rn such that for any x Rn , ˇ ˇ ˇ @fi ˇ @fi ˇ x/ i , i nI ˇ x/ˇˇ aij , i 6D j , i, j n @xi @xj Copyright © 2012 John Wiley & Sons, Ltd n Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc STABILITY OF NONLINEAR TIME-VARYING SYSTEMS To state the next result, we now consider a linear time-varying differential system of the form x.t P / D A.t /x.t /, where A / W RC ! Rn n t> > 0, (10) , k m is a given continuous vector function Corollary 2.4 Let A.t / WD aij t //, t > Suppose there exists a Hurwitz stable Metzler matrix A WD aij / Rn so that for any t > i t / i , i nI n jaij t /j aij , 8i 6D j , i, j n Then, (10) is exponentially stable We illustrate the obtained results by a couple of examples Example 2.5 Consider the nonlinear time-varying differential equation  à 2t x.t P /D x.t / C sin x.t / , t C1 t> > (11) Á t , x/ D Clearly, (11) is of the form (1) with f t , x/ WD 32 x Csin t 22tC1 x Furthermore, f and @f @x Á C t 22tC1 cos t 22tC1 x are continuous on RC R Because @f t , x/ 12 , 8t > 0, 8x R, (11) is @x exponentially stable, by Theorem 2.2 Example 2.6 Consider the nonlinear time-varying differential system x1 t / t2 ˆ < xP t / D 4e x1 t / C 1Cx t / sin x2 t / Á ˆ t / : xP t / D sin 2tx 2e t x2 t / t C1 (12) where t > > Then, (12) can be represented in the form (1) with x1 4e t x1 C 1Cx sin x2 B C T f W RC R2 ! R2 I f t , x/ D t , x/ 7! @ Á A , x D x1 , x2 / R 2tx1 t2 sin t C1 2e x2 , (13) It is clear that f is continuous on RC R2 and f t , / is continuously differentiable on R2 for each t RC and f t , 0/ D 0, 8t RC Furthermore, the Jacobian matrix of f t , / is given by B J.t , x/ D @ x12 4e t C 2t t C1 1Cx12 cos / sin x2 2tx1 t C1 Á x1 x12 C1 cos x2 2e t2 C A, t > 0, x D x1 , x2 /T R2 à is Hurwitz stable and satisfies (4) with n D 2 Thus, (12) is exponentially stable, by Theorem 2.2  It is easy to check that the matrix A WD Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc P H A NGOC STABILITY OF PERTURBED SYSTEMS Suppose all hypotheses of Theorem 2.2 hold Thus, (1) is exponentially stable Consider a perturbed system of the form x.t P / D f t , x.t // C N X Dk t , x.t //Pk t , Ek t , x.t /// , t> > 0, (14) kD1 where N is a given positive integer and Dk W RC Rn ! Rn lk , Ek W RC Rn ! Rqk , (k N ) are given continuous functions and Pk W RC Rqk ! Rlk (k N ) are unknown continuous functions Furthermore, we assume that (H1 ) for each k N , Dk , Pk and Ek (k N ) are locally Lipschitz in the second argument uniformly in t on compact intervals of RC and Pk t , 0/ D 0, Ek t , 0/ D for all t RC ; n l q n l q (H2 ) there exist Dk RC k , Ek RCk and Pk RCk k (k N ) such that 8t RC , x Rn (15) jPk t , y/j Pk jyj, 8t RC , 8y Rqk (16) jDk t , x/j Dk , and jEk t , x/j Ek jxj, 8t RC , 8x Rn I Note that the preceding assumptions imply that the right hand side of (14) is continuous on RC Rn and is locally Lipschitz in the second argument Thus, (14) always has a unique local solution satisfying the initial condition (3) The main problem here is to find a positive number such that an arbitrary perturbed system of the form (14) remains exponentially stable whenever the size of perturbations is less than Remark 3.1 In particular, if Ek t , x/ WD Ek t /xI Pk t , y/ WD Pk t /y, t > 0I x Rn I y Rqk , P PN then perturbation term N kD1 Dk t , x.t //Pk t , Ek t , x.t /// becomes kD1 Dk t /Pk t /Ek t /x.t / The robust stability of linear time-varying system (10) under the time-varying multi-perturbations Dk t , x/ WD Dk t /I A.t / ,! A.t / C N X Dk t /Pk t /Ek t /, (17) kD1 has been analyzed in [7], and an abstract stability bound was given in terms of input–output operators Theorem 3.2 Assume that all hypotheses of Theorem 2.2 hold and A Rn hold and N X kPk k < kD1 maxi ,j 2N kEi A n is as in Theorem 2.2 If (H1 )–.H2 ) 1D jk , (18) then (14) remains exponentially stable Proof We divide the proof into two steps Step We claim that AC PN kD1 Á Dk Pk Ek < Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc STABILITY OF NONLINEAR TIME-VARYING SYSTEMS N P Because A is a Metzler matrix and Dk , Ek , Pk are nonnegative for any k N , A C Dk Pk Ek kD1 ! N P is a Metzler matrix We show that WD AC Dk Pk Ek < Assume on the contrary that kD1 > By the Perron–Frobenius theorem (Theorem 1.1 (i)), there exists x RnC , x 6D such that N X AC ! Dk Pk Ek x D x kD1 Let Q.t / D tIn A, t R Because A/ < 0, Q 0/ Q N X 0/ is invertible It follows that Dk Pk Ek x D x (19) kD1 Let i0 be an index such that kEi0 xk D maxk2N kEk xk It follows from (19) that kEi0 xk > Multiply both sides of (19) from the left by Ei0 to obtain N X Ei0 Q 0/ Dk Pk Ek x D Ei0 x kD1 It follows that N X kEi0 Q 0/ Dk kkPk kkEk xk > kEi0 xk kD1 Thus, max kEi Q 0/ i ,j 2N N X Dj k ! kPk k kEi0 xk > kEi0 xk, kD1 or equivalently, max kEi Q i ,j 2N 0/ Dj k N X kPi k > (20) kD1 On the other hand, the resolvent identity gives Q.0/ Q 0/ D Q.0/ Q 0/ (21) Because A is a Metzler matrix with A/ < and > 0, Theorem 1.1 (iii) yields Q.0/ > and Q / > Then, (21) implies that Q.0/ > Q / > Hence, Ei Q.0/ Dj > Ei Q / Dj > 0, for any i, j N By (2), kEi Q.0/ Dj k > kEi Q / Dj k, for any i, j N Thus, (20) implies that N X kPk k > kD1 maxi ,j 2N kEi Q.0/ 1D jk However, this conflicts with (18) Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc P H A NGOC Step The proof of Step is similar to that of Theorem 2.2 Let x0 Rn be given, and let x.t / WD x.t I , x0 /, t Œ , / be a noncontinuable solution of (14) satisfying the initial condition (3) We first show that there exists ˇ > such that for any > and any r > and any x0 Br , we have kx.t I , x0 /k Ke ˇ t / , 8t Œ , /, (22) where K is independent of t , , x0 By (ii) of Theorem 1.2, there exists p WD ˛1 , ˛2 , : : : , ˛n /T , ˛i > 0, 8i n such that ! N X Dk Pk Ek p AC (23) kD1 Furthermore, (23) implies that AC N X ! Dk Pk Ek p ˇp D ˇ.˛1 , : : : , ˛n /T , (24) kD1 for some sufficiently small ˇ > Fix r > and choose K > such that jx0 j Kp for any x0 Br Define u.t / WD Ke ˇ t / p, t Œ , 1/ Set x.t / WD x.t I , x0 /, t Œ , / Note that jx /j D jx0 j u / D Kp We show that jx.t /j u.t / for any t Œ , / P n n Taking (15)–(16) into account, we obtain the following Let N kD1 Dk Pk Ek WD bij / R estimate n n X X d jaij jjxj t /j C bij jxj t /j, (25) jxi t /j i jxi t /j C dt j D1 j D1,j 6Di for almost any t Œ , / The remainder of the proof is similar to that of the proof of Theorem 2.2 Suppose all hypotheses of Corollary 2.4 hold Thus, (10) is exponentially stable Consider a perturbed system of the form x.t P / D A.t /x.t / C N X Dk t , x.t //Pk t , Ek t , x.t /// , t> > 0, (26) kD1 where Dk , Pk and Ek , (k N ) are as previously The following is immediate from Theorem 3.2 Corollary 3.3 Let A Rn n be as in