Revista Colombiana de Matemáticas Volumen 33 (1999), páginas 1–8

A note on the asymptotic stability in the whole of non-autonomous systems

Juan E. Nápoles Valdés

Universidad de la Cuenca del Plata, Corrientes, Argentina

Abstract. In this paper we present some results on the global stability of the trivial solutions x ≡ 0 of the system x′ = f(t, x). Our main results are then applied to various systems. Keywords and phrases. Stability, asymptotic stability, asymptotic stability in the whole, Liapunov’s second method.

1991 Mathematics Subject Classification. Primary 34D99.

1. Introduction

Liapunov’s principal theorems give sufficient conditions for the stability, asymptotic stability and instability of systems. In the last few years these stability concepts have been refined and further generalized in several directions, one of these being the asymptotic stability in the whole.

This paper is concerned with sufficient conditions guaranteeing that the trivial solution x ≡ 0 of the system

x′ = f(t, x),  (1)

where the prime marks indicate differentiation with respect to t, is asymptotically stable in the whole.

Throughout this paper, the following notations will be used. With I we denote the interval 0 ≤ t < ∞,and ℝ^{m} will stand for Euclidean m-space; ∥ · ∥ will be an arbitrary norm in ℝ^{m}, and S_{r} = {x ∈ ℝ^{m} : ∥x∥ < r}.

A solution of (1) through a point (t_{0}, x_{0}) in I × ℝ^{m} will be denoted by x(t; x_{0}, t_{0}),with x(t_{0}; x_{0}, t_{0}) = x_{0}.

With C(ℝ) and CI(ℝ) we respectively denote the families of continuous functions and increasing continuous functions defined on ℝ, and

CS(ℝ) = {h ∈ C(ℝ) : xh(x) > 0 for x ≠ 0},

CC(ℝ) =CI(ℝ)∩ CS(ℝ),

CP_{b}^{K}(ℝ) = {h ∈ C^{k}(ℝ) : h(x) ≥ b > 0 for all x}, CP_{b} := CP_{b}^{0}.

Finally, by F we denote the class of non decreasing continuous functions φ on I such φ(u) > 0 for all u ∈ I and

∫_{0}^{∞} \frac{du}{φ(μ)} = ∞.

We consider system (1) for f ∈ C(I × S_{r}) and f(t, 0) ≡ 0.

With V (t, x) we denote an arbitrary continuous scalar function defined on an open set S ⊂ I × ℝ^{m}. In all what follows it is assumed that all these functions V (t, x) have continuous partial derivatives with respect to all arguments. These functions will be called Liapunov’s functions. Corresponding to V (t, x), we define the function

V′_{(1)}(t, x) := lim sup_{h→0+} \frac{V (t + h, x + hf(t, x)) − V (t, x)}{h},  (2)

called the total derivative of V (t, x) for system (1). Under the above conditions,

V′_{(1)}(t, x) = \frac{∂V}{∂t} + \frac{∂V}{∂x} f(t, x).

We need the following definitions (cf. [3], [14]).

Definition 1. The solution x(t) ≡ 0 of (1) is stable if for any ε > 0 and any t_{0} ∈ I there exists a δ(t_{0}, ε) < 0 such that if x_{0} ∈ S_{δ(t_{0^{,}} ε)} then x(t; x_{0}, t_{0}) ∈ S_{ε} for all t ≥ t_{0}.

Definition 2. asymptotically stable if it is stable and there exists a δ(t_{0}) > 0 such that ∥x(t; x_{0}, t_{0})∥ ⟶ 0 as t ⟶ ∞ for all x_{0} ∈ S_{δ(t_{0}).}

Definition 3. The solution x(t) ≡ 0 of (1) is asymptotically stable in the whole if it is stable and every solution of (1) tends to zero as t ⟶ ∞.

Definition 4. The solution x(t) ≡ 0 of (1) is quasi equiasymptotically stable in the whole if for any α > 0, any ε > 0 and any t_{0} ∈ I, there exists T (t_{0}, ε, α) > 0 such that if x_{0} ∈ S_{α} then x(t; x_{0}, t_{0}) ∈ S_{ε} for all t ≥ t_{0} + T (t_{0}, ε, α).

