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UDC 517.934
ON STABILITY IN THE CRITICAL CASE OF A DOUBLE ZERO ROOT FOR SYSTEMS WITH AFTEREFFECT
V. P. PROKOP'EV, S. N. SHIMANOV
The paper gives a method for solving the problem of investigating the unperturbed motion for systems with aftereffect in the critical case of a double zero root, where two groups of solutions of the first-approximation equations correspond to the double zero root.
1. STATEMENT OF THE PROBLEM
Consider the equation
\[ \frac{d x_i(t)}{dt} = \sum_{j=1}^{n}\int_{-\tau}^{0} x_j(t+\theta)\,d\eta_{ij}(\theta) + X_i\bigl(x_1(t+\theta),\ldots,x_n(t+\theta)\bigr) \qquad (1.1) \]
\[ (i=1,\ldots,n), \]
where the integrals are understood in the Stieltjes sense [1]; \(X_i\) are functionals defined on piecewise-continuous functions \(x_i(\theta)\) of the argument \(\theta\) \((-\tau \le \theta \le 0)\) and satisfying the Lipschitz conditions
\[ \left|X_i(x'')-X_i(x')\right|<L\|x''-x'\|,\qquad L=L_1\{\|x''\|+\|x'\|\}^{\alpha_1}, \]
\[ \|x(\theta)\|=\sup\bigl(|x_1(\theta)|,\ldots,|x_n(\theta)|\bigr) \quad \text{for } -\tau\le \theta\le 0, \qquad (1.2) \]
where \(L_1\) and \(\alpha_1\) are positive numbers; \(X_i(0,\ldots,0)=0\). The motion \(x=0\) will be called the unperturbed motion of system (1.1). We shall assume that, when any analytic function \(x(y,\theta)\) of \(y\) is substituted into the functional \(X_i\), an analytic function of \(y\) is obtained.
Consider the characteristic equation of the system of the first approximation
\[ \Delta(\lambda)= \left|-\delta_{ij}\lambda+\int_{-\tau}^{0}e^{\lambda\theta}\,d\eta_{ij}(\theta)\right|=0. \qquad (1.3) \]
Suppose that among the countable set of roots \(\lambda_1,\lambda_2,\ldots,\lambda_n,\ldots\) of equation (1.3) there are two roots equal to zero \((\lambda_1=\lambda_2=0)\), while the remaining roots have negative real parts
\[ \operatorname{Re}\lambda_j\le -2\sigma \qquad (j>2;\ \sigma>0). \qquad (1.4) \]
Then the critical case of a double zero root occurs. In solving the problem it is assumed that two groups of solutions of the first-approximation equations correspond to the double zero root, i.e., all minors of order \((n-1)\) of the characteristic determinant (1.3)
for \(\lambda=0\) are equal to zero. The stability study of system (1.1) will proceed in the same way as in [2].
Let us consider, as an element of the trajectory \(x_i(t)\) of system (1.1), the segment of the trajectory \(x_i(t+\theta)\) \((-\tau \leq \theta \leq 0)\). In the functional space \(\{x_i(\theta)\}\) \((-\tau \leq \theta \leq 0)\), system (1.1) corresponds to the system of ordinary differential-operator equations [1]
\[ \frac{d x_t(\theta)}{dt}=A x_t(\theta)+R\bigl(x_t(\theta)\bigr), \tag{1.5} \]
\[ x_t(\theta)=\{x_1(t+\theta),\ldots,x_n(t+\theta)\} =\{x_{1t}(\theta),\ldots,x_{nt}(\theta)\}, \]
\[ A x_t(\theta)= \begin{cases} \dfrac{d x_i(\theta)}{d\theta}, & (-\tau \leq \theta < 0),\\[1.2em] \displaystyle \sum_{j=1}^{n}\int_{-\tau}^{0} x_j(\theta)\,d\eta_{ij}(\theta), & (\theta=0), \end{cases} \qquad (i=1,\ldots,n), \]
\[ R\bigl(x_t(\theta)\bigr)= \begin{cases} 0, & (-\tau \leq \theta < 0),\\ X_i\bigl(x_{1t}(\theta),\ldots,x_{nt}(\theta)\bigr), & (\theta=0)\quad (i=1,\ldots,n). \end{cases} \]
The problem of studying the stability of the unperturbed motion of system (1.5), \(x_t(\theta)=0\), is equivalent to the corresponding problem for system (1.1), since \(x_t(\theta)=x(t+\theta)\). We shall assume the initial functions to be differentiable and shall consider the operator \(A\) for \(t \geq 0\).
