MATHEMATICS

V. I. ZUBOV

INVESTIGATION OF THE STABILITY PROBLEM FOR SYSTEMS OF EQUATIONS WITH HOMOGENEOUS RIGHT-HAND SIDES

(Presented by Academician V. I. Smirnov, January 7, 1957)

In the present paper conditions are derived for the asymptotic stability of the zero solution of a system of ordinary differential equations with homogeneous right-hand sides, and exact estimates are given for the distance of an integral curve from the equilibrium position; a number of applications of the results obtained is also set forth.

Definition 1. A real single-valued continuous function \(X(x_1,\ldots,x_n)\), defined in \(E_n\), will be called homogeneous of order \(\mu\) and denoted by \(X^{(\mu)}\), \(\mu=p/q\), where \(p\) and \(q\) are natural numbers, \(q\) odd, if for any real constant \(c\) the equality
\[ X(cx_1,\ldots,cx_n)=c^\mu X(x_1,\ldots,x_n) \]
holds; here the quantity \(c^\mu\) will be regarded as positive when \(p\) is even, and when \(p\) is odd—as real, having the sign of \(c\).

Definition 2. A real single-valued continuous function \(V(x_1,\ldots,x_n)\), defined in \(E_n\), will be called positive-homogeneous of order \(m>0\) and denoted by \(V^{[m]}\), if for any real quantity \(c\) the equality
\[ V(cx_1,\ldots,cx_n)=|c|^m V(x_1,\ldots,x_n) \]
holds; here the quantity \(|c|^m\) is regarded as positive.

Consider the system of ordinary differential equations
\[ \frac{dx_s}{dt}=X_s^{(\mu)}(x_1,\ldots,x_n)\qquad (s=1,\ldots,n). \tag{1} \]

Further, by
\[ X=X(t,X^{(0)}) \tag{2} \]
we shall denote an integral curve of system (1) such that \(X(0,X^{(0)})=X^{(0)}\), where \(X\) is a real \(n\)-dimensional vector \((x_1,\ldots,x_n)\). It is clear that, together with the integral curve (2), system (1) has a family of integral curves, depending on one arbitrary real constant \(c\), representable in the form
\[ X=cX(c^{\mu-1}t,X^{(0)})=X(t,cX^{(0)}). \]

Theorem 1. 1) For \(p\) even, the zero solution of system (1) cannot be asymptotically stable.

2) If \(p\) is odd and \(\mu\ne 1\), then the zero solution of system (1) can be asymptotically stable only under real perturbations.

3) If \(\mu=1\), then the zero solution of system (1) can be asymptotically stable under arbitrary complex perturbations. With the aid of the results contained in work (1), one can show that the integral curves of system (1) satisfy the inequality
\[ \left|X(t,X^{(0)})\right|<At^{-\alpha} \tag{3} \]
for \(\left|X^{(0)}\right|=1\), where \(A\) and \(\alpha\) are positive constants, if only

the zero solution of system (1) is asymptotically stable and \(X_s^{(\mu)}\in C_\nu\), i.e., the functions \(X_s^{(\mu)}\) are continuously differentiable with respect to all their arguments in \(E_n\) up to order \(\nu\geqslant 1\), inclusive.

Thus, in what follows, inequality (3) for \(X_s^{(\mu)}\in C_\nu,\ \nu\geqslant 1\), should be regarded as equivalent to asymptotic stability.

Theorem 2. If the integral curves of system (1) satisfy inequality (3), then there exist a constant \(m>\mu-1\) and two functions \(W^{[m]}\) and \(V^{[m+1-\mu]}\) possessing the following properties:

1) the functions \(V^{[m-\mu+1]}\) and \(-W^{[m]}\) are positive definite;

2) the function \(V^{[m-\mu+1]}(X(t,X^{(0)}))\) is continuously differentiable with respect to \(t\), and the equality
\[ \frac{dV^{[m-\mu+1]}}{dt}=W^{[m]}. \]

holds.

If, in addition, \(X_s^{(\mu)}\in C_\nu,\ \nu\geqslant 1\), then the function \(W^{[m]}\) can be chosen so that the function \(V^{[m-\mu+1]}\in C_\nu\) is the unique solution of the system
\[ \sum_{i=1}^{n}\frac{\partial V}{\partial x_i}X_i^{(\mu)}=W^{[m]};\qquad \sum_{i=1}^{n}x_i\frac{\partial V}{\partial x_i}=(m+1-\mu)V, \tag{4} \]
determined by the condition \(V(0,\ldots,0)=0\).

Theorem 3. In order that inequality (3) hold, it is necessary and sufficient that the following inequalities be satisfied for the solution of system (1):

1) for \(\mu=1\),
\[ p_1|X^{(0)}|e^{-p_2t}\leqslant |X(t,X^{(0)})|\leqslant q_1|X^{(0)}|e^{-q_2t}; \]

2) for \(\mu>1\) and \(\mu\ne 2k/q\),
\[ \frac{c_1|X^{(0)}|}{\sqrt[\mu-1]{1+c_2t\,|X^{(0)}|^{\mu-1}}} \leqslant |X(t,X^{(0)})|\leqslant \frac{d_1|X^{(0)}|}{\sqrt[\mu-1]{1+d_2t\,|X^{(0)}|^{\mu-1}}}; \]

3) for \(\mu\in(0,1)\), every solution of system (1), if (3) is assumed, after a finite interval of time reaches the origin. Until that moment the inequalities
\[ \sqrt[1-\mu]{a_1|X^{(0)}|^{1-\mu}-a_2t} \leqslant |X(t,X^{(0)})|\leqslant \sqrt[1-\mu]{b_1|X^{(0)}|^{1-\mu}-b_2t}. \]

hold.