Definition 5. The solution x(t) ≡ 0 of (1) is equiasymtotically stable in the whole if it is stable and quasi-equiasymptotically stable in the whole.

The distinction between asymptotic stability in the large and asymptotic stability in the whole has often been obliterated by inaccurate translation of the Russian terminology.

Definition 6. A solution x(t; x_{0}, t_{0}) of (1) is bounded if there exists a β > 0 such that x(t; x_{0}, t_{0}) ∈ S_{β} for all t ≥ t_{0′}, where β may depend on the solution.

Definition 7. The solutions of (1) are equibounded if for any α > 0 and t_{0} ∈ I there exists β(t_{0}, α) > 0 such that if x_{0} ∈ S_{α} then x(t; x_{0}, t_{0}) ∈ S_{β}(t_{0}, α) for all t ≥ t_{0}.

We now mention some theorems which will play an important role in the proofs of our main results.

Theorem A. Suppose that there exists a Liapunov’s function V (t, x) defined on I × S_{r} satisfying the following conditions:

1. V (t, 0) ≡ 0.

2. a(∥x∥) ≤ V (t, x), where a(t) is a positive definite function in CI(ℝ).

3. V′_{(1)}(t, x) ≤ 0.

Then, the trivial solution of (1) is stable.

Theorem B. Suppose that there exists a Liapunov’s function V (t, x) defined on I × ℝ^{m} which satisfies the following conditions:

1. a(∥x∥) ≤ V (t, x),where a(r) ∈ CC(ℝ) and a(r) ⟶ ∞ as r ⟶ ∞.

2. V′_{(1)}(t, x) ≤ 0. Then, the solutions of (1) are equi-bounded.

For the proof of the above results, see Theorems 8.1 and 10.1 of [14].

Theorem C (Barbashin and Krasovskii). If there exists a function V (t, x) which is everywhere positive definite, radially unbounded, decreasing, and whose total derivative (2) for system (1) is negative definite, then solution x(t) ≡ 0 of (1) is asymptotically stable in the whole.

For the proof, see [3, p. 248].

2. Main results

Theorem 1. Suppose that there exists a continuous, positive definite function a(r) such that a(r)⟶ ∞ as r ⟶ ∞, and that the following conditions are fulfilled:

1. There exists a Liapunov’s function V (t, x) such that a(x) ≤ V (t, x).

2. V′_{(1)}(t, x) ≤ −λ(t)V (t, x) − μ(t)W (t, x), where λ, μ ∈ CP_{0}(I) and W (t, x) is a positive definite function.

Then, the trivial solution x ≡ 0 of (1) is equi-asymptotically stable in the whole.

Proof. The conditions of Theorems A and B are satisfied, so that the solutions of (1) are stable and equibounded.

Let x(t; t_{0} x_{0}) be a unique solution of (1) satisfying x_{0} ∈ S_{ε} for ε sufficiently small.

Since μ is non-negative, we have

V′_{(1)}(t, x) ≤ −λ(t)V (t, x).

From the above inequality and the comparison theorem we obtain that

V (t, x(t; t_{0,}x_{0})) ≤ V (t_{0,}x_{0}) exp(−L(t)),

where

L(t) = ∫_{t_{0}}^{t} λ(s) ds.

Now define

M(t_{0}, ε) = max {V (t_{0}, x) : ∥x∥ ≤ ε}

and

T (t_{0}, τ, ε) = L^{−1} (1n\frac{M(t_{0}, ε)}{a(τ)}) .

Observe that since L(t) is strictly increasing then L^{−1} exists. Then we have, for t > t_{0} + T (t_{0}, τ, ε),

V (t, x(t; t_{0}, x_{0})) < M(t_{0}, ε)\frac{a(τ)}{M(t_{0}, ε)} = a(τ).

