Full Text
UDC 517.917
ON DETECTING ASYMPTOTIC STABILITY BY MEANS OF SMALL \(vd\)-NUMBERS
Yu. S. Bogdanov
In the \(n\)-dimensional real Euclidean space \(E_0=\{\xi\}\), let us single out a topological ball \(\overline E\) containing the origin \(O_0\). Suppose that the \(n\)-dimensional vector function \(f(\xi)\) is defined and satisfies the Lipschitz condition on \(\overline E\), with \(f(O_0)=O_0\). We investigate the equilibrium point \(x=O_0\) of the system of equations
\[ dx/d\tau=f(x) \tag{1} \]
for stability under the assumption that all solutions of (1) are indefinitely continuable in both directions.
Suppose that on the domain \(\dot E=\overline E\setminus\{O_0\}\) there is defined a function \(v\) of type \(L\): continuous, positive, and such that \(\lim_{\xi\to O_0}v(\xi)=0\) and \(\lim_{m\to\infty}v(\xi_m)=+\infty\) for every sequence \(\{\xi_m\}\) tending to the boundary of \(\overline E\) (in any compact subset of \(\overline E\) there is contained all but a finite number of the terms of the sequence \(\{\xi_m\}\)). In addition, consider a real function \(d(\gamma_1,\gamma_2)\), defined and continuous for all positive values of the arguments, and such that for all \(\gamma_1,\gamma_2,\gamma_3,\gamma\), with \(0<\gamma_1<\gamma_2<\gamma_3,\ 0<\gamma\), the following hold:
\[ \begin{aligned} d_1)&\quad d(\gamma_2,\gamma_1)=-d(\gamma_1,\gamma_2),\\ d_2)&\quad d(\gamma_2,\gamma)>d(\gamma_1,\gamma),\\ d_3)&\quad d(\gamma_3,\gamma_2)+d(\gamma_2,\gamma_1)>d(\gamma_3,\gamma_1),\\ d_4)&\quad U\{d(\gamma,\gamma_1)\}=(+\infty,-\infty). \end{aligned} \]
For every nonzero solution \(x_\xi,\ x_\xi(0)=\xi\) of system (1), one can define the small \(vd\)-number \(\overline\Omega vdx_\xi\) by the formula
\[ \overline\Omega vdx_\xi = \max\left\{ \lim_{\tau\to+\infty}\frac{1}{\tau}d[v(x_\xi(\tau)),v(\xi)], -\lim_{\tau\to+\infty}\frac{1}{\tau}d[v(x_\xi(-\tau)),v(\xi)] \right\}. \tag{2} \]
If \(\overline\Omega vdx_\xi<0\) for all \(\xi\in\dot E\), then the equilibrium point \(O_0\) is asymptotically stable with respect to system (1), and \(\overline E\) is the domain of attraction of this equilibrium point (a sufficient condition for asymptotic stability). On the other hand, if \(O_0\) is an asymptotically stable equilibrium point of system (1) with domain of attraction \(\overline E\), then always \(\overline\Omega vdx_\xi\le 0\), and there exist functions \(v_0,d_0\) such that \(\overline\Omega v_0d_0x_\xi<0\) for all \(\xi\in\dot E\) (see [1, 2]). In the present note it is shown that the asymptotic stability of \(O_0\) can be detected by means of a sufficient condition for asymptotic stability for any prescribed function \(v\) by a suitable choice of \(d\), or, for any \(v\) and \(d\), by changing the scale of \(v\), or finally, for any \(v\) and \(d\), by replacing the argument \(\tau\) in (1).
