UDC 517.916

ON THE ASYMPTOTIC STABILITY OF THE ZERO SOLUTION OF A NONAUTONOMOUS SYSTEM OF TWO DIFFERENTIAL EQUATIONS. I

G. S. KRECHETOV

1. Consider the system

\[ \frac{dx}{dt}=X(x,y,t),\qquad \frac{dy}{dt}=Y(x,y,t), \tag{1.1} \]

where \(X, Y\) are holomorphic functions of the variables \(x\) and \(y\) in the domain
\(H\{(x,y)\mid \|x,y\|<h\}\) \((h=\mathrm{const}>0,\ \|x,y\|=\max(|x|,|y|))\) for all \(t\geqslant 0\), whose expansions in series in powers of \(x,y\) begin with terms of degree not lower than the first with respect to \(x,y\); the coefficients of these expansions are real, continuous, and bounded functions of \(t\) on the half-axis \([0,\infty)\).

Moreover, there exists a constant \(K>0\) such that

\[ |X(x,y,t)|\leqslant K,\qquad |Y(x,y,t)|\leqslant K \tag{1.2} \]

for \(\|x,y\|<h,\ t\geqslant 0\).

We shall compare system (1.1) with the system

\[ \frac{dx}{dt}=y+X^*(x,y),\qquad \frac{dy}{dt}=Y^*(x,y), \tag{1.3} \]

in which \(X^*\) and \(Y^*\) are holomorphic functions of the variables \(x,y\) in the domain \(H\), not depending explicitly on \(t\), whose expansions in series begin with terms of degree not lower than the second with respect to \(x,y\). Here \(X^*\) is such that

\[ X^*(x,0)\equiv 0. \tag{1.4} \]

Writing \(Y^*(x,y)\) in the form

\[ Y^*(x,y)=f(x)+\varphi(x)y+\psi(x,y)y^2, \]

where \(\psi(x,y)\) is a holomorphic function in \(H\),

\[ f(x)=ax^\alpha+a_1x^{\alpha+1}+\cdots,\qquad \varphi(x)=bx^\beta+b_1x^{\beta+1}+\cdots, \]

we shall also assume

\[ \begin{gathered} a<0,\qquad \alpha=2n-1\text{ is an odd number }(n\geqslant 2),\\ \beta>n-1\qquad (\text{or } \varphi(x)\equiv 0). \end{gathered} \tag{1.5} \]

Lyapunov showed how the problem of the stability of the zero solution of system (1.3) is solved in the case when conditions (1.4) and (1.5) are satisfied (see [1], pp. 289–305). Therefore we shall assume that this problem has been solved, and moreover in the positive sense: namely, that the trivial solution of system (1.3) is asymptotically stable.

Setting

\[ X(x,y,t)-y-X^*(x,y)=X_1(x,y,t), \]
\[ Y(x,y,t)-Y^*(x,y)=Y_1(x,y,t), \]

we shall seek conditions (as weak as possible) which the functions \(X_1\) and \(Y_1\) must satisfy so that the asymptotic stability of the zero solution of (1.3) implies the asymptotic stability of the zero solution of system (1.1).

Obviously, the functions \(X_1\) and \(Y_1\) in the domain

\[ H_1\{(x,y)\mid |x,y|<h_1\}\qquad (h_1=\text{const}>0,\quad h_1<h) \]

for \(t\geq 0\) possess the same properties as \(X,Y\). Therefore, in the domain \(H_1\) we have:

\[ \begin{aligned} X_1(x,y,t)&=\sum_{i,j=0}^{\infty}\varepsilon_{ij}(t)x^i y^j,\\ Y_1(x,y,t)&=\sum_{k,l=0}^{\infty}\varepsilon_{kl}(t)x^k y^l . \end{aligned} \tag{1.6} \]

Put

\[ a=-1 \quad \text{(see [1], p. 293),} \]
\[ \varepsilon_{i0}=0 \quad \text{for } t\geq 0,\quad 1\leq i\leq n-1, \tag{1.7} \]

and, having this, make the change of variables

\[ x=r\operatorname{Cs}(\theta),\qquad y=-r^n\operatorname{Sn}(\theta), \tag{1.8} \]

where \(r\) and \(\theta\) are real variables; \(\operatorname{Cs}(\theta)\), \(\operatorname{Sn}(\theta)\) are real continuous periodic functions of \(\theta\) with period \(2\omega\), giving a solution of the system of ordinary differential equations

\[ \frac{d\operatorname{Cs}(\theta)}{d\theta}=-\operatorname{Sn}(\theta),\qquad \frac{d\operatorname{Sn}(\theta)}{d\theta}=\operatorname{Cs}^{2n-1}(\theta) \]

with initial conditions \(\operatorname{Cs}(0)=1\), \(\operatorname{Sn}(0)=0\) (see [1], pp. 289–292).

In this case the problem of stability with respect to \(x,y\) is equivalent to the problem of stability with respect to \(r\). After the substitution (1.8), system (1.1) takes the form

\[ \frac{dr}{dt} = \frac{-r^n X(x,y,t)\operatorname{Cs}^{2n-1}(\theta)+rY(x,y,t)\operatorname{Sn}(\theta)} {-r^n}, \]
\[ \frac{d\theta}{dt} = \frac{Y(x,y,t)\operatorname{Cs}(\theta)+nr^{\,n-1}X(x,y,t)\operatorname{Sn}(\theta)} {-r^n}, \tag{1.9} \]

where by \(x,y\) we mean their expressions (1.8), and system (1.3) passes into the system

\[ \frac{dr}{dt}=r^{n+1}R,\qquad \frac{d\theta}{dt}=r^{n-1}+r^n\vartheta, \tag{1.10} \]

where \(R,\vartheta\) are series arranged in integral positive powers of \(r\), whose coefficients are integral rational functions of \(\operatorname{Cs}(\theta)\) and \(\operatorname{Sn}(\theta)\). These series converge uniformly and absolutely for all \(\theta\in(-\infty,+\infty)\), \(r\in(-h_2,h_2)\),

\[ h_2=\min\left(h_1,\,(nh_1^2)^{\frac1{2n}}\right). \]

Using (1.10), we write system (1.9) in the form

\[ \frac{dr}{dt}=r^{n+1}R+R_1(r,\theta,t), \]

\[ \frac{d\theta}{dt}=r^{\,n-1}+r^n\vartheta+\vartheta_1(r,\theta,t), \tag{1.11} \]

where

\[ R_1(r,\theta,t)= \frac{-r^nX_1(x,y,t)\operatorname{Cs}^{2n-1}(\theta)+Y_1(x,y,t)\,r\operatorname{Sn}(\theta)} {-r^n}, \]

\[ \vartheta_1(r,\theta,t)= \frac{Y_1(x,y,t)\operatorname{Cs}(\theta)+X_1(x,y,t)r^{\,n-1}\operatorname{Sn}(\theta)} {-r^n}, \]

where, as before, by \(x,y\) we mean their expressions (1.8).