Since a(r) is increasing, and by condition 1 we have

∥x(t; t_{0}, x_{0})∥ < τ for t > t_{0} + T (t_{0}, τ, ε),

which implies the equi-asymptotic stability in the whole of the trivial solution, the theorem is proved. □ ✓

Theorem 2. Suppose that the functions a, b defined on S_{r} and W (t, x) defined on I × ℝ^{m}are positive definite and that the following two conditions hold:

1. a(∥x∥) ≤ V (t, x) ≤ b(∥x∥), a(r) ⟶ ∞ as r ⟶ ∞.

2. V′_{(1)}(t, x) ≤ −λ(t)ϕ(V (t, x)) − μ(t))W (t, x), where ϕ ∈ CI(ℝ^{+}) and is

positive definite. Then the trivial solution x ≡ 0 of the system (1) is asymptotically stable in the whole.

Proof. If

U(t, x) := ∫_{0}^{V (t,x)} du/ψ(u),

where ψ ∈ F(ℝ^{+}), then U(t, x) satisfies the conditions of the Barbashin-Krasovski theorem (Theorem C above), and the proof of the theorem is complete. □ ✓

Remark 1. Observe the advantage of resorting to the inequalities for V′_{(1)}(t, x) obtained in Theorems 1 and 2, not only for applications (Theorems 3 and 4 below), but also to render account of previous results (see [2], [3], [9] and [14]).

3. Applications and related results

The damped linear oscillator of one degree of freedom is described by the second order differential equation

x′′ + h(t)x′ + m^{2}x = 0, t ∈ I,  (3)

where m > 0 is a constant, the “damping” coefficient h : I ⟶ I being measurable and locally integrable.

It is an old problem to find conditions on h guaranteeing the asymptotic stability of the equilibrium x = x′ = 0, which means that for every solution of (3),

lim_{t→∞} x(t) = lim_{t→∞} x′(t) = 0  (4)

holds. It is known (see e.g., [5], [13]) that the condition

∫_{0}^{∞} h(t)dt = ∞  (5)

is necessary for asymptotic stability. It is also known that if h(t) ≡ h_{0} > 0, where h_{0} is constant, then the equilibrium is asymptotically stable. Results in [1], [4], [7], [8], [10] [11], [13] show that the condition

h ∈ CP_{b}(I)  (6)

is not necessary for asymptotic stability: it can be essentially weakened. Hale in [4] proved that if h(t) = 2 + e^{t},the equilibrium of (2) with k ≡ 1 is not asymptotically stable.

It is therefore natural to pose the problem of when the null solution of (3) is asymptotically stable in the large, or, in other words, when each solution curve of (3) approaches 0 as t ⟶ ∞. This problem is of paramount importance for applications of stability theory.

A. Castro and R. Alonso [2] considered the special case

x′′ + h(t)x′ + x = 0,  (7)

of equation (3), under condition (6) with b = 2a (a sufficiently small, i.e., a ≪ \frac{1}{2}) and h ∈ C^{1}(I). Letting y = x′,we obtain the system

\begin{aligned}y & = x′ \\ y′ & = −x − h(t)y, \end{aligned}  (8)

defined on I × S_{r}, with r > 0. Further they required that the condition

ah′(t) + 2h(t) ≤ 4a

be fulfilled, and applying Theorem 4 of [2] obtained results on the asymptotic stability of the trivial solution of (8) (and consequently of (7)). In the next theorem we shall prove that condition (6) with b = 2a, h ∈ C(I) and

h < \frac{\sqrt{1 − a^{3}} + 1}{a},  (9)

are sufficient for the asymptotic stability in the whole of the trivial solution of (8).

We consider the functions

H(t) = exp (− ∫_{0}^{t} h(s) ds) ,

R(x, y) = \frac{x^{2} + y^{2}}{2}

V (t, x, y) = (H(t) + 2)R(x, y) + axy.

Taking a(x, y) = R(x, y), b(x, y) = 3R(x, y), λ(t) ≡ a, μ(t) = (H(t)+1)h(t) and c(x, y) = 2R(x, y), all assumptions of Theorem 2 with ϕ(u) = u are satisfied. Thus, we obtain the following result.

Theorem 3. Assume that h ∈ C(I) and that conditions (5) and (9) hold. Then, the equilibrium of (3) is asymptotically stable in the whole.

