DSpace at VNU: Exponential Stability of Functional Differential Systems

DSpace at VNU: Exponential Stability of Functional Differential Systems tài liệu, giáo án, bài giảng , luận văn, luận án...

Vietnam J Math DOI 10.1007/s10013-016-0193-z Exponential Stability of Functional Differential Systems Pham Huu Anh Ngoc1 · Cao Thanh Tinh2 Received: 13 April 2015 / Accepted: 22 September 2015 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016 Abstract We present a novel approach to exponential stability of functional differential systems Our approach is relied upon the theory of positive linear functional differential systems and a comparison principle Consequently, we get some comparison tests and explicit criteria for exponential stability of functional differential systems Two examples are given to illustrate the obtained results Keywords Functional differential systems · Exponential stability Mathematics Subject Classification (2010) Primary 34K20 · Secondary 34 K15 Introduction Functional differential systems have numerous applications in science and engineering They are used as models for a variety of phenomena in the life sciences, physics and technology, chemistry, and economics, see, e.g., [10, 15, 26] Problems of stability of functional differential systems have been studied intensively during the past decades, see, e.g., [1, 5–20, 26–29] and the references therein Recently, problems of exponential stability of functional differential systems have attracted much attention from researchers, see, e.g., [1, 12, 16, 18–23, 27–29] Pham Huu Anh Ngoc phangoc@hcmiu.edu.vn Cao Thanh Tinh tinhct@uit.edu.vn Department of Mathematics, International University, Vietnam National University-HCMC, Thu Duc District, Ho Chi Minh City, Vietnam Department of Mathematics, University of Information Technology, Vietnam National University-HCMC, Thu Duc District, Ho Chi Minh City, Vietnam P.H.A Ngoc, C.T Tinh The traditional approaches to analyze the stability of delay differential systems are Lyapunov’s method and its variants (Razumikhin-type theorems, Lyapunov–Krasovskii functional techniques), see, e.g., [6–13, 27–29] That is why most existing stability criteria in the literature for delay differential systems (even for linear delay time-invariant systems) are given in terms of matrix inequalities or differential inequalities To the best of our knowledge, there are not many explicit criteria for exponential stability of time-varying (linear and nonlinear) delay differential systems Furthermore, in general, it is difficult to construct Lyapunov functions for delay differential systems In this paper, we present a novel approach to the exponential stability of functional differential systems Our approach is based on the theory of positive linear functional differential systems (see [17]) and the comparison principle Consequently, we get some comparison tests for exponential stability of functional differential systems which are analogs of comparison test for convergence of infinite series In particular, we obtain some new explicit criteria for exponential stability of functional differential systems Roughly speaking, the main result of this paper (Theorem 6) says that “If a nonlinear (time-varying) functional differential system is bounded above by a positive linear time-invariant differential system and the linear system is exponentially stable then the nonlinear system is exponentially stable too” This is a nice surprise because it is very similar to the well-known Weierstrass M-test in the theory of infinite series of functions, see, e.g., [25] To the best of our knowledge, Theorem of this paper