MATHEMATICS

V. A. PLISS

NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL STABILITY FOR A SYSTEM OF THREE DIFFERENTIAL EQUATIONS

(Presented by Academician V. I. Smirnov, 24 I 1958)

In the present note we consider the system of three equations

\[ \frac{dx}{dt}=y-ax-f(x),\qquad \frac{dy}{dt}=z-bf(x),\qquad \frac{dz}{dt}=-cf(x), \tag{1} \]

where the constants \(a\), \(b\), and \(c\) satisfy the inequalities

\[ ab>c,\qquad b>0,\qquad c>0, \tag{2} \]

and the continuous function \(f(x)\) satisfies the condition for uniqueness of solutions of system (1) and the generalized Hurwitz condition

\[ xf(x)>0\quad \text{for } x\ne 0;\qquad f(0)=0. \tag{3} \]

System (1) under conditions (2) and (3) was considered by A. P. Tuzov (¹), who indicated conditions sufficient for global stability of the zero solution. The aim of the present work is to establish necessary and sufficient conditions for global stability.

Theorem. In order that the zero solution of system (1) be globally stable, it is necessary and sufficient that the following conditions be fulfilled:

\[ \lim_{x\to+\infty}\left(f(x)+\int_0^x f(x)\,dx\right)=+\infty, \tag{4} \]

\[ \lim_{x\to-\infty}\left(-f(x)+\int_0^x f(x)\,dx\right)=+\infty. \tag{5} \]

The proof of sufficiency is based on the Lyapunov function constructed in (¹) and on the theorem formulated in note (²).

Let us prove necessity. Suppose, for definiteness, that condition (4) is violated. Then there exists an \(M>0\) such that

\[ f(x)<M\quad \text{for } x\geq 0. \tag{6} \]

Put

\[ \int_0^{+\infty} f(x)\,dx=I<+\infty. \tag{7} \]

Let \(H=a^2-ab+c\).

First consider the case \(H\geq 0\). Let the point \(p\) of phase space have coordinates \(x=0,\ y=y_0,\ z=z_0\), and suppose that the inequalities

\[ ay_0-z_0>2aM+\frac{cl}{M}. \tag{8} \]

Through the point \(p\) at \(t=0\) draw the trajectory \(\varphi(p,t)\) of system (1). Next introduce for consideration the function

\[ u=a^2x-ay+z. \tag{9} \]

The derivative of this function with respect to \(t\), taken by virtue of system (1), has the form

\[ \dot u=-au-Hf(x). \tag{10} \]

From equality (10) it follows that on \(\varphi(p,t)\) the relation

\[ u=e^{-at}\left(u_0-H\int_0^t fe^{at}\,dt\right), \tag{11} \]

holds, where \(u_0=z_0-ay_0\).

We shall show that on \(\varphi(p,t)\), for \(t\ge 0\), the inequality

\[ z>2aM \tag{12} \]

is satisfied.

For \(t=0\) this inequality is satisfied, as follows from (8). Suppose that it is violated at \(t=t_1\). Then, by continuity, we may assume that \(z(t_1)=2aM\) and that for \(t\in[0,t_1)\) inequality (12) is satisfied. We shall show that then on \(\varphi(p,t)\), for \(t\in[0,t_1]\), it will turn out that

\[ y-ax>\frac{z}{a}. \tag{13} \]

Indeed, for \(t=0\) this inequality is satisfied. Suppose that there exists a \(t^*\in(0,t_1]\) such that on the trajectory \(\varphi(p,t)\) one has

\[ y(t^*)-ax(t^*)=\frac{z(t^*)}{a}, \]

and inequality (13) is satisfied for \(t\in[0,t^*)\). But for \(t\in[0,t_1]\), by assumption, \(z\ge 2aM\); consequently,

\[ y-ax\ge 2M>f(x) \tag{14} \]

on \(\varphi(p,t)\) for \(t\in[0,t^*]\). Hence, for such \(t\), \(x\) increases along \(\varphi(p,t)\), and therefore for \(t\in(0,t^*]\), \(x>0\) on \(\varphi(p,t)\). But, by assumption, \(H\ge 0\), whence from (11) it follows that for \(t\in[0,t^*]\) on \(\varphi(p,t)\), \(u<0\). And this means that for \(t\in[0,t^*]\) (13) is satisfied. The contradiction obtained proves that inequality (13) is satisfied for all \(t\in[0,t_1]\).

