LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS

Abstract

The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.

1. INTRODUCTION

The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi [8] . For linear systems, the notions of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. In fact, uniformly Lipschitz stability lies somewhere between uniformly stability on one side and the notions of asmptotic stability in variation of Brauer[4] and uniformly stability in variation of Brauer and Strauss[3] on the other side. Gonzalez and Pinto[9] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems.

In this paper, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear differential systems. To do this we need some integral inequalities. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method.

 

2. PRELIMINARIES

We consider the nonlinear nonautonomous differential system

where and is the Euclidean n-space. We assume that the Jacobian matrix fx = ∂ f / ∂x exists and is continuous on and f(t, 0) = 0. Also, consider the perturbed differential system of (2.1)

where , g(t, 0) = 0. For , let For an n × n matrix A, define the norm |A| of A by |A| = sup|x|≤1 |Ax|.

Let x(t, t0, x0) denote the unique solution of (2.1) with x(t0, t0, x0) = x0, existing on [t0,∞). Then we can consider the associated variational systems around the zero solution of (2.1) and around x(t), respectively,

and

The fundamental matrix Φ(t, t0, x0) of (2.4) is given by

and Φ(t, t0, 0) is the fundamental matrix of (2.3).

Before giving further details, we give some of the main definitions that we need in the sequel[8].

Definition 2.1. The system (2.1) (the zero solution x = 0 of (2.1)) is called

(S) stable if for any 𝜖 > 0 and t0 ≥ 0, there exists 𝛿 = 𝛿(t0, 𝜖) > 0 such that if |x0| < 𝛿, then |x(t)| < 𝜖 for all t ≥ t0 ≥ 0,

(US) uniformly stable if the 𝛿 in (S) is independent of the time t0,

(ULS) uniformly Lipschitz stable if there exist M > 0 and 𝛿 > 0 such that |x(t)| ≤ M|x0| whenever |x0| ≤ 𝛿 and t ≥ t0 ≥ 0

(ULSV) uniformly Lipschitz stable in variation if there exist M > 0 and 𝛿 > 0 such that |Φ(t, t0, x0)| ≤ M for |x0| ≤ 𝛿 and t ≥ t0 ≥ 0,

(EAS) exponentially asymptotically stable if there exist constants K > 0 , c > 0, and 𝛿 > 0 such that

|x(t)| ≤ K |x0|e-c(t-t0), 0 ≤ t0 ≤ t

provided that |x0| < 𝛿,

(EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that

|Φ(t,t0,x0)| ≤ K e-c(t-t0), 0 ≤ t0 ≤ t

provided that |x0| < ∞.

We give some related properties that we need in the sequel.

We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system

where and g(t, 0) = 0. Let y(t) = y(t, t0, y0) denote the solution of (2.5) passing through the point (t0, y0) in × .

The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1].

Lemma 2.2. Let x and y be a solution of (.1) and (.5), respectively. If then for all t such that

Lemma 2.3 ([7]). Let u, λ1, λ2, w(u) be nondecreasing in u and w(u) ≤ w() for some v > 0. If , for some c > 0,

then

where u > 0, u0 > 0 W-1(u) is the inverse of W(u) and

Lemma 2.4 ([10]). Let u, p, q,w, and r ∈ C () and suppose that, for some c ≥ 0, we have

Then

Lemma 2.5 ([15]). Let u(t), f(t), and g(t) be real-valued nonnegative continuous functions defined on , for which the inequality

holds, where u0 is a nonnegative constant. Then,

Lemma 2.6 ([12]). Let u, λ1, λ2, λ3 ∈ C(), w ∈ C((0,∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

where W, W -1 are the same functions as in Lemma 2.3 and

Lemma 2.7 ([13]). Let u, p, q,w, r ∈ C(), w ∈ C((0,∞)) and w(u) be nondecreasing in u. Suppose that for some c ≥ 0,

Then

where and

Lemma 2.8 ([14]). Let the following condition hold for functions u(t), v(t) ∈ C[[t0,∞)) and k(t, u) ∈ C[[t0,∞) × , ):

t ≥ t0 and k(s, u) is strictly increasing in u for each fixed s ≥ 0. If u(t0) < v(t0), then u(t) < v(t), t ≥ t0 ≥ 0.

Lemma 2.9 ([5]). Let u, λ1, λ2, λ3 ∈ C(), w ∈ C((0,∞)) and w(u) be nondecreasing in u. Suppose that for some c > 0,

Then

where u > 0, u0 > 0, W-1(u) is the inverse of W(u) and

 

3. MAIN RESULTS

In this section, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear perturbed differential systems.

Theorem 3.1. Assume that x = 0 of (.1) is ULS. Let the following condition hold for (.):

where W(t, u) ∈ C( × , ) is monotone nondecreasing in u with W(t, 0) = 0. Suppose that u(t) is any solution of the scalar differential equation

existing on such that m(t0) < u(t0). If u = 0 of (3.1) is ULS, then y = 0 of (.) is also ULS whenever M|y0| < u0.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. Using the variation of constants formula, we have

where Φ(t, t0, y0) is the fundemental matrix of (2.4). Since x = 0 of (2.1) is ULS, it is ULSV by Corollary 3.6[5]. Thus there exist M > 0 and 𝛿 > 0 such that |Φ(t, t0, y0)| ≤ M for t ≥ t0 ≥ 0. Therefore, by the assmption, we have

Hence |y(t)| < u(t) by Lemma 2.8. Since u = 0 of (3.1) is ULS, it easily follows that y = 0 of (2.2) is ULS.