Corollary 2.4 Suppose H1 )–.H2 ) hold If (18) holds, then (26) is exponentially stable Corollary 3.4 Let A Rn n be a Hurwitz stable Metzler matrix Suppose Dk W RC ! Rn lk , Ek W RC ! Rqk n k N / are given continuous functions and Pk W RC ! Rlk qk is an unknown continuous function n l q n l q If there exist Dk RC k , Ek RCk and Pk RCk k k N / such that jDk t /j Dk I jEk t /j Ek I jPk t /j Pk , and (18) holds, then the perturbed system x.t P /D AC N X 8t > 0, ! Dk t /Pk t /Ek t / x.t /, t> > 0, (27) kD1 is exponentially stable Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc STABILITY OF NONLINEAR TIME-VARYING SYSTEMS Remark 3.5 When A Rn n is a Metzler matrix, the system x.t P / D Ax.t /, t > 0, (28) is positive That is, for any initial state x0 RnC , the corresponding trajectory of the system x.t , x0 / remains in RnC for all t > Positive dynamical systems play an important role in modeling of dynamical phenomena whose variables are restricted to be nonnegative They are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modeling transport and accumulation phenomena of substances (see, e.g [5, 23]) In particular, the problem of robust stability of the positive linear differential system (28) under the time-invariant structured perturbations A ,! A C DE has been studied in [9, 21, 24] For example, it has been shown in [9, Theorem 5] that if (28) is exponentially stable and positive and D, E are given nonnegative matrices, then a perturbed system of the form x.t P / D A C DE/x.t /, t > 0, remains exponentially stable whenever kk < kEA Dk This result has been extended to various classes of positive differential systems such as positive linear time-delay differential systems, positive linear functional differential systems, positive linear Volterra integro-differential systems, and so on (see, e.g [25–31]) Furthermore, the problem of robust stability of the positive system (28) under the time-invariant multi-perturbations A ,! A C N X Di i Ei , i D1 has been analyzed in [9] by techniques of -analysis Although there are many works devoted to the study of robust stability of differential systems, to the best of our knowledge, the problem of robust stability of the positive system (28) under the time-varying multi-perturbations A ,! A C N X Dk t /Pk t /Ek t /, kD1 has not yet been studied, and a result like Corollary 3.4 cannot be found in the literature We illustrate the obtained results by a couple of examples Example 3.6 We now reconsider (11) As shown in Example 2.5, (11) is exponentially stable Consider a perturbed equation given by  à  à 2t t2 x.t P /D x.t / C sin C be x.t / C sin.ax.t //, t > > (29) t C1 where a, b R are parameters Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc P H A NGOC Note that jbe t xj jbjjxj and j sin.ax/j jajjxj, for all t > 0, x R Thus, by Theorem (3.2), (29) is exponentially stable if jaj C jbj 12 Example 3.7 Consider a linear differential equation in R2 defined by x.t P / D Ax.t /, t > 0, (30) where  1 A WD à Clearly, (30) is positive and exponentially stable Consider a perturbed system given by x.t P / D A C D1 t /P1 t /E1 t / C D2 t /P2 t /E2 t // x.t /, (31) where  D1 t / WD E1 t / WD e sin t t2 à  , à cos2 t C1 ! 