2. SPLITTING OF THE LINEAR OPERATOR
Since equation (1.3) has a double zero root, \(\Delta(0)\) and \(\Delta'(0)\) are equal to zero, while the second derivative of \(\Delta(\lambda)\) at \(\lambda=0\) is nonzero and has the form
\[ \Delta''(0)= \sum_{i,j=1}^{n} \left\{ \left[-\delta_{ij}+\int_{-\tau}^{0}\theta\,d\eta_{ij}(\theta)\right] \sum_{k,l=1}^{n} \left[-\delta_{kl}+ \int_{-\tau}^{0}\theta\,d\eta_{kl}(\theta)\right] \Delta_{ij,kl}(0) \right\} \neq 0 \tag{2.1} \]
\[ (i\neq k;\quad j\neq l), \]
where \(\Delta_{ij,kl}(0)\) is the minor obtained by deleting in \(\Delta(0)\) the \(i\)-th and \(k\)-th rows and the \(j\)-th and \(l\)-th columns, and taken with the sign \((-1)^{i+j+k+l}\).
Let \(b^{(1)}(\theta)=(b^{(1)}_1,\ldots,b^{(1)}_n)\), \(b^{(2)}(\theta)=(b^{(2)}_1,\ldots,b^{(2)}_n)\) be two linearly independent solutions of system (1.1) corresponding to the double zero root, where
\[ b^{(1)}_j=x^{(1)}_{jt} = \sum_{v,l=1}^{n} \left[-\delta_{vl}+\int_{-\tau}^{0}\theta\,d\eta_{vl}(\theta)\right] \Delta_{1j,2l}(0)\, \frac{1}{\Delta_{11,22}(0)\Delta''(0)} \tag{2.2} \]
\[ (j=1,\ldots,n;\quad j\neq l), \]
\[ b_l^{(2)}=\chi_{lt}^{(2)}= \sum_{\substack{p,j=1\\}}^{n} \left[ -\delta_{\mu i}+\int_{-\tau}^{0}\theta\,d\eta_{\mu j}(\theta) \right]\Delta_{1l,2j}(0)\, \frac{1}{\Delta_{11,22}(0)\Delta''(0)} \tag{2.3} \]
\[ (l=1,\ldots,n;\quad l\ne j). \]
Here it is assumed that \(\Delta_{11,22}(0)\) is nonzero. It is obvious that \(b^{(1)}(\theta)\) and \(b^{(2)}(\theta)\) are constant in \(\theta\).
Let us now consider two functionals
\[ f_1\left|x_t(\theta)\right|= \sum_{i=1}^{n} \left\{ \left[ -x_{it}(0)+ \sum_{j=1}^{n}\int_{-\tau}^{0}\int_{0}^{\theta} x_{jt}(\xi)\,d\xi\,d\eta_{ij}(\theta) \right] \sum_{k=1}^{n}\Delta_{i1,k2}(0) \right\} \tag{2.4} \]
\[ (i\ne k), \]
\[ f_2\left|x_t(\theta)\right|= \sum_{k=1}^{n} \left\{ \left[ -x_{kt}(0)+ \sum_{l=1}^{n}\int_{-\tau}^{0}\int_{0}^{\theta} x_{lt}(\xi)\,d\xi\,d\eta_{kl}(\theta) \right] \sum_{i=1}^{n}\Delta_{k1,i2}(0) \right\} \tag{2.5} \]
\[ (k\ne i), \]
defined on differentiable functions \(x_t(\theta)\) \((-\tau\le \theta\le 0)\). These functionals have the following properties [3]:
\[ \begin{aligned} f_1\left|b^{(1)}(\theta)\right|&=1, & f_1\left|b^{(2)}(\theta)\right|&=0,\\ f_2\left|b^{(1)}(\theta)\right|&=0, & f_2\left|b^{(2)}(\theta)\right|&=1. \end{aligned} \tag{2.6} \]
One can also verify the identities
\[ f_1\left|Ax_t(\theta)\right|=0,\qquad f_2\left|Ax_t(\theta)\right|=0, \tag{2.7} \]
from which it follows that \(f_1|x_t(\theta)|\) and \(f_2|x_t(\theta)|\) are functional integrals. Then