is original Furthermore, Theorem has potential applications, for example, it can be used to study the exponential stability of equilibria of various classes of neural networks such as Cohen–Grossberg neural networks, Hopfield-type neural networks and cellular neural networks, etc We proceed as follows In the next section, we give notation and preliminary results which will be used in what follows In Section 3, we present some comparison tests for exponential stability of functional differential systems Consequently, we derive explicit criteria for exponential stability of functional differential systems A discussion to the obtained results and two illustrative examples are given Preliminaries Let R be the set of all real numbers and let C be the set of all complex numbers For a complex number z, denote by z the real part of z For a natural number n, let n := {1, 2, , n} Let Rl×q be the set of all l × q-matrices with real entries In what follows, inequalities between real matrices or vectors will be understood componentwise, i.e., for two real l × q-matrices A = (aij ) and B = (bij ), the inequality A ≥ B means aij ≥ bij for l×q i = 1, , l, j = 1, , q The set of all nonnegative l × q-matrices is denoted by R+ n l×q For x ∈ R and P ∈ R , we define |x| = (|xi |) and |P | = (|pij |) A norm · on Rn is said to be monotonic if x ≤ y whenever x, y ∈ Rn , |x| ≤ |y| Every p-norm on Rn ( x p = (|x1 |p + |x2 |p + · · · + |xn |p ) p , ≤ p < ∞ and x ∞ = maxi=1,2, ,n |xi |) are monotonic A matrix A ∈ Rn×n is called a Metzler matrix if all the off-diagonal entries of A are nonnegative With a given matrix A = (aij ) ∈ Rn×n , one associates the Metzler matrix M(A) := (aˆ ij ) ∈ Rn×n , where aˆ ij := |aij |, i = j , i, j ∈ n; aˆ ii := aii , i ∈ n For a square matrix A ∈ Rn×n , the spectral abscissa of A is denoted by μ(A) = max{ λ : det(λIn − A) = 0} Exponential Stability of Functional Differential Systems We now summarize some properties of Metzler matrices which will be used in what follows Theorem [24] Suppose M ∈ Rn×n is a Metzler matrix Then (i) (ii) (iii) (iv) (Perron–Frobenius) μ(M) is an eigenvalue of M and there exists a nonnegative eigenvector x = such that Mx = μ(M)x Given α ∈ R, there exists a nonzero vector x ≥ such that Mx ≥ αx if and only if μ(M) ≥ α (tIn − M)−1 exists and is nonnegative if and only if t > μ(M) n×n Then Given B ∈ Rn×n + ,C ∈ C |C| ≤ B ⇒ μ(M + C) ≤ μ(M + B) The following is immediate from Theorem and is used in what follows Theorem Let M ∈ Rn×n be a Metzler matrix Then the following statements are equivalent (i) (ii) (iii) (iv) (v) μ(M) < 0; Mp for some p ∈ Rn , p 0; M is invertible and M −1 ≤ 0; 0, there exists x ∈ Rn+ such that Mx + b = 0; For given b ∈ Rn , b n For any x ∈ R+ \ {0}, the row vector x T M has at least one negative entry Let Rn be endowed with a vector norm · and C := C([−h, 0], Rn ) be a Banach space of all continuous functions on [−h, 0] with values in Rn normed by the maximum norm φ = maxθ∈[−h,0] φ(θ) Denote C+ := {ϕ ∈ C : ϕ(θ) ∈ Rn+ ∀θ ∈ [−h, 0]}, the positive convex cone of C For ϕ ∈ C , let |ϕ| ∈ C+ be defined by |ϕ|(s) := |ϕ(s)|, s ∈ [−h, 0] An operator L : C → Rn is called positive if Lϕ ∈ Rn+ for any ϕ ∈ C+ Let N BV ([−h, 0], Rn×n ) denote the Banach space of all matrix functions η(·) of bounded variation on [−h, 0], continuous from the left on [−h, 0], satisfying η(−h) = and endowed with the norm η = Var(η; −h, 0) Finally, η(·) is said to be increasing if η(θ2 ) ≥ η(θ1 ) for −h ≤ θ1 ≤ θ2 ≤ Exponential Stability of Functional Differential Systems Consider a linear functional differential system of the form x(t) ˙ = Ax(t) + Lxt , t ≥ 0, where for