1. The function \(\chi(\gamma)\)
Suppose that \(O_0\) is asymptotically stable with respect to (1), and that \(\Xi\) is the domain of attraction for \(O_0\). Consider on \(\Xi\) a function \(v\) of type \(L\) with level sets \(\Gamma_\gamma\). Take two numbers \(\bar\gamma>\gamma>0\). For \(\xi\in\Gamma_\gamma\) define the set \(T(\xi,\gamma,\bar\gamma)\) by the rule
\[ T(\xi,\gamma,\bar\gamma)=\{\tau\mid \tau\geq 0,\ x_\xi(\tau)\in\Gamma_{[\gamma,\bar\gamma]}\}, \]
where \(\Gamma_{[\gamma,\bar\gamma]}\) is the union of all \(\Gamma_\gamma\), \(\gamma\in[\gamma,\bar\gamma]\). For fixed \(\gamma\) and \(\bar\gamma\), and for arbitrary \(\xi\in\Gamma_\gamma\), the set \(T(\xi,\gamma,\bar\gamma)\) is bounded above. Indeed, choose \(\varepsilon\) so small that the entire ball \(\Sigma_\varepsilon^*\) of radius \(\varepsilon\) with center at \(O_0\) lies strictly inside \(\Gamma^*=\Gamma_{(0,\gamma)}\). System (1) is autonomous; therefore (see [3, p. 32]) the asymptotic stability of \(O_0\) is necessarily uniform in \(\tau_0\) and in the domain \(\Xi\). Consequently, for the compact set \(\Gamma_\gamma\subset\Xi\) there is a \(\tau'\) such that \(x_\xi(\tau)\in\Sigma_\varepsilon^*\) for all \(\tau\geq\tau'\), \(\xi\in\Gamma_\gamma\), and therefore
\[ \sup_{\xi\in\Gamma_\gamma} T(\xi,\gamma,\bar\gamma)\leq \tau'<+\infty . \]
Set
\[ \chi(\gamma,\bar\gamma)=\sup_{\xi\in\Gamma_\gamma} T(\xi,\gamma,\bar\gamma). \]
It is obvious that the function \(\chi(\gamma,\bar\gamma)\) does not increase with respect to \(\bar\gamma\) and does not decrease with respect to \(\gamma\), and moreover
\[ \lim_{\bar\gamma\to +0}\chi(\gamma_1,\bar\gamma) = \lim_{\gamma\to +\infty}\chi(\gamma,\gamma_2) = +\infty . \]
Set
\[ \chi(\gamma)= \begin{cases} \chi(1,\gamma), & \text{for } 0<\gamma<1,\\ 0, & \text{for } \gamma=1,\\ -\chi(\gamma,1), & \text{for } \gamma>1. \end{cases} \]
The function \(\chi\) is defined for all positive \(\gamma\) and does not increase; moreover,
\[ \lim_{\gamma\to +0}\chi(\gamma) = -\lim_{\gamma\to +\infty}\chi(\gamma) = +\infty . \]
2. The function \(\bar d(\gamma_1,\gamma_2)\)
For any natural number \(m>1\), set
\[ d(m)=\lim_{\gamma\to m+1+0}\chi(\gamma)-m,\qquad d\!\left(\frac{1}{m}\right)= \lim_{\gamma\to \frac{1}{m+1}-0}\chi(\gamma)+\frac{1}{m}. \]
In addition, set \(d(1)=0\). Finally, at all other points of the positive \(\gamma\)-axis, we linearly interpolate the function \(d(\gamma)\). The function \(d\) is continuous and decreases for all \(\gamma>0\). From the definition of \(d\) it follows that, for \(\gamma\in[m,m+1]\),
\[ d(\gamma)\leq d(m)<-\lim_{\gamma\to m+1+0}\chi(\gamma)\leq \chi(\gamma) \]
and, for \(\gamma\in\left[\frac{1}{m+1},\frac{1}{m}\right]\)
\[ d(\gamma) \geq d\left(\frac{1}{m}\right) \geq \lim_{\gamma \to \frac{1}{m+1}-0} \chi(\overline{\gamma}) \geq \chi(\gamma). \]
Consequently, \(d(\gamma) < \chi(\gamma)\) for all \(\gamma \geq 2\), and \(d(\overline{\gamma}) > \chi(\overline{\gamma})\) for all \(\overline{\gamma} \leq \frac{1}{2}\). Define \(\overline{d}(\gamma_1,\gamma_2)\) by the rule
\[ \overline{d}(\gamma_1,\gamma_2)=d(\gamma_2)-d(\gamma_1). \tag{3} \]
The function \(\overline{d}\) is continuous and satisfies the requirements \(d_{1-4}\).