In view of condition (1.7), these functions are expanded in series in positive integral powers of \(r\), converging uniformly and absolutely for all \(\theta\in(-\infty,+\infty)\), \(t\in[0,\infty)\), \(r\in(-h_2,h_2)\). The coefficients of these expansions are continuous and bounded functions of \(t\) and \(\theta\) \((t\geq0,\ -\infty<\theta<+\infty)\), periodic in \(\theta\) with period \(2\omega\), obtained in the known way from the expressions for \(R_1\) and \(\vartheta_1\).

Since

\[ R_1(0,\theta,t)\equiv0 \qquad (t\geq0,\ -\infty<\theta<+\infty), \]

the problem of stability with respect to \(r\) splits into two stability problems, one of which corresponds to \(r>0\), and the other to \(r<0\). Since, upon replacing \(r\) by \(-r\) and \(\theta\) by \(\omega-(-1)^n\theta\), system (1.11) is unchanged, a solution of the first of these problems in any sense implies a solution of the second problem in the same sense, and conversely. Therefore in what follows we shall assume \(r>0\).

For what follows we present a number of calculations, following Lyapunov. In system (1.10) we eliminate \(dt\), and seek the solution of the resulting equation

\[ \frac{dr}{d\theta}=\frac{r^2R}{1+r\vartheta}, \tag{1.12} \]

after first expanding its right-hand side in powers of \(r\), in the form of the series

\[ r=c+u_2c^2+u_3c^3+\cdots, \]

arranged in powers of an arbitrary constant \(c\). The functions \(u_2,u_3,\ldots\) are found successively and will either all be periodic, or periodic up to a certain number \(m\), so that \(u_m\) is a nonperiodic function and, consequently, \(u_m=g\theta+v\), where \(v\) is a periodic function and \(g\) is a constant. It is precisely this latter case that interests us. In this case the question of stability is decided by the sign of the constant \(g\) in the positive sense if \(g<0\), and in the negative sense if \(g>0\).

In view of the assumption made earlier, we shall take \(g<0\). Transform equation (1.12) by the substitution

\[ r=z+u_2z^2+u_3z^3+\cdots+u_{m-1}z^{m-1}+vz^m, \tag{1.13} \]

and obtain

\[ \frac{dz}{d\theta}=gz^m+\cdots. \tag{1.14} \]

(The dots denote terms of order higher than \(m\).)

Equality (1.13) shows that there exists a positive constant \(h_3 \leqslant h_2\) such that for \(|r| < h_3\) we have

\[ z=r+v_2 r^2+v_3 r^3+\cdots, \tag{1.15} \]

where \(v_2, v_3,\ldots\) are continuous periodic functions with period \(2\omega\). We shall assume \(h_3\) sufficiently small so that the series on the right-hand sides of (1.14) and (1.15) converge uniformly and absolutely for all \(\theta \in (-\infty,+\infty)\), \(r \in (-h_3,h_3)\), \(z \in (-h_3,h_3)\), and so that

\[ \frac{\partial r}{\partial z}>0,\qquad \frac{\partial z}{\partial r}>0 \tag{1.16} \]

for all \(r,z\) in the interval \((-h_3,h_3)\) and all \(\theta \in (-\infty,+\infty)\).

Let us write (1.14), taking (1.15) into account, in the form

\[ \frac{\partial z}{\partial r}\frac{dr}{d\theta} +\frac{\partial z}{\partial \theta} = gz^m+\cdots . \]

Multiplying this equation term by term by the second equation of system (1.10), taking (1.13) into account, we find

\[ \begin{aligned} \frac{dz}{dt}&=gz^{m+n-1}+\cdots,\\ \frac{d\theta}{dt}&=z^{n-1}+z^n\vartheta, \end{aligned} \tag{1.17} \]

where \(\vartheta\) has the same character as \(\vartheta\).

1. Finally, let us indicate one simple inequality. Suppose a Bernoulli differential equation is given,

\[ \frac{dx}{dt}=a(t)x-gx^m, \]

where \(a(t)\) is a continuous, nonnegative function on the interval \([t_0,t_1]\) \((0\leqslant t_0<t_1<+\infty)\), and \(g\) and \(m\) are positive numbers, of which \(m>1\). Then, if \(x(t)\) is a solution of this equation with initial value \(x(t_0)>0\), we have the inequality

\[ x(t_1)\leqslant \exp\left(\int_{t_0}^{t_1} a(t)\,dt\right) \left[x_0^{\,1-m}+(m-1)g(t_1-t_0)\right]^{\frac{1}{1-m}}, \tag{2.1} \]

which is obtained directly from the formula for the solution \(x(t)\).

Denote

\[ a(t)=a_1(t)+\frac{|\varepsilon_{n,0}(t)|}{2\sqrt n}, \]

where

\[ a_1(t)=\max\left(|e_{1,0}(t)|,\frac{|\varepsilon_{0,1}(t)|}{n}\right) \]

for every value \(t \in [0,\infty)\).

Theorem 1. Let the initial value \(t=t_0\geqslant 0\) be given, and suppose condition (1.7) is satisfied. Then, if there exists a value \(t=t^*>1\) such that

\[ \int_0^t a(s)\,ds \leq \frac{\xi}{m+n-2}\ln t \quad \text{for } t \geq t^*, \tag{2.2} \]

\[ \int_0^t |\varepsilon_{i,j}(s)|\,ds \leq \frac{\xi \rho^k}{m+n-2}\ln t \quad \text{for } t \geq t^*, \tag{2.3} \]

\(i, j\) satisfy the relation \(i+nj=k+n\) \((k=1,\ldots,m+n-2)\),

\[ \int_0^t |e_{i,j}(s)|\,ds \leq \frac{\xi \rho^k}{m+n-2}\ln t \quad \text{for } t \geq t^*, \tag{2.4} \]

\(i, j\) satisfy the relation \(i+nj=k+1\) \((k=1,\ldots,m+n-2)\), and \(\xi\) and \(\rho\) are arbitrarily prescribed values respectively from the intervals \((0,1)\) and \([1,+\infty)\), then the zero solution of system (1.1) is asymptotically stable.