each t ≥ 0, xt ∈ C defined by xt (θ) = x(t + θ), θ ∈ [−h, 0] and A ∈ L : C → Rn is a bounded linear operator given by Lϕ := −h d[η(θ )]ϕ(θ), (1) Rn×n and ϕ ∈ C, with η(·) ∈ N BV ([−h, 0], Rn×n ) For given ϕ ∈ C , (1) always has a unique solution x(·, ϕ) such that x(θ) = ϕ(θ), θ ∈ [−h, 0], (2) P.H.A Ngoc, C.T Tinh see, e.g., [11] Then (1) is said to be exponentially stable, if there are constants M ≥ and α > such that ∀t ≥ 0, ∀ϕ ∈ C x(t, ϕ) ≤ Me−αt ϕ It is well-known that (1) is exponentially stable if and only if sup z : det zIn − A − −h ezθ d[η(θ )] = < 0, see, e.g., [11] Clearly, it is not easy to verify this condition So it is very interesting to find a subclass of systems, for which stability criteria are explicit and easy to verify in practice Definition [17] The system (1) is said to be positive if for any initial function ϕ ∈ C+ , the corresponding solution x(·, ϕ) of (1)–(2) satisfies x(t, ϕ) ∈ Rn+ for every t ≥ Theorem [17] The following statements are equivalent: (i) (1) is positive; (ii) A ∈ Rn×n is a Metzler matrix and L is positive; (iii) A ∈ Rn×n is a Metzler matrix and η(·) is increasing Let (1) be positive Then (1) is exponentially stable if and only if μ(A + η(0)) < (a) (b) Definition Let A0 ∈ Rn×n and η0 (·) ∈ N BV ([−h, 0], Rn×n ) be given The system x(t) ˙ = A0 x(t) + L0 xt , where L0 ϕ := −h d[η0 (θ)]ϕ(θ), t ≥ 0, (3) ϕ ∈ C , is said to be bounded above by (1) if A0 ≤ A and ∀ϕ ∈ C+ L0 ϕ ≤ Lϕ (4) For example, it is easy to check that the linear time delay differential system m x(t) ˙ = A0 x(t) + Ai x(t − hi ) + i=1 −h C(s)x(t + s)ds is bounded above by m x(t) ˙ = B0 x(t) + Bi x(t − hi ) + i=1 −h D(s)x(t + s)ds if and only if A i ≤ Bi , i ∈ {0, 1, , m} and C(s) ≤ D(s) ∀s ∈ [−h, 0] We are now in the position to state the first result of this paper Theorem (Comparison stability test for positive functional differential systems) Suppose (1) and (3) are positive and (3) is bounded above by (1) Then (i) (ii) If (1) is exponentially stable then (3) is exponentially stable If (3) is not exponentially stable then (1) is not exponentially stable Proof Since (1) and (3) are positive systems, it remains to show that μ(A0 + η0 (0)) ≤ μ(A + η(0)), by Theorem Fix x ∈ Rn+ and let ϕ0 (θ) = x ∀θ ∈ [−h, 0] Clearly, ϕ0 ∈ C+ Since (3) is bounded above by (1), (4) implies L0 ϕ0 ≤ Lϕ0 This gives (η(0)−η0 (0))x ≥ Exponential Stability of Functional Differential Systems Since this holds for any x ∈ Rn+ , η(0) ≥ η0 (0) Thus, A + η(0) = A0 + (A − A0 ) + η(0) ≥ A0 +η0 (0) By Theorem 1(iv), μ(A0 +η0 (0)) ≤ μ(A+η(0)) This completes the proof Remark (i) In the theory of infinite series, the comparison test for convergence of infinite series with nonnegative terms (see, e.g., [25]) asserts that if ≤ an ≤ bn for sufficiently large n ∈ N, then – – ∞ n=1 an ∞ n=1 bn converges if so does diverges if so does ∞ n=1 bn ; ∞ n=1 an Clearly, Theorem is an analog of the comparison test for convergence of infinite series for positive linear functional differential systems (ii) The assumption of positivity of systems imposed in Theorem cannot be removed To see it, we consider the following systems: x(t) ˙ = −x(t) + x(t − 1), t ≥ 0, x(t) ˙ = −2x(t) − bx(t − 1), t ≥ 0, (5) (6) where b > Clearly, (5) is positive and exponentially stable, by Theorem Furthermore, (6) is bounded above by (5) However, (6) may not be exponentially stable In fact, the characteristic equation of (6) is given by z + + be−z = 0, which is equivalent to (z + 2)ez + b = This equation has a root z0 with z0 ≥ if b > π + 2, see [11, Theorem A.5, p 416] Thus, (6) is not exponentially stable if b > π + Note that (6) is not a positive equation, by