\[ f_1\left|x_t(\theta)\right|=f_1\left|x_0(\theta)\right|, \qquad f_2\left|x_t(\theta)\right|=f_2\left|x_0(\theta)\right| \quad \text{for } t\ge 0, \tag{2.8} \]
where \(x_0(\theta)\) is the initial function for the solution \(x_t(\theta)\), chosen so that
\[ f_1\left|x_0(\theta)\right|=0,\qquad f_2\left|x_0(\theta)\right|=0. \tag{2.9} \]
Then, provided conditions (1.4), (2.8), (2.9) are satisfied, all solutions that decrease asymptotically according to an exponential law will be located in the functional space \(\{x_t(\theta)\}\) in the plane \(L\) [2]
\[ f_1\left|x_t(\theta)\right|=0,\qquad f_2\left|x_t(\theta)\right|=0. \tag{2.10} \]
In the functional space \(\{x_t(\theta)\}\), let us decompose an arbitrary element \(x(\theta)\) into three summands
\[ x(\theta)=z(\theta)+b^{(1)}(\theta)y_1+b^{(2)}(\theta)y_2, \tag{2.11} \]
where
\[ y_1=f_1\left|x(\theta)\right|,\qquad y_2=f_2\left|x(\theta)\right|. \tag{2.12} \]
It can be shown that the element \(z(\theta)\) lies in the plane \(L\) and that the decomposition of the functions \(x(\theta)\) in this case will be unique. Taking into account (2.6), (2.7), (2.11), (2.12), one can write system (1.5) in the variables \(z_t(\theta), y_1, y_2\):
\[ \frac{d y_1}{d t}=Y_1\bigl(y_1,y_2,z_t(\theta)\bigr), \]
\[ \frac{d y_2}{d t}=Y_2\bigl(y_1,y_2,z_t(\theta)\bigr), \tag{2.13} \]
\[ \frac{d z_t(\theta)}{d t}=A z_t(\theta)+Z\bigl(y_1,y_2,z_t(\theta),\theta\bigr), \]
where
\[
Y_1\bigl(y_1,y_2,z_t(\theta)\bigr)=f_1\bigl[R(x_t(\theta))\bigr],\qquad
Y_2\bigl(y_1,y_2,z_t(\theta)\bigr)=f_2\bigl[R(x_t(\theta))\bigr],
\]
and \(Z\bigl(y_1,y_2,z_t(\theta),\theta\bigr)\) is an operator defined as follows:
\[
Z\bigl(y_1,y_2,z_t(\theta),\theta\bigr)=
\begin{cases}
-b_k^{(1)}Y_1-b_k^{(2)}Y_2, & (-\tau\leqslant \theta<0),\\[4pt]
X_k-b_k^{(1)}Y_1-b_k^{(2)}Y_2, & (\theta=0),
\end{cases}
\qquad (k=1,\ldots,n).
\tag{2.14}
\]
Let us note that \(Z\) is a function of \(\theta\) belonging to \(L\), since \(f_1[Z]=0\) and \(f_2[Z]=0\). Obviously,
\[
Y_1(0,0,0)\equiv 0,\qquad Y_2(0,0,0)\equiv 0,\qquad Z(0,0,0,\theta)\equiv 0.
\tag{2.15}
\]
The functionals \(Y_1\) and \(Y_2\) and the operator \(Z\) satisfy Lipschitz conditions of type (1.2).
For the initial condition \(x_0(\theta)\) in a neighborhood of the origin \(x=0\), system (1.5) admits a unique solution \(x_t(\theta)\); then \(y_1, y_2, z_t(\theta)\) form a solution of system (2.13) with initial conditions \(y_1(0), y_2(0), z_0(\theta)\), and the solution of system (2.13) is unique.