Corollary 3.2. Assume that x = 0 of (.1) is ULS. Consider the scalar differential equation

where u0 ≥ 1, K ≥ 1 and a, k ∈ C() satisfy the conditions

(a) where

(b)

Then y = 0 of (.) is ULS.

Proof. Let u(t) = u(t, t0, x0) be any solution of (3.2). Then, by Lemma 2.5 , we have

Hence u = 0 of (3.2) is ULS. This implies that the solution y = 0 of (2.2) is ULS by Theorem 3.1.

Remark 3.3. In Corollary 3.2, it is needed that b1 = ∞. The condition W(∞) = ∞ is too strong and it represents situations which are not stable. For example, if w(u) = u𝛼, then only 𝛼 ≤ 1 satisfies W(∞) = ∞ and 𝛼 < 1 is not stable. See [18].

Corollary 3.4. Assume that x = 0 of (.1) is ULS. Consider the scalar differential equation

where u0 ≥ 1, K ≥ 1, u,w ∈ C(), w(u) be nondecreasing in u and w(u)≤w() for some v > 0, and a, k ∈ C() satisfy the conditions

(a) where

(b) and a, k ∈ L1(). Then y = 0 of (.) is ULS.

Proof. Let u(t) = u(t, t0, x0) be any solution of (3.3). Then, by Lemma 2.3, we have

Hence u = 0 of (3.3) is ULS. By Theorem 3.1, the solution y = 0 of (2.2) is ULS.

Corollary 3.5. Assume that x = 0 of (.1) is ULS. Consider the scalar differential equation

where w ∈ C((0,∞), w(u) is nondecreasing on u and u ≤ w(u), u0 ≥ 1, K ≥ 1 and a, b, k ∈ C() satisfy the conditions

(a) where

(b) L1(). Then y = 0 of (.) is ULS.

Proof. Let u(t) = u(t, t0, x0) be any solution of (3.4). Then, Lemma 2.6, we have

Hence u = 0 of (3.4) is ULS, and so by Theorem 3.1, the solution y = 0 of (2.2) is ULS. □

Theorem 3.6. For the perturbed (.), we asssume that

where a, b, k ∈ C(), a, b, k ∈ L1(), w ∈ C((0,∞), and w(u) is nondecreasing in u,u ≤ w(u), and w(u) ≤ w() for some v > 0,

where M(t0) < ∞ and b1 = ∞. Then the zero solution of (.) is ULS whenever the zero solution of (.1) is ULSV.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. Since x = 0 of (2.1) is ULSV, it is ULS by Theorem 3.3[8]. Applying Lemma 2.2, we have

Set u(t) = |y(t)||y0|-1. Now an application of Lemma 2.6 yields

Hence we have |y(t)| ≤ M(t0)|y0| for some M(t0) > 0 whenever |y0| < 𝛿. This completes the proof. □

Theorem 3.7. For the perturbed (.), we asssume that

where a, b, k ∈ C(), a, b, k ∈ L1(), w ∈ C((0,∞), and w(u) is nondecreasing in u,u ≤ w(u), and w(u) ≤ w() for some v > 0,

where M(t0) < ∞ and b1 = ∞. Then the zero solution of (.) is ULS whenever the zero solution of (.1) is ULSV.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. Using the nonlinear variation of constants formula and the ULSV condition of x = 0 of (2.1), we have

Set u(t) = |y(t)||y0|-1. Now an application of Lemma 2.7 yields

Thus we have |y(t)| ≤ M(t0)|y0| for some M(t0) > 0 whenever |y0| < 𝛿, and so the proof is complete. □

Theorem 3.8. Let the solution x = 0 of (.1) be EAS. Suppose that the perturbing term g(t, y) satisfies

where 𝛼 > 0, a, b, k ∈ C(), a, b, k ∈ L1(), w(u) is nondecreasing in u, and w(u) ≤ w() for some v > 0. If

where c = |y0|Me𝛼t0 , then all solutions of (.) approch zero as t → ∞

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. Since the solution x = 0 of (2.1) is EAS, we have |Φ(t, t0, x0)| ≤ Me-𝛼(t-t0) for some M > 0 and c > 0(Theorem 2[2]). Using Lemma 2.2, we have

since e𝛼t is increasing. Set u(t) = |y(t)|e𝛼t. An application of Lemma 2.4 obtains

The above estimation yields the desired result. □

Theorem 3.9. Let the solution x = 0 of (.1) be EAS. Suppose that the perturbing term g(t, y) satisfies

where 𝛼 > 0, a, b, k, w ∈ C(), a, b, k ∈ L1() and w(u) is nondecreasing in u. If

where c = M|y0|e𝛼t0 , then all solutions of (.) approch zero as t → ∞

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. Using Lemma 2.2 and the assmptions, we have

Set u(t) = |y(t)|e𝛼t. Since w(u) is nondecreasing, an application of Lemma 2.9 obtains

where c = M|y0|e𝛼t0 . From the above estimation, we obtains the desired result. □