Theorem As shown in Remark 1, (3) may not be exponentially stable although it is bounded above by a positive and exponentially stable system Actually, (3) is exponentially stable under a slightly stronger condition Theorem (Comparison stability test for functional differential systems) Let A0 ∈ Rn×n and η0 (·) ∈ N BV ([−h, 0], Rn×n ) be given Suppose L : C → Rn , Lϕ := −h d[η(θ )]ϕ(θ), with η(·) ∈ N BV ([−h, 0], Rn×n ), is positive If |L0 ϕ| ≤ L|ϕ| for any ϕ ∈ C and x(t) ˙ = M(A0 )x(t) + Lxt , t ≥0 is exponentially stable then (3) is exponentially stable In other words, (3) is exponentially stable if μ(M(A0 ) + η(0)) < Proof Theorem is just a particular case of Theorem given below So, we omit the proof Remark It is well known that if |an | ≤ bn for sufficiently large n ∈ N and ∞ n=1 bn converges then ∞ n=1 an (absolutely) converges (see, e.g., [25]) Theorem gives an analog of the comparison test for convergence of infinite series, for exponential stability of linear functional differential systems To end this paper, we state and prove an extension of Theorem to nonlinear functional differential systems Consider a nonlinear time-varying functional differential equation of the form x(t) ˙ = f (t, x(t)) + g(t; xt ), t ≥ σ, (7) P.H.A Ngoc, C.T Tinh where (i) (ii) (iii) For any t ∈ R+ , xt (·) ∈ C is defined by xt (θ) := x(t + θ), θ ∈ [−h, 0] for given h > 0; f (·, ·) : R+ × Rn → Rn is a given continuous function and is locally Lipschitz in the second argument, uniformly in t on compact intervals of R+ and f (t, 0) = for all t ∈ R+ ; g(·; ·) : R+ × C → Rn is a given continuous function such that g(t; 0) = ∀t ∈ R+ and g(t; u) is (locally) Lipschitz continuous with respect to u on each compact subset of R+ × C It is well-known that for fixed σ ∈ R+ and given ϕ ∈ C , there exists a unique local solution of (7), denoted by x(·, σ, ϕ) satisfying the initial value condition xσ (s) = ϕ(s), s ∈ [−h, 0], (8) see, e.g., [11] This solution is defined and continuous on [σ − h, γ ) for some γ > σ and satisfies (7) for every t ∈ [σ, γ ) see, e.g., [11, p 44] Furthermore, if the interval [σ − h, γ ) is the maximum interval of existence of the solution x(·, σ, ϕ) then x(·, σ, ϕ) 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 The zero solution of (7) is said to be (globally) exponentially stable if there exist positive numbers K, β such that for each σ ∈ R+ and each ϕ ∈ C , the solution x(·, σ, ϕ) of (7)–(8) exists on [σ − h, +∞) and furthermore satisfies x(t, σ, ϕ) ≤ Ke−β(t−σ ) ϕ ∀t ≥ σ Theorem Let for each t ∈ R+ , f (t, ·) be continuously differentiable on Rn Suppose there exist a matrix A := (aij ) ∈ Rn×n and a positive linear bounded operator L : C → Rn defined by Lϕ = −h d[η(θ )]ϕ(θ), with η(·) ∈ N BV ([−h, 0], Rn×n ) such that ∂fi (t, x) ≤ aii ∂xi ∀i ∈ n; ∂fi (t, x) ≤ aij ∂xj ∀i, j ∈ n, i = j, (9) for any t ∈ R+ and for any x ∈ Rn and |g(t; ϕ)| ≤ L|ϕ| ∀t ∈ R+ , ∀ϕ ∈ C (10) The zero solution of (7) is exponentially stable provided (1) is exponentially stable In other words, the zero solution of (7) is exponentially stable if μ(A + η(0)) < Remark Theorem is an analog of the Weierstrass M-test for infinite series of functions (see, e.g., [25, Theorem 7.10, p 134]) Roughly speaking, if the nonlinear functional differential system (7) is “bounded above” by the positive linear functional differential system (1) then the zero solution of (7) is exponentially stable provided so is (1) Proof of Theorem Since A ∈ Rn×n is a Metzler matrix and L is a positive linear bounded operator, the system (1) is positive Therefore, η(0) ≥ η(−h) = and thus A + η(0) is also a Metzler matrix It follows from μ(A + η(0)) < that (A + η(0))p (11) Exponential Stability of Functional Differential Systems for some p := (α1 , α2 , , αn )T ∈ Rn , αi > ∀i ∈ n, by Theorem For given ϕ ∈ C , let x(t) := x(t, σ, ϕ), t ∈ [σ − h, γ ) be a noncontinuable solution of (7)–(8) and let y(t) := y(t, |ϕ|), t ∈ [0, ∞) be the solution of (1) such that y(s) = |ϕ|(s), s ∈ [−h, 0] We show that |x(t + σ )| ≤ y(t) ∀t ∈ [−h, γ − σ ) Fix ζ > and define u(t) := y(t) + ζp ∀t ∈ [−h, ∞) Clearly, |x(t + σ )| = |ϕ(t)| = y(t) y(t) + ζp = u(t) ∀t ∈ [−h, 0] We show that |x(t + σ )| ≤ u(t) ∀t ∈ [0, γ − σ ) Assume on the contrary that there exists t∗ ∈ [0, γ − σ ) such that |x(t∗ + σ )| u(t∗ ) Set tb := inf{t ∈ [0, γ − σ ) : |x(t + σ )| u(t)} By continuity, tb > and there is i0 ∈ n such that |x(t + σ )| ≤ u(t) ∀t ∈ [0, tb ); |xi0 (tb + σ )| = ui0 (tb ); |xi0 (τk + σ )| > ui0 (τk ), τk ∈ (tb , tb + k1 ), (12) for some τk ∈ (tb , tb + k1 ), k ∈ N By the mean value theorem [3], we have for each t ∈ [σ, γ ) and for each i ∈ n n x˙i (t) = (fi (t, x(t)) − fi (t, 0)) + gi (t, xt ) = j =1 dfi (t, sx(t))ds xj (t) + gi (t, xt ) dxj Thus, df d |xi (t)| = sgn(xi (t))x˙i (t) ≤ dt i dxi (t, sx(t))ds |xi (t)| n + j =1,j =i dfi (t, sx(t)) ds |xj (t)| + |gi (t, xt )| dxj for almost any t ∈ [σ, γ ) Taking (9) and (10) into account, we obtain d |xi (t)| ≤ aii |xi (t)| + dt n n aij |xj (t)| + j =1,j =i j =1 −h d[ηij (θ)]|xj (t + θ)| for almost any t ∈ [σ, γ ) It follows that for any t ∈ [σ, γ ) D + |xi (t)| := lim sup |xi (t + )| − |xi (t)| →0+ = lim sup →0+ n n ≤ aii |xi (t)| + aij |xj (t)| + j =1,j =i t+ j =1 −h t d |xi (s)|ds ds d[ηij (θ)]|xj (t + θ)|, where D + denotes the Dini upper-right derivative In particular, it follows from (11)–(12) that D + |xi0 (tb + σ )| ≤ n n ai0 j |xj (tb + σ )| + j =1 (12) j =1 −h n ≤ n ai0 j (yj (tb ) + ζ αj ) + j =1 ⎛ ≤ D + yi0 (tb ) + ζ ⎝ d[ηi0 j (θ)]|xj (tb + σ + θ)| j =1 −h n d[ηi0 j (θ)](yj (tb + θ) + ζ αj ) n ai0 j αj + j =1 j =1 ⎞ (11) (ηi0 j (0))αj ⎠ < D + yi0 (tb ) P.H.A Ngoc, C.T Tinh However, this conflicts with (12) Therefore, |x(t + σ )| ≤ y(t) + ζp ∀t ∈ [0, γ − σ ) Without loss of generality, suppose Rn is endowed with a monotonic norm This implies x(t + σ ) = |x(t + σ )| ≤ u(t) = y(t) + ζp ≤ y(t) + ζ p ∀t ∈ [0, γ − σ ) (13) Letting ζ tend to zero in (13), we obtain x(t + σ ) = |x(t + σ )| ≤ y(t) ∀t ∈ [0, γ − σ ) (14) Since (1) is exponentially stable, there are K ≥ and α > such that y(t) ≤ Ke−αt ϕ ∀t ≥ 0, ∀ϕ ∈ C (15) ∀t ∈ [σ, γ ), ∀ϕ ∈ C (16) Then (14) and (15) imply x(t) ≤ Ke−α(t−σ ) ϕ Finally, we show that γ = ∞ and so the zero of (7) is exponentially stable Seeing a contradiction, we assume that γ < ∞ Then it follows from (16) that x(·, σ, ϕ) is bounded on [σ, γ ) Furthermore, this together with (7), (9), and (10) imply that x(·) ˙ is bounded on [σ, γ ) Thus, x(·) is uniformly continuous on [σ, γ ) Therefore, limt→γ − x(t) exists and x(·) can be extended to a continuous function on [σ, γ ] Moreover, the closure of {xt : t ∈ [σ, γ )} is a compact set in C by the Arzela–Ascoli theorem [3] Note that {(t, xt ) : t ∈ [σ, γ )} ⊂ [σ, γ ] × the closure of {xt : t ∈ [σ, γ )} Thus, the closure of {(t, xt ) : t ∈ [σ, γ )} is a compact set in R × C Since (γ , xγ ) belongs to this compact set, one can find a solution of (7) through this point to the right of γ This contradicts the noncontinuability hypothesis on x(·) Thus, γ must be equal to ∞ This completes the proof Discussion and Illustrative Examples Consider the linear delay differential equation x(t) ˙ = −ax(t) + b(t)x(t − h), t ∈ R+ , (17) where a > 0, h > 0, and b(·) is a bounded continuous function on R+ By applying a Razumikhin-type theorem to (17), it has been