From inequalities (6), (12), and (13) it follows that for \(t\in[0,t_1]\) on \(\varphi(p,t)\) the relation

\[ \frac{dz}{dx}>-\frac{cf(x)}{M}. \tag{15} \]

holds. Integrating this inequality, we obtain

\[ z(t_1)-z_0>-\frac{cl}{M}, \]

and hence from (8) it follows that \(z(t_1)>2aM\), which contradicts the definition of the time \(t_1\) \((z(t_1)=2aM)\). The contradiction obtained proves that inequality (12) is satisfied on the trajectory \(\varphi(p,t)\) for all \(t\ge 0\). Since \(a>0\) and \(M>0\), it follows from this that \(\varphi(p,t)\) does not approach the origin. Thus, in the case under consideration, stability in the large is absent.

Let us now turn to the case \(H<0\). In this case let the point \(p\) have coordinates \(x=0,\ y=y_0,\ z=z_0\), and let

\[ z_0=ay_0>2aM+(c-H)\frac{l}{M}. \tag{16} \]

We shall show that on the trajectory \(\varphi(p,t)\) of system (1), for \(t \geq 0\), the inequality

\[ y-ax>2M. \tag{17} \]

holds. For \(t=0\) this inequality is satisfied. Suppose that it is violated at \(t=t_1>0\), so that

\[ y(t_1)-ax(t_1)=2M \tag{18} \]

and inequality (17) is satisfied for \(t\in[0,t_1)\).

From inequalities (6) and (17) it follows that on the trajectory \(\varphi(p,t)\), for \(t\in[0,t_1]\), the relation

\[ \frac{dz}{dx}>-\frac{cf(x)}{M}. \]

holds. Integrating this inequality, we obtain

\[ z(t_1)-z_0>-\frac{cI}{M}, \]

whence, also from (16), we infer

\[ z(t_1)>2aM-\frac{HI}{M}. \tag{19} \]

Let us again turn to equality (10). From this equality, in view of (16), we obtain for the trajectory \(\varphi(p,t)\)

\[ u=-He^{-at}\int_0^t f e^{at}\,dt; \tag{20} \]

hence, and from (17), just as in the case \(H\geq 0\), we establish that for \(t\in[0,t_1]\) on the trajectory \(\varphi(p,t)\) one has \(u\geq 0\). Dividing equality (10) by the first of the equations of system (1), we then obtain

\[ \frac{du}{dx}=\frac{-au-Hf(x)}{y-ax-f(x)}; \]

hence, from \(u\geq 0\), we obtain the inequality

\[ \frac{du}{dx}\leq-\frac{Hf(x)}{y-ax-f(x)}, \]

which is valid on \(\varphi(p,t)\) for \(t\in[0,t_1]\). The last inequality, in view of (6) and (17), gives

\[ \frac{du}{dx}\leq-\frac{Hf(x)}{M}. \]

Integrating this inequality along the trajectory \(\varphi(p,t)\) from \(t=0\) to \(t=t_1\), we obtain

\[ u(t_1)-u(0)<-\frac{HI}{M}. \]

But from (16) it follows that \(u(0)=0\). Therefore, from the last inequality and from (9), we obtain

\[ a\,[ax(t_1)-y(t_1)]+z(t_1)<-\frac{HI}{M}. \]

Hence, and from (9), we infer

\[ y(t_1)-ax(t_1)>2M, \]

which contradicts the definition of the time \(t_1\) (equality (18)). The contradiction obtained proves that on the trajectory \(\varphi(p,t)\), for \(t\geq 0\), inequality (17) is satisfied. It follows that the trajectory \(\varphi(p,t)\) does not tend to the origin as \(t\to+\infty\). Consequently, in this case as well there is no stability in the large. The necessity is proved.

Received
19 I 1958

REFERENCES

\(^1\) A. P. Tuzov, Vestn. LGU, No. 2 (1955). \(\quad\) \(^2\) V. A. Pliss, DAN, 101, No. 1 (1955).