Denote by \(Y_1^0(y_1,y_2)\), \(Y_2^0(y_1,y_2)\), and \(Z^0(y_1,y_2,\theta)\) the analytic functions
\[
Y_1^0(y_1,y_2)\equiv Y_1(y_1,y_2,0)
=Y_1^{(m)}(y_1,y_2)+Y_1^{(m+1)}(y_1,y_2)+\cdots
\]
\[
Y_2^0(y_1,y_2)\equiv Y_2(y_1,y_2,0)
=Y_2^{(m)}(y_1,y_2)+Y_2^{(m+1)}(y_1,y_2)+\cdots
\tag{2.16}
\]
\[
Z^0(y_1,y_2,\theta)\equiv Z(y_1,y_2,0,\theta)
=Z^{(s)}(y_1,y_2,\theta)+Z^{(s+1)}(y_1,y_2,\theta)+\cdots,
\]
where \(Y_1^{(m)}\), \(Y_2^{(m)}\), and \(Z^{(s)}\) are, respectively, the collections of terms of the \(m\)-th and \(s\)-th orders, with \(Y_1^{(m)}\) and \(Y_2^{(m)}\) forms with constant coefficients; \(m\geq 2\) and \(s\geq 2\).
By means of a Lyapunov transformation of the variable \(z\) one can ensure that \(s>m\) [2, 4]. To this end we consider the system of equations
\[
A u(\theta)+Z\bigl(y_1,y_2,u(\theta),\theta\bigr)=0,
\tag{2.17}
\]
where \(Z\in L\). This system admits a solution \(u^*(y_1,y_2,\theta)\in L\); \(u^*(0,0,\theta)=0\). In system (2.13) we make the change of variable according to the formula
\[
z_t(\theta)=z_{1t}(\theta)+u^*(y_1,y_2,\theta).
\tag{2.18}
\]
The transformed system will have the same form as (2.13). Therefore we assume that in (2.13) the corresponding change of variables has been made and that for (2.13) \(s>m\) holds.
3. INVESTIGATION OF THE STABILITY OF THE SYSTEM
We shall investigate the stability of the unperturbed motion of the system (2.13) in the domain \(\|x_t(\theta)\|<H\). As will be shown below, the principal role in determining the stability of the motion will be played by the sign of the expression
\[ P(y_1,y_2)=y_1^m Y_1^{(m)}(y_1,y_2)+y_2^m Y_2^{(m)}(y_1,y_2). \tag{3.1} \]
Accordingly, the following cases will be considered: a) \(P(y_1,y_2)<0\); b) \(P(y_1,y_2)>0\) at least on one straight line; c) \(P(y_1,y_2)<0\) everywhere in the domain \(\|x_t(\theta)\|<H\), with the exception of one or several straight lines, where \(P(y_1,y_2)=0\); d) \(P(y_1,y_2)=0\).
Let us consider the first case. In the functional space \(\{z_t(\theta)\}\) on the plane \(L\), for the linear system
\[ \frac{d z_t(\theta)}{dt}=A z_t(\theta),\qquad f_1\{z_t(\theta)\}=0,\qquad f_2\{z_t(\theta)\}=0 \tag{3.2} \]
one can construct, on the basis of the results of [1], a functional \(v_2(z_t(\theta),t)\) satisfying the following conditions:
\[ c_1\|z_t(\theta)\|<v_2(z_t(\theta),t)<c_2\|z_t(\theta)\|, \tag{3.3} \]
\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta v_2}{\Delta t}\right)<-c_3\|z_t(\theta)\|, \tag{3.4} \]
\[ \left|v_2(z_t''(\theta),t)-v_2(z_t'(\theta),t)\right| <c_4\|z_t''-z_t'\|, \tag{3.5} \]
where \(c_1,c_2,c_3,c_4\) are positive constants.