Proof. Subjecting system (1.11) to the transformation (1.13), taking (1.17) into account, we write it in the form

\[ \begin{aligned} \frac{dz}{dt} &= g z^{m+n-1}+g_1(\theta,t)z^{m+n}+g_2(\theta,t)z^{m+n+1}+\cdots \\ &\quad +p_1(\theta,t)z+p_2(\theta,t)z^2+\cdots+p_{m+n-1}(\theta,t)z^{m+n-1}, \end{aligned} \tag{2.5} \]

\[ \frac{d\theta}{dt} = z^{m-1}+z^n \vartheta+\Psi_0(\theta,t)+\Psi_1(\theta,t)z+\Psi_2(\theta,t)z^2+\cdots, \]

where \(g_1, g_2,\ldots, p_1, p_2,\ldots,p_{m+n-1}, \Psi_0, \Psi_1,\ldots\) are continuous functions of \(\theta\) and \(t\), periodic in \(\theta\) with period \(2\omega\), and bounded in the domain \(t\geq 0\), \(-\infty<\theta<+\infty\).

It follows from (1.13) and (1.15) that the problem of stability with respect to \(r\) is equivalent to the problem of stability with respect to \(z\). At the same time condition (1.16) means that \(z>0\) for all \(r\in(0,h_3)\). Therefore, in view of the assumption made concerning the sign of \(r\), one should take \(z>0\). Moreover, we shall take \(h_3\) so small that for all \(z\in(0,h_3)\), \(\theta\in(-\infty,+\infty)\), \(t\in[0,+\infty)\), the right-hand sides of (2.5) are holomorphic functions of \(z\).

Regarding each time in (2.5) \(\theta\) as a prescribed, but completely arbitrary, function of \(t\) \((t\geq t_0)\), we restrict ourselves to considering only the first equation (2.5) (we note that in every interval \(t_0\leq t\leq t_1\) in which \(z(t)\) does not exceed a certain limit different from 0 and smaller than \(h_3\), the function \(\theta(t)\) is defined and continuous).

Let an arbitrarily small positive number \(\varepsilon<h_3\) be given. We shall show that there exists a positive number \(\delta(\varepsilon,t_0)<\varepsilon\) such that, if \(z(t_0)<\delta\), then for all \(t>t_0\) we have:

\[ z(t)<\varepsilon, \tag{2.6} \]

\[ \lim_{t\to\infty} z(t)=0. \tag{2.7} \]

From the character of the functions \(p_1, p_2,\ldots,p_{m+n-1}\) it follows that there exist continuous, nonnegative functions \(q_1,q_2,\ldots,q_{m+n-1}\), depending only on \(t\) \((t\geq 0)\), bounded on the half-axis \(t\geq 0\), such that

\[ |p_i(\theta,t)|\leq q_i(t) \quad \text{for } t\geq 0,\quad i=1,2,\ldots,m+n-1, \tag{2.8} \]

\[ \int_0^t q_i(s)\,ds \leqslant \frac{\xi l_i p^{i-1}}{m+n-2}\ln t,\quad t \geqslant t^* \quad (i=1,2,\ldots,m+n-1), \tag{2.9} \]

where \(l_i\) \((i=1,2,\ldots,m+n-1)\) are certain positive constants, independent of \(\varepsilon\); for \(i=1\) we set \(l_1=1,\ q_1(t)\equiv a(t)\).

Let \(\varepsilon\) be so small that

\[ \xi \sum_{i=1}^{m+n-1} l_i p^{i-1}\varepsilon^{i-1}=\gamma<1, \tag{2.10} \]

\[ g z^{n+m-1}+g_1(\theta,t)z^{n+m}+g_2(\theta,t)z^{m+n+1}+\cdots<-\eta z^{n+m-1} \tag{2.11} \]

for \(z\in[0,\varepsilon],\ t\in[0,+\infty),\ \theta\in(-\infty,+\infty)\), where \(\eta=\mathrm{const}>0\).

Choose the number \(t_1\) sufficiently large \((t_1\geqslant t^*)\) so that the inequality

\[ t^{\frac{\gamma}{n+m-2}}\,[\eta(m+n-2)(t-t_0)]^{\frac{1}{2-n-m}}<\varepsilon \tag{2.12} \]

holds for all \(t\geqslant t_1\), and consider the differential equation

\[ \frac{du}{dt}=q(t)u-\eta u^{m+n-1}, \tag{2.13} \]

in which

\[ q(t)=\sum_{i=1}^{n+m-1} q_i(t)\varepsilon^{i-1}. \]

From conditions (2.8) and (2.11) it follows that the right-hand side of (2.13) is greater than the right-hand side of the first equation of system (2.5), when \(z\) and \(u\) do not exceed \(\varepsilon\). Therefore, choosing any solution of equation (2.13) with initial value \(u(t_0)=u_0>0\) and choosing \(0<\delta<\varepsilon\) so that \(u(t)\) remains less than \(\varepsilon\) for all \(u_0\in(0,\delta)\), \(t\in[t_0,t_1]\), we may assert that (2.6) holds for \(t\in[t_0,t_1]\), if we set \(z(t_0)=u_0\).

Further, applying formula (2.1), we obtain

\[ u(t)\leqslant \exp\left(\int_{t_0}^{t} q(s)\,ds\right) \left[u_0^{2-m-n}+\eta(m+n-2)(t-t_0)\right]^{\frac{1}{2-m-n}}< \]

\[ <\exp\left(\int_0^{t} q(s)\,ds\right) \left[\eta(m+n-2)(t-t_0)\right]^{\frac{1}{2-m-n}}, \quad t\geqslant t_1. \]

But (2.9) and (2.10) give

\[ \int_0^t q(s)\,ds< \frac{\gamma}{m+n-2}\ln t\quad (t\geqslant t_1), \]

as a result of which we arrive at the inequality

\[ u(t)<t^{\frac{\gamma}{m+n-2}} \left[\eta(m+n-2)(t-t_0)\right]^{\frac{1}{2-m-n}}, \tag{2.14} \]

from which, on the basis of (2.12), we obtain \(u(t)<\varepsilon\) for \(t\geqslant t_1\).