2t t C1 t > 0I D2 t / WD , 1Ct t > 0I E2 t / WD , t > 0I ! , t C1 t > 0, and P1 t / WD a.t /, b.t // R1 , t > and P2 t / WD c.t /, d.t // R1 , t > are unknown Note that for any t > 0, we have  à  à jD1 t /j D1 WD I jD2 t /j D2 WD I 1  jE1 t /j E1 WD 1 à  I jE2 t /j E2 WD 0 à and  E1 A D1 D E1 A D2 D E2 A D1 D E2 A    D2 D 1 1 1 0 1 0 àààà1 1 1 1 àààà1 1 1 à  D à  D à  D à  D 1 à I à I à I à Let R2 be endowed with 2-norm By Corollary 3.4, (31) is exponentially stable if a /, b /, c /, d / are continuous, bounded and satisfy r r sup ja.t /j/2 C sup jb.t /j/2 C sup jc.t /j/2 C sup jd.t /j/2 < p 34 t 2RC t 2RC t 2RC t 2RC Copyright © 2012 John Wiley & Sons, Ltd Int J Robust Nonlinear Control (2012) DOI: 10.1002/rnc STABILITY OF NONLINEAR TIME-VARYING SYSTEMS GENERALIZED AIZERMAN CONJECTURE As an application, we now deal with an extension of the classical Aizerman conjecture Generalized Aizerman conjecture: Let A Rn n , D Rn l , E Rq n be given For any the linear systems x.t P / D A C DE/x.t /,  Rl q >0 kk < , , (32) are asymptotically stable if and only if the origin is globally asymptotically stable for all nonlinear systems x.t P / D Ax.t / C DN t , Ex.t //, (33) where N W RC Rq ! Rl , N t , 0/ D 0, 8t > is continuous and is locally Lipschitz in the second argument, uniformly in t on any compact interval of RC and satisfies jN t , y/j P jyj, 8t > 0, 8y Rq and P Rl q , kP k < (34) In particular, when N W R ! R, y 7! N y/ is a scalar function and D, E R , the generalized Aizerman conjecture reduces to an equivalent form of the classical Aizerman conjecture (see, e.g [8, page 702]) The original Aizerman conjecture is stated first in [3] It is well known that in general, the Aizerman’s conjecture does not hold (see, e.g [32]) So a natural question is that under what conditions of A, D, E and N does the Aizerman’s conjecture hold? T n Theorem 4.1 If A Rn n is a Metzler matrix and D RnC l , E RqC n , then the generalized Aizerman conjecture holds In other words, if D RnC l , E RqC n , then the generalized Aizerman conjecture holds for linear positive time-varying differential systems Proof Suppose (32) is asymptotically stable for any  Rl q , kk < , for some > In particular, the unperturbed system x.t P / D Ax.t / is asymptotically stable It follows from Corollary 3.4 that (32) is asymptotically stable for any  Rl q , kk < kEA1 Dk (see also Remark 3.5) Furthermore, there exists 0 RlC q , k0 k D kEA1 Dk such that (32) is not asymptotically stable for  WD 0 (see, e.g [9]) It remains to show that (33) is globally asymptotically stable for any nonlinearity N satisfying (34) with WD kEA1 Dk Let N satisfy (34) with WD kEA1 Dk Because P Rl q , kP k < kEA1 Dk , (33) is globally asymptotically stable, by Corollary 3.3 Conversely, assume that (33) is globally asymptotically stable for any nonlinearity N satisfying (34) for some > Then, the unperturbed system x.t P / D Ax.t / is asymptotically stable As mentioned earlier, (32) is asymptotically stable for any  Rl q , kk < kEA1 Dk So we assume that > kEA Dk Note that (32) is not asymptotically stable for some 0 RlC q , k0 k D Dk This means that (33) is not globally asymptotically stable for N defined by N t , y/ WD 0 y, t > 0, y Rq This completes the proof kEA REFERENCES Aeyels D, Peuteman J A new asymptotic stability criterion for nonlinear time-variant differential equations IEEE Transactions on Automatic Control 1998; 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