We form the functional \(v^*(x_t(\theta),t)\) for the system (2.13) (or (1.5)) in the form
\[ v^*(x_t(\theta),t)=v_2^2(z_t(\theta),t)+\frac{1}{m+1}\left(y_1^{m+1}+y_2^{m+1}\right). \tag{3.6} \]
Let \(m\) be an odd number. There exist positive constants \(B_1,B_2,B_3,B_4\) such that the inequalities
\[ |y_1|<B_1\|x_t(\theta)\|,\qquad |y_2|<B_2\|x_t(\theta)\|,\qquad \|z_t(\theta)\|<B_3\|x_t(\theta)\|, \]
\[ \|x_t(\theta)\|<B_4\left\{|y_1|+|y_2|+\|z_t(\theta)\|\right\} \tag{3.7} \]
hold.
Let us estimate \(v^*(x_t(\theta),t)\) in the domain \(\|x_t(\theta)\|<H\) from above:
\[ \|v^*\|< \left|c_2^2B_3^2H+\frac{H^m}{m+1}\left(B_1^{m+1}+B_2^{m+1}\right)\right| \|x_t(\theta)\|. \tag{3.8} \]
We also have
\[ v^*\ge \frac{1}{m+1}\left(|y_1|^{m+1}+|y_2|^{m+1}\right) -c_1^2\|z_t(\theta)\|^2 \ge \omega\|x_t(\theta)\|. \tag{3.9} \]
Let us compute along the trajectory of the system (1.5)
\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta v^*}{\Delta t}\right)_{(1.5)} = y_1^mY_1^{(m)}+y_2^mY_2^{(m)} +y_1^mY_1^{(m+1)} \]
\[ +y_2^mY_2^{(m+1)}+\ldots +y_1^m\left|Y_1-Y_1^{(0)}\right| +y_2^m\left|Y_2-Y_2^{(0)}\right|+ \]
\[ +2|v_2|\limsup_{\Delta t\to +0}\left(\frac{\Delta v_2}{\Delta t}\right)_{(3.2)}+ \]
\[ +2|v_2|\limsup_{\Delta t\to +0}\left[\left(\frac{\Delta v_2}{\Delta t}\right)_{(1.5)}-\left(\frac{\Delta v_2}{\Delta t}\right)_{(3.2)}\right]. \]
Choose \(\tilde H_1<\tilde H\) so small that, taking into account (3.3), (3.4), (3.5), the inequalities
\[ \frac{1}{3}\left(y_1^mY_1^{(m)}+y_2^mY_2^{(m)}\right)+y_1^mY_1^{(m+1)}+y_2^mY_2^{(m+1)}+\ldots<0, \]
\[ \frac{2}{3}c_2c_3-2c_2c_4q_1(\|y_1\|,\|y_2\|,\|z_t(\theta)\|)>0, \]
\[ \frac{1}{3}\left(y_1^mY_1^{(m)}+y_2^mY_2^{(m)}\right)-\frac{2}{3}c_2c_3\|z_t(\theta)\|^2+ \]
\[ +2c_2c_4\|z_t(\theta)\|\left|y_1^k y_2^l\right|q_2(\|y_1\|,\|y_2\|)+y_1^m\|z_t(\theta)\|q_3(\|y_1\|,\|y_2\|,\|z_t(\theta)\|)+ \]
\[ +y_2^m\|z_t(\theta)\|q_4(\|y_1\|,\|y_2\|,\|z_t(\theta)\|)<0 \]
\[ (k+l=m). \]
The fulfillment of these three inequalities can always be achieved, since \(P(y_1,y_2)<0\), while the functions \(q_1,q_2,q_3,q_4\) tend to zero as \(y_1,y_2,z_t(\theta)\) tend to zero. Thus we obtain
\[ \limsup_{\Delta t\to +0}\left(\frac{\Delta v^*}{\Delta t}\right)_{(1.5)} < \frac{1}{3}\left[y_1^mY_1^{(m)}+y_2^mY_2^{(m)}-2c_2c_3\|z_t(\theta)\|^2\right]<0. \]
Then, analogously to how this was done when estimating \(v^*\) from below, there is a continuous, monotonically increasing positive function \(\omega_1\) such that
\[ \limsup_{\Delta t\to +0}\left(\frac{\Delta v^*}{\Delta t}\right)_{(1.5)} \le -\omega_1(\|x_t(\theta)\|). \tag{3.10} \]
From conditions (3.8), (3.9), (3.10), by Theorem 30.1 of [1], the motion \(x_t(\theta)=0\) of system (1.5) (and consequently also of (1.1)) will be asymptotically stable under the conditions: \(P(y_1,y_2)<0\) and \(m\) is an odd number.