Thus, (2.6) is also satisfied for \(t>t_1\), if \(z(t_0)=u_0\). Moreover, from (2.14) it follows that \(\lim_{t\to\infty}u(t)=0\) and, consequently, \(\lim_{t\to\infty}z(t)=0\), if \(z(t_0)=u_0\).

Hence (2.7) is also satisfied, which proves the theorem.

Remark. In the case under consideration, as is seen from inequality (2.14), system (1.1) has property \((D)\) ([2], p. 64). However, property \((C)\) ([2], p. 58), generally speaking, does not hold.

Let the functions \(\varepsilon_{ij}(t)\) and \(e_{ij}(t)\) in (1.6) have the form:
\[ \varepsilon_{ij}(t)=\alpha_{ij}(t)+\beta_{ij}(t), \]
\[ e_{ij}(t)=a_{ij}(t)+b_{ij}(t)\quad (i,j=1,2,\ldots;\ i+j\geqslant 1), \]
where \(\alpha_{ij}, \beta_{ij}, a_{ij}, b_{ij}\) are functions of the same character as \(\varepsilon_{ij}\) and \(e_{ij}\). Denote
\[ a(t)=a_1(t)+\frac{|\alpha_{n,0}(t)|}{2\sqrt n}, \]
where
\[ a_1(t)=\max\left(|a_{10}(t)|,\frac{|\alpha_{01}(t)|}{n}\right) \]
for every value \(t\in[0,\infty)\).

Theorem 2. Let us assign an arbitrary number \(t_0\geqslant 0\) and suppose that the following conditions are satisfied:
\[ \alpha_{i0}(t)\equiv 0,\quad t\geqslant 0,\quad 1\leqslant i\leqslant n-1; \tag{2.15} \]
there exist numbers \(T>1\) (independent of \(t_0\)) and \(\xi\in(0,1)\) such that
\[ \int_{t_0}^{t} a(s)\,ds\leqslant \frac{\xi}{m+n-2}\ln(t-t_0),\quad t\geqslant t_0+T, \tag{2.16} \]
\[ \int_{t_0}^{t}|\alpha_{ij}(s)|\,ds\leqslant \frac{\xi p^k}{m+n-2}\ln(t-t_0),\quad t\geqslant t_0+T, \tag{2.17} \]
where \(i,j\) satisfy the relation \(i+nj=k+n\) \((k=1,2,\ldots,n+m-2)\),
\[ \int_{t_0}^{t}|a_{ij}(s)|\,ds\leqslant \frac{\xi p^k}{m+n-2}\ln(t-t_0) \tag{2.18} \]
for \(t>t_0+T\), with \(i,j\) such that \(i+nj=k+n\) \((k=1,2,\ldots,m+n-2)\);
\[ \lim_{t\to\infty}\int_t^{t+1}|\beta_{ij}(s)|\,ds=0\quad (1\leqslant i+nj\leqslant m+2n-2), \]
\[ \tag{2.19} \lim_{t\to\infty}\int_t^{t+1}|b_{ij}(s)|\,ds=0\quad (1\leqslant i+nj\leqslant m+n-1), \]
where \(p=\mathrm{const}>1\). Then the trivial solution of (1.1) is asymptotically stable uniformly with respect to \(t_0\) and the initial values \(x_0,y_0\).

Proof. From conditions (2.15), (2.16), (2.17), and (2.18) it follows (by Theorem 1) that the trivial solution of the system
\[ \frac{dx}{dt}=X(x,y,t)-X'(x,y,t), \]

\[ \frac{dy}{dt}=Y(x,y,t)-Y'(x,y,t), \tag{2.20} \]

where

\[ X'(x,y,t)=\sum_{i,j} b_{ij}(t)x^i y^j \quad (1\leq i+nj\leq m+n-1), \]

\[ Y'(x,y,t)=\sum_{i,j} \beta_{ij}(t)x^i y^j \quad (1\leq i+nj\leq m+2n-2) \]

is asymptotically stable. Moreover, the stability is uniform with respect to \(t_0\) and the initial values \(x_0,y_0\).

Indeed, formula (2.14) in this case has the form

\[ u(t)<(t-t_0)^{\frac{\gamma}{m+n-2}} \left[\eta(m+n-2)(t-t_0)\right]^{\frac{1}{2-m-n}}, \quad t\geq t_0+T. \]

Since \(T\) does not depend on \(t_0\), the property of uniform stability is easily established with the aid of this inequality. But in that case, by Theorem (5.1) of [2] (pp. 32–34), for system (2.20), in some domain \(D\)

\[ D\{(x,y)\mid \|x,y\|<d\}\quad (d=\operatorname{const}>0) \]

there exists a Lyapunov function \(V(x,y,t)\) possessing in \(D\) the following properties:

\(V\) is positive definite, admits an infinitely small upper bound, and has in \(D\) continuous partial derivatives of the first order with respect to all arguments, uniformly bounded with respect to \(t\in(0,\infty)\).

The derivative of the function \(V\) by virtue of system (2.20) is negative definite in \(D\).

Thus, there exist continuous functions \(W(x,y)\), \(W'(x,y)\) such that for \((x,y)\in D\) we have

\[ V(x,y,t)\geq W(x,y)>0 \quad \text{for } \|x,y\|\neq 0. \tag{2.21} \]

By virtue of (2.20),

\[ \frac{dV}{dt}\leq -W'(x,y)<0,\quad \|x,y\|\neq 0, \tag{2.22} \]

\[ V(0,0,t)\equiv W(0,0)\equiv W'(0,0)\equiv 0. \tag{2.23} \]

Moreover, there exists a constant \(N\) satisfying the inequalities

\[ \left|\frac{\partial V}{\partial x}\right|\leq N,\quad \left|\frac{\partial V}{\partial y}\right|\leq N,\quad \left|\frac{\partial V}{\partial t}\right|\leq N. \tag{2.24} \]

Let us fix an arbitrary positive \(\varepsilon<d\) and put

\[ l=\min W(x,y)\quad \text{for } \|x,y\|=\varepsilon. \tag{2.25} \]

On the other hand, by virtue of (2.21), (2.23), and (2.24), for \(\delta<d\) we have

\[ W(x,y)\leq V(x,y,t)\leq \sqrt{2}\,N\delta \quad \text{for } \|x,y\|\leq \delta. \tag{2.26} \]

Choose \(\delta\) so that

\[ \sqrt{2}\,N\delta<l \tag{2.27} \]

(from (2.25) and (2.27) it follows that \(\delta<\varepsilon\)) and so that, for every solution \(x(t),y(t)\) of system (1.1) (if such exists) for which \(\|x(t_1),y(t_1)\|=\delta\), \(\|x(t_2),y(t_2)\|=\varepsilon\) \((0\leq t_1<t_2<+\infty)\), the inequality holds

\[ t_2-t_1\geqslant 1 \tag{2.28} \]

under the condition \(\delta \leqslant \|x(t),\, y(t)\|<\varepsilon\) for \(t\in(t_1,t_2)\).