3. ESTIMATE OF \(\overline{\Omega}\,v d x_\xi\)
Take any nonzero solution \(x_\xi\) of system (1). The function \(v[\tau]=v[x(\tau)]\) is continuous, takes all positive values, and tends to \(0\) as \(\tau \to +\infty\); therefore there exist numbers \(\tau_0' < \tau_0 < \tau_0''\) such that
\[ v[\tau_0']=2,\quad v[\tau_0]=1,\quad v[\tau_0'']=\frac{1}{2}, \]
and \(v[\tau_0-\tau] > 2\) for all \(\tau > \tau_0-\tau_0'\), \(v[\tau_0+\overline{\tau}] < \frac{1}{2}\) for all \(\overline{\tau} > \tau_0''-\tau_0\). If \(v[\tau_1]>v[\tau_2]\), then
\[ \chi(v[\tau_1],v[\tau_2])=\chi(v(\xi_0),v[x_{\xi_0}(\tau_2-\tau_1)])>\tau_2-\tau_1, \]
where \(\xi_0=x_\xi(\tau_1)\). Moreover, for \(\tau>\tau_0-\tau_0'\) and \(\overline{\tau}>\tau_0''-\tau_0\), we successively derive
\[ \chi(v[\tau_0-\tau],v[\tau_0])=\chi(v[\tau_0-\tau],1)=-\chi(v[\tau_0-\tau])>\tau, \]
\[ \chi(v[\tau_0],v[\tau_0+\overline{\tau}])=\chi(1,v[\tau_0+\overline{\tau}])=\chi(v[\tau_0+\overline{\tau}])>\overline{\tau}, \]
\[ d(v[\tau_0-\tau])<-\tau,\quad d(v[\tau_0+\overline{\tau}])>\overline{\tau}, \]
\[ \overline{d}(v[x_\xi(\tau_0-\tau)],v[x_\xi(\tau_0)])>\tau, \tag{4} \]
\[ \overline{d}(v[x_\xi(\tau_0+\overline{\tau})],v[x_\xi(\tau_0)])<-\overline{\tau}. \tag{5} \]
In definition (2), the second argument \(v(\xi)\) of the function \(d\), on the basis of \(d_{1-3}\) (see [2]), may be replaced by any other fixed positive number; therefore it follows from (4) and (5) that
\[ \overline{\Omega}\, v d x_\xi \leq -1. \tag{6} \]
4. A NECESSARY CONDITION FOR ASYMPTOTIC STABILITY
From the arguments carried out (see, in particular, estimate (6)) the following proposition follows.
If \(O_0\) is asymptotically stable with respect to system (1), and \(\Xi\) is the domain of attraction \(O_0\), then for any function \(v\) of type \(L\) there exists a function \(d\) with properties \(d_{1-4}\) (and even representable in the form (3)) such that the small \(v d\)-numbers of all nonzero solutions of (1) are essentially negative.
In particular, if \(\Xi\) coincides with the whole space \(E_0\), then one may take \(\|\xi\|\) as \(v(\xi)\). In the latter case the proposition proved takes the form:
If \(O_0\) is asymptotically stable with respect to (1) as a whole, then there exists a function \(\tilde d\) with properties \(d_{1-4}\), with \(d_3\) satisfied with the equality sign, such that for any nonzero solution of (1) the following simultaneously hold:
\[
\lim_{\tau\to+\infty}\frac{1}{\tau}\tilde d(\|x(\tau_0+\tau)\|,\|x(\tau_0)\|)\le -1,
\]
\[
\lim_{\tau\to+\infty}\frac{1}{\tau}\tilde d(\|x(\tau_0-\tau)\|,\|x(\tau_0)\|)\ge 1.
\]
In the case where \(\Xi\) is an open ball of radius \(\rho\) with center at \(O_0\), one may take as \(v(\xi)\) the function \(\|\xi\|/(\rho-\|\xi\|)\).
Remark 1. The function \(d(\gamma)\) constructed in Section 2 is a decreasing piecewise-linear function. It is obvious that there exist arbitrarily smooth decreasing functions \(d^*\) of a positive argument \(\gamma\) such that \(d^*(\gamma)<d(\gamma)\) for all \(\gamma\ge 2\) and \(d^*(\bar\gamma)>d(\bar\gamma)\) for all \(\bar\gamma\le \frac12\). The function
\[
d^*(\gamma_1,\gamma_2)=d^*(\gamma_2)-d^*(\gamma_1)
\]
for positive values of the arguments has partial derivatives of any order. Moreover, it satisfies the conditions \(d_{1-4}\), and for any nonzero solution \(x_\xi\) of system (1) one has
\[
\overline{\Omega}\, v d^* x_\xi \le -1.
\]
Remark 2. The quantities \(\overline{\Omega}\, v d x_\xi\) for some, or even for all, nonzero solutions (1) may turn out to be equal to \(-\infty\). Moreover, there exist systems (1) for which a finite value for small \(vd\)-numbers \(\overline{\Omega}\, v d x_\xi\) can be obtained only by passing from a given function \(v\) to another function of the same type (a replacement of the function \(d\) is insufficient).