Consider the case when \(m\) is an even number. We shall show that, by choosing differentiable initial functions \(x_0(\theta)\), arbitrarily small in norm, the functional \(v^*\) can be made negative. Choose
\[ y_1(0)=-\eta_1<0,\qquad y_2(0)=-\eta_2<0,\qquad z_0(\theta)=0. \]
Substituting \(x_0(\theta)\), obtained from formula (2.11), one can verify that the sign of \(v^*\) will be negative for arbitrarily small \(\|x_t(\theta)\|\). Thus all the conditions of Theorem 2 (the first Lyapunov theorem on instability) [5] are satisfied, and the motion \(x_t(\theta)=0\) of system (1.5) is unstable.
Consider the second case, i.e., \(P(y_1,y_2)>0\) on at least one straight line in the domain \(\|x_t(\theta)\|<H\). The forms \(Y_1^{(m)},Y_2^{(m)}\) and \(P(y_1,y_2)\) can be written as follows:
\[ Y_1^{(m)}(y_1,y_2)=Ay_1^m+A_1y_1^{m-1}y_2+\ldots+A_my_2^m, \]
\[ Y_2^{(m)}(y_1,y_2)=By_1^m+B_1y_1^{m-1}y_2+\ldots+B_my_2^m, \tag{3.11} \]
\[ P(y_1,y_2)=Ay_1^{2m}+A_1y_1^{2m-1}y_2+\ldots+(A_m-B)y_1^my_2^m+\ldots+B_my_2^{2m}. \]
For convenience one may assume that \(P(y_1,y_2)>0\) on the axis \(y_1\), i.e. \(y_2=0\). Obviously, by continuity, \(P(y_1,y_2)\) will retain a positive value also in some small neighborhood of the axis \(y_1\). Let, in this neighborhood, \(\|y_2\|\le |\alpha'|\cdot \|y_1\|\), where the quantity \(|\alpha'|\) is chosen so small that the sign of \(P(y_1,y_2)\), as well as the sign of \(P'(y_1,y_2)=y_1^mY_1^{(m)}-y_2^mY_2^{(m)}\), will be determined only by the term \(Ay_1^{2m}\). Consequently, \(A>0\).
In the case under consideration we take the functional \(v^{**}(x_t(\theta),t)\) in the form
\[ v^{**}(x_t(\theta),t)=-\gamma v_2^2(z_t(\theta),t)+\frac{1}{m+1}\left(y_1^{m+1}-y_2^{m+1}\right), \tag{3.12} \]
where \(v_2(z_t(\theta),t)\) is the functional constructed by virtue of system (3.2) and satisfying conditions (3.3), (3.4), (3.5); \(\gamma\) is an arbitrary positive small constant.
Take \(\alpha\) (\(|\alpha|<|\alpha'|\)) and choose \(\gamma\) so that in a neighborhood of the axis \(y_1\)
\[ \|y_2\|<|\alpha|\cdot \|y_1\| \tag{3.13} \]
in the region \(r<\|x_t(\theta)\|<H\) (where \(r\) can be chosen arbitrarily small), the functional \(v^{**}(x_t(\theta),t)\) is positive definite.
Similarly, condition (3.9) in the region \(r<\|x_t(\theta)\|<H\), for sufficiently small \(\gamma\), can be written, taking (3.13) into account, as
\[ v^{**}\ge -\gamma c_1^2\|z_t(\theta)\|^2+\frac{1-|\alpha|^{m+1}}{m+1}\|y_1\|^{m+1}\ge w_2(\|x_t(\theta)\|). \tag{3.14} \]
We shall show that \(v^{**}\) is bounded above in the region \(v^{**}>0\):
\[ \|v^{**}\|<\left(|\gamma|c_2^2B_3^2H+\frac{1-|\alpha|^{m+1}}{m+1}B_1^{m+1}H^m\right)\cdot \|x_t(\theta)\|. \tag{3.15} \]
Now, carrying out estimates analogous to those made in the first case in obtaining inequality (3.10), for sufficiently small \(\|x_t(\theta)\|\) in the region \(v^{**}>0\) we may write
\[ \liminf_{\Delta t\to 0}\left(\frac{\Delta v^{**}}{\Delta t}\right)_{(1.5)}\ge w_3(\|x_t(\theta)\|). \tag{3.16} \]
Thus, all the conditions of Theorem 2 (Chetaev’s theorem [5]) are fulfilled, and consequently the unperturbed motion \(x_t(\theta)=0\) is unstable.