From conditions (2.19) it follows that there exists a function \(f(t)\), continuous, nonnegative, and bounded on \([0,+\infty)\), such that

\[ |X'(x,y,t)|\leqslant f(t), \tag{2.29} \]

\[ |Y'(x,y,t)|\leqslant f(t) \]

for \(t\geqslant 0,\ \|x,y\|\leqslant \varepsilon\),

\[ \lim_{t\to\infty}\int_t^{t+1} f(s)\,ds=0. \tag{2.30} \]

Putting

\[ l'=\min W'(x,y)\quad \text{for } \delta \leqslant \|x,y\|\leqslant \varepsilon, \tag{2.31} \]

we choose the initial value \(t=t_0'\) so large that the inequality

\[ \frac{2N}{T'}\int_t^{t+T'} f(s)\,ds<l' \quad \text{for all } T'\geqslant 1,\ t\geqslant t_0' \tag{2.32} \]

is satisfied (by virtue of (2.30), such a \(t_0'\) will be found).

We shall show that for any solution \(x(t), y(t)\) of system (1.1) with initial values \(x(t_0'), y(t_0')\) satisfying the condition

\[ \|x(t_0'),\, y(t_0')\|<\delta, \tag{2.33} \]

the following relations hold:

\[ \|x(t),\, y(t)\|<\varepsilon \quad \text{for } t>t_0', \tag{2.34} \]

\[ \lim_{t\to\infty}\|x(t),\, y(t)\|=0. \tag{2.35} \]

Suppose that inequality (2.34) is violated in finite time. Then there exist two instants \(t_1\) and \(t_2\) \((t_0'<t_1<t_2)\) such that (2.28) holds. But in that case, on the interval \([t_1,t_2]\), along the solution under consideration, we shall have

\[ V(t_2)=V(t_1)+\int_{t_1}^{t_2}\frac{dV^{(1.1)}}{dt}\,dt =V(t_1)+\int_{t_1}^{t_2}\frac{dV^{(2.20)}}{dt}\,dt+ \]

\[ +\int_{t_1}^{t_2}\left(\frac{\partial V}{\partial x}X' +\frac{\partial V}{\partial y}Y'\right)\,dt, \]

whence, according to (2.22), (2.31), (2.34), (2.29), the inequality follows:

\[ V(t_2)\leqslant V(t_1)-l'(t_2-t_1)+2N\int_{t_1}^{t_2} f(s)\,ds, \]

from which, in turn, on the basis of (2.32), where we put \(T'=t_2-t_1\), we obtain

\[ V(t_2)<V(t_1). \tag{2.36} \]

Since \(\|x(t_1),\, y(t_1)\|=\delta\), from (2.27), (2.26), and (2.36) we find \(V(t_2)<l\). On the other hand, since \(\|x(t_2),\, y(t_2)\|=\varepsilon\), (2.21) and (2.25) give \(V(t_2)\geq l\). From these contradictory inequalities we conclude that (2.34) holds for all \(t>t'_0\).

To show that (2.35) holds, choose an arbitrary number \(\eta\in(0,\varepsilon)\) and put \(\lambda=\min W(x,y)\) for \(\|x,y\|=\eta\).

Denote by \(\xi\) a number from \((0,\delta)\) satisfying the relations (2.26), (2.27), and (2.28), in which, in place of \(\delta\), we put \(\xi\), in place of \(l\), \(\lambda\), and in place of \(\varepsilon\), \(\eta\) (whence it immediately follows that \(\xi<\eta\)), and put \(\lambda'=\min W'(x,y)\) for \(\xi\leq\|x,y\|\leq\varepsilon\).

Finally, choose the numbers \(\tau_1\) and \(\tau_2\) so that

\[ \tau_2-\tau_1\geq 1,\qquad \tau_1\geq t'_0, \tag{2.37} \]

\[ \frac{2N}{T'}\int_t^{t+T'} f(s)\,ds < \frac{1}{2}\lambda' \quad \text{for all } T'\geq 1,\ t\geq \tau_1, \tag{2.38} \]

\[ \sqrt{2}\,N\varepsilon-\frac{1}{2}\lambda'(\tau_2-\tau_1)\leq 0. \tag{2.39} \]

We shall show that for every solution \(x(t),y(t)\) of system (1.1) satisfying condition (2.33), there exists a value \(t=t^*\) from the interval \((\tau_1,\tau_2)\) such that

\[ \|x(t^*),\, y(t^*)\|<\xi . \tag{2.40} \]

Indeed, suppose that for some such solution (2.40) does not hold; then along this solution we have

\[ V(\tau_2)=V(\tau_1)+\int_{\tau_1}^{\tau_2}\frac{dV^{(2.20)}}{dt}\,dt +\int_{\tau_1}^{\tau_2}\left(\frac{\partial V}{\partial x}X' +\frac{\partial V}{\partial y}Y'\right)dt \leq V(\tau_1)- \]

\[ -\left(\lambda'-\frac{2N}{\tau_2-\tau_1}\int_{\tau_1}^{\tau_2} f(s)\,ds\right)(\tau_2-\tau_1). \]

On the basis of (2.37), (2.38) (in which we put \(T'=\tau_2-\tau_1\)), from the last relation we derive

\[ V(\tau_2)<V(\tau_1)-\frac{1}{2}\lambda'(\tau_2-\tau_1). \]

Since in the domain \(\|x,y\|\leq\varepsilon\), \(V(x,y,t)\leq \sqrt{2}\,N\varepsilon\), from the last inequality, on the basis of (2.39), we obtain the inequality

\[ V(\tau_2)\leq 0, \]

which contradicts the positive definiteness of the function \(V\).