shown in [13, Example 5.1, p 74] that (17) is exponentially stable if b := supt∈R+ |b(t)| < a Note that (17) is bounded above by the positive linear delay differential equation x(t) ˙ = −ax(t) + bx(t − h), t ∈ R+ (18) Because of −a + b < 0, (18) is exponentially stable, by Theorem Therefore, the above assertion is immediate from Theorem More generally, the linear differential equation with delays of the form x(t) ˙ = a(t)x(t) + b(t)x(t − h1 ) + c(t)x(t − h2 ), t ∈ R+ , h1 , h2 > is exponentially stable, by Theorem provided a(·) ∈ C(R+ , R) satisfies a(t) < −a, t ∈ R+ for some a > and b(·), c(·) ∈ C(R+ , R) are bounded such that sup |b(t)| + sup |c(t)| < a t∈R+ t∈R+ Exponential Stability of Functional Differential Systems Next, using a Lyapunov–Krasovskii functional, it has been proven in [10, p 154] that the zero solution of the nonlinear delay differential system x(t) ˙ = Ax(t) + f (x(t − h)) is asymptotically stable provided A ∈ Rn×n is a Metzler matrix and f (·) : Rn → Rn is locally Lipschitz such that f (0) = 0, f (x) ≥ ∀x ∈ Rn and f (x) ≤ γ x ∀x ∈ Rn+ and μ(A + γ In ) < for some γ > (compare with Theorem 6) Furthermore, consider the nonlinear delay differential system x(t) ˙ = Ax(t) + F (t; x(t − h1 ), , x(h − hm )) , t ∈ R+ , (19) Rn×n and F (·) is continuous in all its arguments It has been where h1 , , hm > 0, A ∈ shown in [12, Theorem 3.1] that (19) is exponentially stable if m F (t; u1 , , um ) ≤ ∀t ∈ R+ , ∀u1 , , um ∈ Rn , βi ui (20) i=1 and m γ (A) + βi < 0, i=1 where γ (A) := limε→0+ εA+Iε n −1 is the matrix measure of A The spirit of this result is very close to our ideas Note that (20) means that (19) is bounded above by the scalar positive linear delay differential equation m βi y(t − hi ), y(t) ˙ = γ (A)y(t) + t ∈ R+ , y(t) ∈ R (21) i=1 Clearly, (21) is positive and exponentially stable, by Theorem This ensures that (19) is exponentially stable If, instead of (20), one assumes that m |F (t; u1 , , um )| ≤ βi |ui | ∀t ∈ R+ , ∀u1 , , um ∈ Rn , (22) i=1 then [12, Theorem 3.1] follows directly from Theorem To end this section, we illustrate Theorem by two examples to which Theorem 3.1 of [12] cannot be applied Example Consider the scalar differential equation with delay x(t) ˙ = −2x(t) + sin(t + x(t)) + −h r(t, s, x(t + s))ds, t ≥ 0, (23) where r(·, ·, ·) : R+ × [−h, 0] × R → R is a given continuous function and r(t, s, u) is Lipschitz continuous with respect to u on each compact subset of R+ × [−h, 0] × R and r(t, s, 0) = ∀(t, s) ∈ R+ × [−h, 0] Assume that there exists a continuous function m(·) : [−h, 0] → R+ such that |r(t, s, u)| ≤ m(s)|u| ∀t ∈ R+ , ∀s ∈ [−h, 0], ∀u ∈ R Let f (t, x) := −2x + sin(t + x), t ∈ R+ , x ∈ R, and g(t; ϕ) := −h r(t, s, ϕ(s))ds, t ∈ R+ , s ∈ [−h, 0], ϕ ∈ C([−h, 0], R) P.H.A Ngoc, C.T Tinh Then ∂f (t, x) = −2 + cos(t + x) ≤ −1, ∂x and |g(t; ϕ)| ≤ −h m(s)|ϕ(s)|ds, t ∈ R+ , x ∈ R, ϕ ∈ C([−h, 0], R) Thus, the zero solution of (23) is exponentially stable if −h m(s)ds < 1, by Theorem The next example gives an application of Theorem Example [2] Consider a bidirectional associative memory (BAM) model described by n u˙ i (t) = −di ui (t) + eij rj (uj (t − τij )) + Ii , i ∈ n, (24) j =1 where di > 0, Ii ∈ R, i ∈ n and E := (eij ) ∈ Rn×n and for each j ∈ n, rj : R+ → R is bounded and globally Lipschitz with constant Lj (i.e., |rj (uj ) − rj (vj )| ≤ Lj |uj − vj | for all uj , vj ) Let D := diag(− Ld11 , − dL2 , , − Ldnn ) ∈ Rn×n We show that if μ(D + |E|) < then (24) has a unique equilibrium point u∗ which is globally exponentially stable Consider the continuous function F : Rn → Rn defined by u := (u1 , u2 , , un ) → F (u) := (ξ1 , ξ2 , , ξn ) , T T ⎛ ⎝ ξi := di n ⎞ eij rj (uj ) + Ii ⎠ , i ∈ n j =1 Since for each j ∈ n, rj : R+ → R is bounded, there exist Mi > 0, i ∈ n such that for any u := (u1 , u2 , , un )T ∈ Rn ⎛ ⎞ n ⎝ eij rj (uj ) + Ii ⎠ ≤ Mi ∀i ∈ n di j =1 u∗ By the Brouwer’s fixed-point theorem, there exists u∗ ∈ Rn such that F (u∗ ) = u∗ Thus, is a equilibrium point of (24) Since D + |E| is a Metzler matrix and μ(D + |E|) < 0, it follows from Theorem that − di ζi + Li n |eij |ζj < ∀i ∈ n, j =1 for some (ζ1 , ζ2 , , ζn )T ∈ Rn , ζi > 0, i ∈ n Hence, n − d i ζi + Li |eij |ζj < ∀i ∈ n (25) j =1 By the coordinate translation z(t) = u(t) − u∗ , (24) can be written as n z˙ i (t) = −di zi (t) + eij sj (zj (t − τij )), i ∈ n, (26) j =1 where sj (x) := rj (x+u∗j )−rj (u∗j ), x ∈ R, j ∈ n Furthermore, u∗ is globally exponentially stable for (24) if and only if the trivial solution of (26) is globally exponentially stable Exponential Stability of Functional Differential Systems Note that for each j ∈ n, |sj (x)| = |rj (x +u∗j )−rj (u∗j )| ≤ Lj |x| for any x ∈ R and then (25) implies that the trivial solution of (26) is globally exponentially stable, by Theorem Clearly, u∗ is the unique equilibrium point of (24) which is globally exponentially stable In fact, Theorem can be used to study the exponential stability of equilibria of various classes of neural networks such as Cohen–Grossberg neural networks, Hopfield-type neural networks and cellular neural networks, etc Furthermore, Theorem and its variants include many existing results in the literature on exponential stability of equilibria of various neural networks, see, e.g., [2, 4, 28, 30] The result presented in Example was given firstly in [2] whose proof is based upon a Liapunov function Concluding Remarks A novel approach to exponential stability of functional differential systems is presented Some comparison tests and explicit criteria for exponential stability of functional differential systems are given The obtained results can be used to investigate the exponential stability of equilibria of neural networks Finally, it is worth noticing that the approach utilized in this paper can be developed to study the stability and robust stability problems of various classes of dynamical systems such as singular (delay) differential systems, delay differential systems of neutral type, integro-differential systems, delay integral systems, coupled differential-difference systems, etc Some of them will be carried out in the near future Acknowledgments This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant 101.01-2015.14 The authors would like to thank the anonymous referee for the appropriate comments and constructive suggestions References Berezansky, L., Braverman, E.: On stability of some linear and nonlinear delay differential equations J Math Anal Appl 314, 391–411 (2006) Cao, J., Wang, L.: Exponential stability and periodic oscillatory solution in BAM networks with delays IEEE Trans Neural Netw 13, 457–463 (2002) Dieudonn´e, J.: Foundations of Modern Analysis Academic Press, New York (1988) van den Driessche, P., Zou, X.: Global attractivity in delayed Hopfield neural network models SIAM J Appl Math 58, 1878–1890 (1998) Driver, R.D.: Existence and stability of solutions of a delay differential system Arch Ration Mech Anal 10, 401–426 (1962) Fridman, E.: New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems Syst Control Lett 43, 309–319 (2001) Fridman, E.: Stability of systems with uncertain delays: a new “complete” Lyapunov-Krasovskii functional IEEE Trans Autom Control 51, 885–890 (2006) Erneux, T.: Applied Delay Differential Equations Surveys and Tutorials in the Applied