Consider the third case. \(P(y_1,y_2)<0\) everywhere in the region \(\|x_t(\theta)\|<H\), except for one straight line, where \(P(y_1,y_2)=0\). Let this straight line be the axis \(y_1\) (\(y_2=0\)). Then, taking (3.11) into account, \(A=0\), where \(A\) is the coefficient of \(y_1^{2m}\). Let
\[ P_l(y_1,y_2)=y_1^mY_1^{(m+l)}+y_2^mY_2^{(m+l)}\qquad (l\ge 1) \]
—the first form, different from zero on the axis \(y_1\). If
\[ Y_1^{(m-l)}(y_1,y_2)=C_l y_1^{m+l}+\cdots+C_{m+l}y_2^{m+l}, \]
then \(C\ne 0\).
Let us consider the stability conditions for the motion \(x_t(\theta)=0\) on the axis \(y_1\). Construct the functional \(v(x_t(\theta),t)\)
\[ v(x_t(\theta),t)=v_2'(z_t(\theta),t)-\frac{1}{m+l+1}\left(y_1^{m+l+1}+y_2^{m+l+1}\right). \]
As was shown above, the stability of the motion \(x_t(\theta)=0\) depends on the properties of \(P(y_1,y_2)\), which in the present case has the form
\[ P_1(y_1,y_2)=P_1(y_1,0)=C y_1^{2m+2l}. \]
Then for \(C>0\), or for \(C<0\) and \(m+l\) even, the unperturbed motion \(x_t(\theta)=0\) will be unstable on the axis \(y_1\), while for \(C<0\) and \(m+l\) odd it will be asymptotically stable. It is obvious that these conditions determine the stability of the unperturbed motion \(x_t(\theta)=0\) also in the whole domain \(\|x_t(\theta)\|<H\). The assertion proved is valid if in (2.16) \(s>m+l\), which can always be achieved [6].
Let now \(P(y_1,y_2)=0\) on the line \(y_2=\beta y_1\) \((\beta\ne0)\). In \(P(y_1,y_2)\) make the substitution \(y_2=\beta y_1\) and obtain
\[ P(y_1,\beta y_1)=y_1^{2m}\left(A+A_1\beta+\cdots+(A_m+B)\beta^m+\cdots+B_m\beta^{2m}\right), \]
where the expression in parentheses will be denoted by \(f(\beta)\). By assumption \(f(\beta)=0\). But \(f(\beta)\) is an algebraic equation of degree \(2m\); it can have up to \(2m\) real roots. Consequently, there may be \(k\) lines \((k=1,\ldots,2m)\) on which \(P(y_1,y_2)=0\).
The stability of the unperturbed motion in the presence of \(k\) lines on which \(P(y_1,y_2)=0\) can evidently be investigated as follows: take an arbitrary line, transform it into the axis \(y_1\), and determine the stability of the motion \(x_t(\theta)=0\) on the axis \(y_1\). This can be done with all the lines. The unperturbed motion will be asymptotically stable in the domain \(\|x_t(\theta)\|<H\) only when the conditions for asymptotic stability are satisfied on all \(k\) lines, and it will be unstable if the conditions for instability are satisfied on at least one of these lines.