Thus, inequality (2.40) is proved.

We shall show that after the instant \(t^*\) the solution cannot leave the domain \(\|x,y\|<\eta\). Indeed, if there existed a value \(t=t'\) such that \(\|x(t'),y(t')\|=\eta\), then, applying the arguments used in proving (2.34) (similar arguments are valid by the choice of the numbers \(\lambda,\xi,\lambda'\)), we would arrive at a contradiction; but in that case the zero solution of system (1.1) is asymptotically stable for \(t=t'_0\).

Moreover, the stability is uniform with respect to \(t_0\) and the initial values \(x_0, y_0\). Indeed, choose a positive \(\delta_1<\delta\) so that in every interval \(t_0''\leq t\leq t_0'\) \((0\leq t_0''<t_0')\) the inequality \(\|x,y\|<\delta\) holds for every solution \(x(t),y(t)\) of system (1.1), as soon as \(\|x(t_0''),y(t_0'')\|<\delta_1\) (the existence of such a \(\delta_1\) is guaranteed by the properties of the functions \(X(x,y,t)\), \(Y(x,y,t)\)). But then, understanding by \(t_0\) any value \(t\in[0,\infty)\), we shall have \(\|x(t),y(t)\|<\varepsilon\) for \(t\geq t_0\), \(\|x(t),y(t)\|<\eta\) for \(t\geq t_0+\tau_2(\eta)\), as soon as \(\|x(t_0),y(t_0)\|<\delta_1\).

The theorem is proved.

Let the functions \(\varepsilon_{ij}(t)\) and \(e_{ij}(t)\) in (1.6) have the form:
\[ \varepsilon_{ij}(t)=\alpha_{ij}(t)+\beta_{ij}(t)+\gamma_{ij}(t), \]
\[ e_{ij}(t)=a_{ij}(t)+b_{ij}(t)+c_{ij}(t)\quad (i,j=0,1,\ldots),\ (i+j\geq 1), \]
where \(\alpha_{ij},\beta_{ij},\gamma_{ij},a_{ij},b_{ij},c_{ij}\) are functions of the same character as the functions \(\varepsilon_{ij}, e_{ij}\). Denote
\[ X'(x,y,t)=\sum_{ij} c_{ij}(t)x^i y^j, \]
\[ Y'(x,y,t)=\sum_{kl} \gamma_{kl}(t)x^k y^l \]
\[ (1\leq i+nj\leq n+m-1,\qquad 1\leq k+nl\leq m+2n-2). \]

Theorem 3. Suppose that the functions \(a_{ij}(t), b_{ij}(t), \alpha_{kl}(t), \beta_{kl}(t)\)
\[ (1\leq i+nj\leq m+n-1,\quad 1\leq k+nl\leq m+2n-2) \]
satisfy all the conditions of Theorem 2, and that the functions \(X(x,y,t)-X'(x,y,t)\) and \(Y(x,y,t)-Y'(x,y,t)\) are uniformly continuous in \(t\in(0,\infty)\) in some domain \(D_1\{x,y\mid \|x,y\|\leq d_1\}\) \((0<d_1<d)\); suppose further that the functions \(c_{ij}\) and \(\gamma_{ij}\) possess, respectively, antiderivatives \(f_{ij},\varphi_{kl}\)
\[ (1\leq i+nj\leq m+n-1,\quad 1\leq k+nl\leq m+2n-2), \]
satisfying the following conditions: there exist sequences \(\{t_s\}\) and \(\{t_\sigma\}\) of values of \(t\) \((s,\sigma=1,2,\ldots)\), increasing without bound as \(s\) and \(\sigma\) increase without bound, such that
\[ f_{ij}(t_s)=0,\qquad \varphi_{kl}(t_\sigma)=0\quad (s,\sigma=1,2,\ldots), \tag{2.41} \]
\[ \lim_{s\to\infty}(t_{s+1}-t_s)=0,\qquad \lim_{\sigma\to\infty}(t_{\sigma+1}-t_\sigma)=0, \tag{2.42} \]
(the sequences \(\{t_s\}\) and \(\{t_\sigma\}\) depend, generally speaking, on the choice of the functions \(f_{ij}\) and \(\varphi_{kl}\)). Then the zero solution of system (1.1) is asymptotically stable uniformly with respect to \(t_0\) and the initial values \(x_0,y_0\).

Proof. From the conditions of the theorem it follows that to the system of differential equations
\[ \frac{dx}{dt}=X(x,y,t)-X'(x,y,t), \]
\[ \frac{dy}{dt}=Y(x,y,t)-Y'(x,y,t) \tag{2.43} \]
Theorem 2 and the already mentioned theorem (5.1) are applicable in its full formulation; therefore for system (2.43) there exists a Lyapunov function \(V(x,y,t)\), which in some domain \(D_2\{x,y\mid \|x,y\|<d_2\}\) \((0<d_2<d_1)\), for \(t>0\), possesses the same properties as in Theorem 2 and, moreover,

that is, it has continuous partial derivatives of the 2nd order with respect to all arguments in \(D_2\), uniformly bounded in \(D_2\) for \(t \in (0,\infty)\).

We shall therefore assume that, for system (2.43) in the domain \(D_2\) for \(t>0\), the relations (2.21), (2.22), (2.23), and also (2.24) hold; the latter we shall regard as valid also for the moduli of the second-order partial derivatives with respect to all arguments.

From the conditions (2.42) it follows that, however small a prescribed positive number \(\tau\) may be, there exists a value \(t=t'>0\) such that the inequalities hold:

\[ t_{s+1}-t_s<\frac{1}{2}\tau,\qquad t_{\sigma+1}-t_\sigma<\frac{1}{2}\tau \]

(for those \(s\) and \(\sigma\) for which \(t_s\geq t'\), \(t_\sigma\geq t'\)),

\[ 0\leq t_{s'}-t'<\frac{1}{2}\tau, \]

\[ 0\leq t_{\sigma'}-t'<\frac{1}{2}\tau, \tag{2.44} \]

where \(s'\) and \(\sigma'\) are the least of the mentioned values of \(s\) and \(\sigma\). Moreover, inequality (2.44) is realized independently of the choice of the functions \(f_{ij}(t)\) and \(\varphi_{kl}(t)\) \((1\leq i+nj\leq m+n-1,\quad 1\leq k+nl\leq m+2n-2)\).