Mathematical Sciences, vol Springer, New York (2009) Goubet-Bartholomeuă s, A., Dambrine, M., Richard, J.P.: Stability of perturbed systems with time-varying delays Syst Control Lett 31, 155–163 (1997) 10 Haddad, W.M., Chellaboina, V., Hui, Q.: Nonnegative and Compartmental Dynamical Systems Princeton University Press, New Jersey (2010) 11 Hale, J., Lunel, S.M.V.: Introduction to Functional Differential Equations Springer, New York (1993) 12 Idels, L., Kipnis, M.: Stability criteria for a nonlinear nonautonomous system with delays Appl Math Model 33, 2293–2297 (2009) P.H.A Ngoc, C.T Tinh 13 Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations Academic Press (1986) 14 Kolmanovskii, V.B., Richard, J.P.: Stability of some linear systems with delays IEEE Trans Autom Control 44, 984–989 (1999) 15 Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics Mathematics in Science and Engineering, vol 191 Academic Press (1993) 16 Liu, X., Yu, W., Wang, L.: Stability analysis for continuous-time positive systems with time-varying delays IEEE Trans Autom Control 55, 1024–1028 (2010) 17 Ngoc, P.H.A., Naito, T., Shin, J.S.: Characterizations of positive linear functional differential equations Funkc Ekvacioj 50, 1–17 (2007) 18 Ngoc, P.H.A.: On exponential stability of nonlinear differential systems with time-varying delay Appl Math Lett 25, 1208–1213 (2012) 19 Ngoc, P.H.A.: Novel criteria for exponential stability of functional differential equations Proc Am Math Soc 141, 3083–3091 (2013) 20 Ngoc, P.H.A.: Stability of positive differential systems with delay IEEE Trans Autom Control 58, 203–209 (2013) 21 Ngoc, P.H.A., Tinh, C.T.: New criteria for exponential stability of linear time-varying differential systems with delay Taiwan J Math 18, 1759–1774 (2014) 22 Ngoc, P.H.A.: Stability of nonlinear differential systems with delay Evol Equ Control Theory 4, 493– 505 (2015) 23 Ngoc, P.H.A.: Novel criteria for exponential stability of nonlinear differential systems with delay IEEE Trans Autom Control 60, 485–490 (2015) 24 Son, N.K., Hinrichsen, D.: Robust stability of positive continuous-time systems Numer Funct Anal Optim 17, 649–659 (1996) 25 Rudin, W.: Principles of Mathematical Analysis McGraw-Hill Science, New York (1976) 26 Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences Texts in Applied Mathematics, vol 57 Springer, New York (2011) 27 Xueli, S., Jigen, P.: A novel approach to exponential stability of nonlinear systems with time-varying delays J Comput Appl Math 235, 1700–1705 (2011) 28 Zhang, J.: Globally exponential stability of neural networks with variable delays IEEE Trans Circuits Syst I Fundam Theory Appl 50, 288–290 (2003) 29 Zhang, B., Lam, J., Xu, S., Shu, Z.: Absolute exponential stability criteria for a class of nonlinear timedelay systems Nonlinear Anal RWA 11, 1963–1976 (2010) 30 Zeng, Z., Wang, J., Liao, X.: Global exponential stability of a general class of recurrent neural networks with time-varying delays IEEE Trans Circuits Syst I: Fundam Theory Appl 50, 1353–1358 (2003) ... convergence of infinite series, for exponential stability of linear functional differential systems To end this paper, we state and prove an extension of Theorem to nonlinear functional differential systems. .. approach to exponential stability of functional differential systems is presented Some comparison tests and explicit criteria for exponential stability of functional differential systems are... problems of various classes of dynamical systems such as singular (delay) differential systems, delay differential systems of neutral type, integro -differential systems, delay integral systems,