5. The function \(\bar v\)
Let, as before, the equilibrium point \(O_0\) be asymptotically stable with respect to (1), with domain of attraction \(\Xi\). Take arbitrary functions \(v\) and \(d\). The function \(\delta=d(\gamma,1)\) is continuous, increasing, and assumes all possible real values; therefore there exists an increasing continuous positive inverse function \(\gamma=q(\delta)\), defined for all real \(\delta\):
\[
d[q(\delta),1]=\delta,\qquad -\infty<\delta<+\infty.
\]
Construct, as above, the function \(\bar d(\gamma,1)\), and then form the superposition \(\varphi(\gamma)=q[\bar d(\gamma,1)]\). Obviously
\[
\bar d(\gamma,1)=d[q(\bar d(\gamma,1)),1],\qquad 0<\gamma<+\infty .
\tag{7}
\]
Introduce the function
\[
\bar v(\xi)=\varphi(v(\xi)),\qquad \xi\in\Xi .
\]
The function \(\bar v\) is continuous, positive, tends to \(0\) as \(\xi\to 0\), and tends to \(+\infty\) when approaching the boundary of \(\Xi\). Moreover, on the basis of (7)
\[ \overline d[v(\xi),1]=d[\overline v(\xi),1]. \]
Further,
\[ d(1,1)=0,\qquad q(0)=1,\qquad \varphi(1)=q[\overline d(1,1)]=q(0)=1, \]
and therefore the level sets of the functions \(v\) and \(\overline v\) for \(\gamma=1\) coincide. For every \(\gamma>0\) there exists, and moreover a unique, \(\gamma>0\) such that \(\Gamma_\gamma=\overline\Gamma_\gamma\), where \(\overline\Gamma_\gamma\) is the level set of \(\overline v\) corresponding to the value \(\gamma\). If \(\xi\in\Gamma_\gamma\), then for arbitrary \(\tau\)
\[ \overline d\bigl(v[x_\xi(\tau)],\,v[x_\xi(0)]\bigr) = d\bigl(\overline v[x_\xi(\tau)],\,\overline v[x_\xi(0)]\bigr) \]
and, consequently,
\[ \overline\Omega\,vdx_\xi=\overline\Omega\,\overline v\,dx_\xi, \]
i.e. the small \(\overline{vd}\)- and \(\overline v d\)-numbers of the solution \(x_\xi\) coincide.
6. SECOND NECESSARY CONDITION FOR ASYMPTOTIC STABILITY
From the arguments of Section 5 it follows that, along with the necessary condition of Section 4, another necessary criterion for the asymptotic stability of \(O_0\) is valid.
If \(O_0\) is asymptotically stable with respect to (1), and \(\Xi\) is the domain of attraction of \(O_0\), then for any \(d\) with properties \(d_{1—4}\) and any \(v\) of type Л there exists \(\overline v\) of type Л, the totality of whose level sets coincides with the totality of the level sets of \(v\), such that
\[ \overline\Omega\,\overline v\,dx_\xi\le -1 \]
for every nonzero solution of (1).
Remark 1. The function \(\overline v\) can always be chosen in the class of functions of the same smoothness as \(v\).
Remark 2. If \(O_0\) is asymptotically stable with respect to (1) in the large, then from the second necessary condition for stability there follows the existence of a completely smooth increasing function \(\varphi(\gamma)\) such that, for every nonzero solution (1), simultaneously
\[ \lim_{\tau\to+\infty}\frac1{\tau}\ln\varphi[\|x(\tau)\|]\le -1, \]
\[ \lim_{\tau\to+\infty}\frac1{\tau}\ln\varphi[\|x(-\tau)\|]\ge -1. \]
An analogous result will also be true when \(\Xi\) is a ball with center at \(O_0\).
Remark 3. The passage from \(v\) to \(\overline v\) can be replaced by a transformation of the argument \(\tau\) of system (1).
References
- Bogdanov Yu. S. DAN SSSR, 158, No. 1, 9–12, 1964.
- Bogdanov Yu. S. Differential Equations, vol. I, No. 1, 41–52, 1965.
- Krasovskii N. N. Some problems in the theory of stability of motion. Moscow, Fizmatgiz, 1959, p. 211.
Received by the editors
24 November 1965
Belorussian State University