Choose arbitrarily a number \(\varepsilon\) from \((0,d_2)\) \((\varepsilon<1)\) and put

\[ L=\min W(x,y)\quad \text{for } \|(x,y)\|=\varepsilon . \tag{2.45} \]

On the other hand, from (2.24), (2.21), and (2.23) we have, for \(\delta<d_2\),

\[ W(x,y)\leq V(x,y,t)\leq \sqrt{2}\,N\delta \quad \text{for } \|(x,y)\|\leq \delta . \tag{2.46} \]

Let \(C\) be a positive constant satisfying the relation

\[ |c_{ij}(t)|\leq C,\qquad |\gamma_{kl}(t)|\leq C \quad \text{for } t\geq 0 \tag{2.47} \]

\[ (1\leq i+nj\leq m+n-1,\qquad 1\leq k+nl\leq m+2n-2), \]

and let

\[ N_1=\frac{1}{2}(m+2n-2+A_1)(q_1+1),\qquad N_2=\frac{1}{2}(m+n-1+A_2)\times \]

\[ \times(q_2+1), \]
where \(A_1,q_1,A_2,q_2\) are nonnegative integers such that
\(A_1<q_1\), \(A_2<q_2\), and
\(m+2n-2=nq_1+A_1,\quad m+n-1=nq_2+A_2\).

Subjecting the numbers \(\delta\) and \(\tau\) to the condition

\[ \sqrt{2}\,N\delta+(N_1+N_2)NC\varepsilon\tau<L \tag{2.48} \]

and putting

\[ L'=\min W'(x,y)\quad \text{for } \delta\leq \|(x,y)\|\leq \varepsilon, \tag{2.49} \]

we shall show that, for any solution \(x(t),y(t)\) of system (1.1) with initial conditions \(x(t_0'),y(t_0')\) satisfying the inequality

\[ \|x(t_0'),\,y(t_0')\|<\delta, \tag{2.50} \]

the following relations hold:

\[ \|x(t),\,y(t)\|<\varepsilon\quad \text{for } t>t_0', \tag{2.51} \]

\[ \lim_{t\to\infty}\|x(t),\,y(t)\|=0, \tag{2.52} \]

if we set \(t_0'=t'\).

Assume that for an arbitrarily large \(t'\) (and, consequently, for an arbitrarily small \(\tau\)) there exists a solution of system (1.1) for which, at \(t=t'_0\), (2.50) holds, but (2.51) is violated in a finite time. Then for such a solution there will be values \(t_1\) and \(t_2\) \((t_2>t_1\geq t'_0)\) such that

\[ \delta < \|x(t),\,y(t)\| < \varepsilon \quad \text{for } t\in (t_1,t_2), \qquad \|x(t_1),\,y(t_1)\|=\delta,\quad \|x(t_2),\,y(t_2)\|=\varepsilon . \]

But in this case, along this solution we have:

\[ \begin{aligned} V(t_2) &= V(t_1)+\int_{t_1}^{t_2}\frac{dV^{(2.43)}}{dt}\,dt +\int_{t_1}^{t_2}\left(\frac{\partial V}{\partial x}X' +\frac{\partial V}{\partial y}Y'\right)\,dt \\ &= V(t_1)+\int_{t_1}^{t_2}\frac{dV^{(2.43)}}{dt}\,dt +\sum_{i,j}\int_{t_1}^{t_2}\frac{\partial V}{\partial x}\,c_{ij}x^i y^j\,dt \\ &\quad +\sum_{k,l}\int_{t_1}^{t_2}\frac{\partial V}{\partial y}\,\gamma_{kl}x^k y^l\,dt \quad (1\leq i+nj\leq m+n-1, \tag{2.53} \end{aligned} \]

\[ 1\leq k+nl\leq m+2n-2). \]

Let \(t_{\bar s}, t_{\bar{s}+1}, \ldots, t_{\bar{s}+r}, t_{\bar\sigma}, t_{\bar\sigma+1}, \ldots, t_{\bar\sigma+\rho}\) be the values, respectively, of the sequences \(\{t_s\}\) and \(\{t_\sigma\}\) falling in the interval \([t_1,t_2]\) (obviously, \(t_{\bar s}\geq t_{s'}\), \(t_{\bar\sigma}\geq t_{\sigma'}\)). Using (2.46), (2.22), and (2.49), from (2.53) we find

\[ \begin{aligned} V(t_2) &\leq \sqrt{2}\,N\delta - L'(t_2-t_1) +\sum_{i,j}\left( \int_{t_1}^{t_{\bar s}}\frac{\partial V}{\partial x}\,c_{ij}x^i y^j\,dt +\int_{t_{\bar{s}+r}}^{t_2}\frac{\partial V}{\partial x}\,c_{ij}x^i y^j\,dt \right) \\ &\quad +\sum_{k,l}\left( \int_{t_1}^{t_{\bar\sigma}}\frac{\partial V}{\partial y}\,\gamma_{kl}x^k y^l\,dt +\int_{t_{\bar\sigma+\rho}}^{t_2}\frac{\partial V}{\partial y}\,\gamma_{kl}x^k y^l\,dt \right) \\ &\quad +\sum_{i,j}\sum_{s=\bar s}^{\bar{s}+r-1} \int_{t_s}^{t_{s+1}}\frac{\partial V}{\partial x}\,c_{ij}x^i y^j\,dt +\sum_{k,l}\sum_{\sigma=\bar\sigma}^{\bar\sigma+\rho-1} \int_{t_\sigma}^{t_{\sigma+1}}\frac{\partial V}{\partial y}\,\gamma_{kl}x^k y^l\,dt \tag{2.54} \end{aligned} \]

\[ (1\leq i+nj\leq m+n-1,\quad 1\leq k+nl\leq m+2n-2). \]