Remark 2. In the case of asymptotic stability, our conditions(6) and (9) and Theorem 3 are consistent with results in [2], [6], [7], [8], [9], [11], [13] and [14].

We now consider the system

x′ = α(y) − β(y)f(x)  (10)
y′ = −a(t)g(x)

where α, β, f and g are continuous real valued functions and a(t) is a positive continuously differentiable function on I. We define

G(x) = ∫_{0}^{x} g(r) dr, A(y) = ∫_{0}^{y} α(s) ds,  (11)

and assume that the conditions

(a) α ∈ CC(ℝ),

(b) β ∈ CP_{b}(ℝ),

(c) f, g ∈ CS(ℝ),

(d) a ∈ CP_{0}^{1}(ℝ) hold. In [12] we prove that under these assumptions all solutions of (10) are continuable toward the future.

If α(y) = y, β(y) ≡ 1, a(t) ≡ 1, then system (10) is the well known Liénard’s equation.

Theorem 4. Under conditions (a)–(d) above, suppose in addition that 1. G(x) ≤ bf(x)g(x), for all x,

2. G(_{−}^{+}∞) = ±∞,

3. \frac{a′(t)}{a(t)} ≥ 1 for all t,

where G is as in (11). Then the trivial solution of (10) is equi-asymptotically stable in the whole.

Proof. Let a(x, y) = β(x), b(x, y) = \frac{A(y)}{a(0)} + β(x), V (t, x, y) = A(y) + a(t)G(x)

and b(t) = \frac{a(0)}{a(t)}. Let W (t, x, y) = (b(t) + 1)V (t, x, y). We observe that

W′_{(10)}(t, x, y) ≤ −W (t, x, y) − b(t)V (t, x, y)

where

W′_{(10)}(t, x, y) = − \frac{a(0)}{a(t)}b(t)V (t, x, y)
+ (b(t) + 1)(α(y)y′ + a′(t)b(x) + a(t)g(x)x′)

(see [9], [12] and [14] for more details) satisfies the requirements of Theorem 1 with λ(t) ≡ 1 and μ(t) = b(t). Thus, we obtain the desired result. □ ✓

Remark 3. In the special case α(y) = y, β(y) ≡ 1, f(x) = x, a(t) ≡ 1 and g(x) = x, of (10), conditions 1, 2 and 3 are easily verified, and system (10) becomes x′′ + x′ + x = 0, which satisfies the Routh-Hurwitz criterion for asymptotic stability in the whole.

Remark 4. From condition 3 of Theorem 4, we have that a′(t) > 0, for all t ∈ I. Thus, this theorem gives the author previous result [9, Theorem 2] on asymptotic stability in the whole.

Acknowledgment. We gratefully acknowledge the critical comments and helpful suggestions of the referee. They enabled us to considerably improve the paper.

References

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[9] J. E. Nápoles, and J. A. Repilado, On the boundedness and asymptotic stability in the whole of the solutions of a system of differential equations, Rev. Ciencias Matemáticas, Univ. of Habana, Vol. XVI, No. 1 (1995), 83–86 (Spanish).

[10] P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta Math. to appear.

[11] P/ Pucci and J. Serrin Precise damping conditions for global asymptotic stability for nonlinear second order systems (II), J. Differential Equations, submitted.

[12] J. A. Repilado and J. E. Nápoles, Continuability, oscillability and boundedness of solutions of bidimensional non-autonomous differential system, Rev. Ciencias Matemáticas, Univ. of Habana, Vol. XV, No. 2-3 (1994), 167–179 (Spanish).

[13] R. A. Smith, Asymptotic stability for x′′ + a(t)x′ + x′ = 0, Quart. J. Math,. Oxford, Ser. (2), 12 (1961), 123–126.

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(Recibido en noviembre de 1995; revisado por el autor en agosto de 1998 y marzo de 1999)

UTN, UDB Matemáticas, French 414 (3500) Resistencia, Chaco, Argentina

Universidad de la Cuenca del Plata

Plácido Martínez 964

(3400) Corrientes, Argentina