In the indicated inequalities \(a_i,\ b_i,\ p_i,\ q_i,\ c_i,\ d_i\ (i=1,2)\) are positive constants.

Remark 1. The possibility of case 3) was pointed out in [2].

Remark 2. The estimates given in Theorem 3, in a certain sense, cannot be improved, i.e., the method of computing the constants \(a_i,\ b_i,\ c_i,\ d_i,\ p_i,\ q_i\ (i=1,2)\) is such that examples can be given in which these inequalities become equalities.

Consider the system
\[ \frac{dx_s}{dt}=X_s^{(\mu)}+f_s(x_1,\ldots,x_n,t)\quad (s=1,\ldots,n). \tag{5} \]

We shall assume that system (5) has a solution \(X=X(t,X^{(0)},t_0)\) for every \(X^{(0)}\in E_n\) and \(t_0\in(-\infty,+\infty)\).

Theorem 4. Let the functions \(X_s^{(\mu)}\in C_\nu\). In order that all solutions of system (5), for any choice of functions \(f_s\), \(|f_s|\leqslant c|X|^\lambda\) for \(|X|\geqslant R,\ \lambda\leqslant\mu\), where \(R\) is a sufficiently large positive quantity, \(c>0\) is sufficiently small when \(\lambda=\mu\), be bounded for \(t\geqslant t_0\), it is necessary and sufficient that (3) hold for every solution of system (1).

Theorem 5. If \(X_s^{(\mu)} \in C\), and (3) holds, then the zero solution of system (5) will be asymptotically stable for any functions \(f_s\) such that
\[ |f_s(X,t)| \leq c_1 |X|^\lambda \]
for \(|X| \leq h\), where \(\lambda \geq \mu\), \(h>0\) is sufficiently small, and \(c_1>0\) is sufficiently small when \(\lambda=\mu\). Moreover, any solution of system (5) beginning for sufficiently small \(|X^{(0)}|\), for all \(t \geq t_0\), satisfies estimates of the same form as in Theorem 3, if in them \(t\) is replaced by \(t-t_0\), and \(X(t,X^{(0)})\) by \(X(t,X^{(0)},t_0)\).

The study of system (1) makes it possible to advance the solution of the stability question in a number of doubtful cases.

Consider the system
\[ \frac{dx_s}{dt}=X_s(x_1,\ldots,x_k,y_1,\ldots,y_n)\quad (s=1,\ldots,k); \]
\[ \frac{dy_j}{dt}=\sum_{i=1}^{n}p_{ji}y_i+Y_j(x_1,\ldots,x_k,y_1,\ldots,y_n)= \]
\[ =\Phi_j(x_1,\ldots,x_k,y_1,\ldots,y_n)\quad (j=1,\ldots,n), \tag{6} \]
where the functions \(X_s, Y_j\) are holomorphic in a neighborhood of
\[ x_1=\cdots=x_k=y_1=\cdots=y_n=0 \]
and their expansions contain no terms linear with respect to \(x_s,y_j\).

Suppose that the roots of the equation \(|P-\lambda E|=0\), \(\{P\}_{ik}=p_{ik}\), have negative real parts. Denote by
\[ u_j(x_1,\ldots,x_k) \]
the analytic functions satisfying the system of equations
\[ \Phi_j(x_1,\ldots,x_k,y_1,\ldots,y_n)=0\quad (j=1,\ldots,n). \]

Theorem 6. If
\[ X_s(x_1,\ldots,x_k,u_1,\ldots,u_n)\equiv 0\quad (s=1,\ldots,k), \]
then system (6) has \(k\) holomorphic integrals and the zero solution of this system is stable in the sense of Lyapunov.

Denote by
\[ \overline{X}_s^{(\mu)}(x_1,\ldots,x_k) \]
the forms of lowest degree in the expansion of the functions
\[ X_s(x_1,\ldots,x_k,u_1,\ldots,u_n) \]
in powers of the quantities \(x_1,\ldots,x_k\).

Theorem 7. If the zero solution of the system
\[ \frac{dx_s}{dt}=\overline{X}_s^{(\mu)}\quad (s=1,\ldots,k) \]
is asymptotically stable, then the zero solution of system (6) is also asymptotically stable, and any solution beginning in a sufficiently small neighborhood of the point
\[ x_1=\cdots=x_k=y_1=\cdots=y_n=0 \]
satisfies the inequalities
\[ p_1|Z^{(0)}|e^{-p_2t}\leq |Z(t,Z^{(0)})|\leq \frac{c_1|Z^{(0)}|}{\sqrt[\mu-1]{\,1+c_2t|Z^{(0)}|^{\mu-1}\,}} \quad \text{for } t>0; \]
where \(c_i,p_i>0\) \((i=1,2)\) are constants;
\[ Z(t,Z^{(0)})=(x_1,\ldots,x_k,y_1,\ldots,y_n) \]
is a solution of system (6);
\[ Z^{(0)}=(x_1^{(0)},\ldots,x_k^{(0)},y_1^{(0)},\ldots,y_n^{(0)}). \]

We note that Theorem 6 is established with the aid of the results contained in § 65 of work \((^3)\).

Leningrad State University
named after A. A. Zhdanov

Received
2 I 1957

REFERENCES

1. N. N. Krasovskii, Prikl. matem. i mekh., 19, no. 5 (1955).
2. V. I. Zubov, Prikl. matem. i mekh., 17, no. 4 (1953).
3. A. M. Lyapunov, Complete collected works, 2, Publishing House of the Academy of Sciences of the USSR, 1956.