Estimating the third and fourth terms on the right-hand side of (2.54) by means of (2.24), (2.47), and (2.44), taking (2.41) into account, we find:

\[ \begin{aligned} V(t_2) &< \sqrt{2}\,N\delta - L'(t_2-t_1) + (N_1+N_2)NC\varepsilon\tau \\ &\quad +\sum_{i,j}\sum_{s=\bar s}^{\bar{s}+r-1} \int_{t_s}^{t_{s+1}} \left\{ \frac{\partial V(x(t),y(t),t)}{\partial x}x^i(t)y^j(t) -\frac{\partial V(x(t_s),y(t_s),t_s)}{\partial x} x^i(t_s)y^j(t_s) \right\}c_{ij}(t)\,dt +\; . \end{aligned} \]

\[ + \sum_{k,l}\sum_{\sigma=\bar{\sigma}}^{\bar{\sigma}+\rho-1} \int_{t_\sigma}^{t_\sigma+1} \left\{ \frac{\partial V(x(t),\,y(t),\,t)}{\partial y}x^k(t)y^l(t) -\frac{\partial V(x(t_\sigma),\,y(t_\sigma),\,t_\sigma)}{dy} x^k(t_\sigma)y^l(t_\sigma) \right\}\gamma_{kl}(t)\,dt \tag{2.55} \]

\[ (1 \leq i+nj \leq m+n-1,\quad 1 \leq k+nl \leq m+2n-2). \]

Applying the finite-increment formulas, by the choice of \(\tau\) from (2.55) we obtain

\[ V(t_2)<\sqrt{2}\,N\delta+(N_1+N_2)NC\varepsilon\tau -\left[L'-\frac12 Q\tau(N_1+N_2)\right](t_2-t_1), \tag{2.56} \]

where \(Q=\mathrm{const}>0\) depends on \(K\) (see (1.2)), \(N\), \(C\), \(\varepsilon\), and does not depend on \(t_0'\). Let us subject \(\tau\) to the condition

\[ L'-\frac12 Q\tau(N_1+N_2)>0, \]

which can be achieved by the choice of \(t_0'\). Then, according to (2.48), from (2.56) we obtain

\[ V(t_2)<L. \]

On the other hand, from (2.21) and (2.45) it follows that \(V(t_2)\geq L\). From the contradiction obtained we conclude that the numbers \(\delta\) and \(t_0'\) can be chosen so that (2.51) holds for arbitrary initial values \(x(t_0')\), \(y(t_0')\) satisfying the inequality (2.50).

Assuming \(\delta\) and \(t_0'\) to be specified precisely in this way, let us show that relation (2.52) is satisfied.

Indeed, given any positive number \(\eta<\varepsilon\), put
\[ \lambda=\min W'(x,y)\quad \text{for } \|x,y\|=\eta. \]
Let, further, the positive numbers \(\xi\) and \(\tau'\) satisfy the inequalities:

\[ \sqrt{2}\,N\xi+(N_1+N_2)NC\eta\tau'<\lambda, \tag{2.57} \]

\[ \lambda'-\frac12 Q\tau'(N_1+N_2)>\frac12\lambda', \tag{2.58} \]

where
\[ \lambda'=\min W'(x,y)\quad \text{for } \xi\leq \|x,y\|\leq \varepsilon\;(\tau'\leq\tau). \]

According to (2.42), there exists a number \(\tau_1>t_0'\) such that the inequalities hold:

\[ t_{s+1}-t_s<\frac12\tau', \]

\[ t_{\sigma+1}-t_\sigma<\frac12\tau' \]

(for those \(s\) and \(\sigma\) for which \(t_s\geq\tau_1,\; t_\sigma\geq\tau_1\)),

\[ t_s-\tau_1<\frac12\tau',\qquad t_{\bar{\sigma}}-\tau_1<\frac12\tau', \]

where \(\bar{s}\) and \(\bar{\sigma}\) are the smallest of the mentioned values of \(s\) and \(\sigma\), and these inequalities are satisfied independently of the choice of the functions

\[ f_{ij}(t)\quad \text{and}\quad \varphi_{kl}(t)\qquad (1\le i+nj\le n+m-1,\; 1\le k+nl\le m+2n-2). \]

Finally, let the number \(\tau_2>\tau_1\) satisfy the relation

\[ \sqrt{2}\,N\varepsilon+(N_1+N_2)NC\varepsilon\tau' -\frac{1}{2}\lambda'(\tau_2-\tau_1)=0. \tag{2.59} \]

We shall show that for every solution \(x(t), y(t)\) of system (1.1) for which (2.50) is satisfied, there exists a value \(t=t^*\) in \([\tau_1,\tau_2]\) such that

\[ \|x(t^*),\,y(t^*)\|<\xi . \tag{2.60} \]

Indeed, suppose that for one of such solutions (2.60) does not hold; then along this solution we have

\[ V(\tau_2)=V(\tau_1)+\int_{\tau_1}^{\tau_2}\frac{dV^{(2.43)}}{dt}\,dt +\int_{\tau_1}^{\tau_2} \left(\frac{\partial V}{\partial x}X' +\frac{\partial V}{\partial y}Y'\right)\,dt . \]

Applying to this relation arguments analogous to those used in establishing (2.51), we find

\[ V(\tau_2)<\sqrt{2}\,N\varepsilon+(N_1+N_2)NC\varepsilon\tau' -\left[\lambda'-\frac{1}{2}Q\tau'(N_1+N_2)\right](\tau_2-\tau_1). \tag{2.61} \]

On the basis of (2.58), from (2.61) we derive the inequality

\[ V(\tau_2)<\sqrt{2}\,N\varepsilon+(N_1+N_2)NC\varepsilon\tau' -\frac{1}{2}\lambda'(\tau_2-\tau_1), \]

from which, by virtue of (2.59), we obtain

\[ V(\tau_2)<0, \]

which contradicts the positive definiteness of the function \(V\).

Thus inequality (2.60) is proved. But in that case, after the instant \(t^*\), the solution cannot leave the region \(\|x,y\|<\eta\).

Indeed, relations (2.57) and (2.58) mean that in this case we are under the same conditions under which the fulfillment of property (2.51) was shown, with the only difference that the role of the numbers \(\varepsilon,\delta,\tau,L,L'\) is played, respectively, by the numbers \(\eta,\xi,\tau',\lambda,\lambda'\).

Thus property (2.52) also holds and, consequently, the trivial solution of system (1.1) with the initial value \(t=t'_0\) is asymptotically stable.

Moreover, it follows at once from the arguments given that the stability is uniform with respect to \(t_0\) and the initial values \(x_0,y_0\). The proof of this fact is the same as at the end of the preceding theorem.

References

1. A. M. Lyapunov. Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1956.
2. N. N. Krasovskii. Some Problems in the Theory of Stability of Motion. Moscow, GIFML, 1959.

Received by the editors
January 20, 1967

Institute of Mathematics
Academy of Sciences of the BSSR