Let us consider the last case: \(P(y_1,y_2)\equiv0\). Since \(Y_1^{(0)}\) and \(Y_2^{(0)}\) are not identically zero, \(P(y_1,y_2)\equiv0\) only when \(A_m=-B\), while the remaining coefficients in \(Y_1^{(m)}\) and \(Y_2^{(m)}\) are equal to zero. In this case one can make the change of variables
\[ \begin{aligned} y_1&=a_1u+a_2V,\\ y_2&=b_1u+b_2V, \end{aligned} \qquad \left|\begin{matrix} a_1 & a_2\\ b_1 & b_2 \end{matrix}\right|\ne0 \]
and consider the system of equations in the space \(\{u,V,z_t(\theta)\}\). The coefficients of the transformation are chosen so that \(P(u,V)\not\equiv0\). Then the investigation of the stability of the motion \(x_t(\theta)=0\) is reduced to one of the three preceding cases. Thus we arrive at the following theorem:
Theorem 3.1. Let in the domain \(\|x_t(\theta)\|<H\) we have the system of equations (2.13) and suppose that the conditions (2.16) are satisfied.
Consider a form of order \(2m\)
\[ P(y_1,y_2)=y_1^m Y_1^{(m)}(y_1,y_2)+y_2^m Y_2^{(m)}(y_1,y_2): \]
a) If \(P(y_1,y_2)<0\) and the number \(m\) is odd, then the unperturbed motion is asymptotically stable; if \(P(y_1,y_2)<0\) and the number \(m\) is even, then the motion \(x_t(\theta)=0\) is unstable.
b) If \(P(y_1,y_2)>0\) on at least one straight line, then the unperturbed motion is unstable.
c) If \(P(y_1,y_2)<0\) everywhere in the region \(\|x_t(\theta)\|<H\), with the exception of several (but not more than \(2m\)) straight lines where \(P(y_1,y_2)=0\), then we take an arbitrary straight line and transform it into the axis \(y_1\). Let
\[ P_l(y_1,y_2)=P_l(y_1,0)=Cy_1^{2m+l}\qquad (l\geqslant 1) \]
be the first form different from zero on the axis \(y_1\). Then the motion \(x_t(\theta)=0\) will be unstable on the axis \(y_1\) for \(C>0\), or for \(C<0\) and \(m+l\) even, and will be asymptotically stable for \(C<0\) and \(m+l\) odd. The unperturbed motion will be asymptotically stable in the region under consideration only when the conditions for asymptotic stability are satisfied on all straight lines where \(P(y_1,y_2)=0\), and it will be unstable if the conditions for instability are satisfied on at least one of these straight lines.
d) If \(P(y_1,y_2)\equiv0\), then by a nonsingular transformation of variables one can arrange that \(P(u,V)\) is not identically equal to zero, and then, consequently, the problem of investigating stability is reduced to one of the preceding cases.
4. INVESTIGATION OF STABILITY IN A SPECIAL CASE
Consider the special case, i.e., \(Y_1^0=0,\; Y_2^0=0,\; Z^0=0\) [4, 7]. In this case system (2.13) admits the solution
\[ y_1=c_1,\qquad y_2=c_2,\qquad z_t(\theta)=0, \tag{4.1} \]
where \(c_1\) and \(c_2\) are arbitrary constants.
In the special case \(u(c_1,c_2,\theta)\) does not depend on the parameter \(\theta\), since
\[ \frac{\partial u}{\partial \theta}=0. \]
Then there exists the solution
\[ x_t(\theta)=b^{(1)}c_1+b^{(2)}c_2+u(c_1,c_2). \tag{4.2} \]
The trivial solution \(x_t(\theta)=0\), whose stability is being studied, is contained in the family (4.1) and (4.2) and corresponds to the zero value of the arbitrary constants. Consequently, in the special case the unperturbed motion belongs to the family of established motions.
Theorem 4.1. If the differential equations of the perturbed motion have the form (2.13), with \(Y_1^0=0,\; Y_2^0=0,\; Z^0=0\), then the unperturbed motion \(x_t(\theta)=0\) is stable. Moreover, every perturbed motion sufficiently close to the unperturbed one tends, as time increases without bound, to one of the established motions of the family (4.2).
The theorem is proved in exactly the same way as the analogous theorem in [7].
5. SOME ADDITIONS
Consider system (2.13). Let
\[ Y_1^0 = Y_1^{(m)} + Y_1^{(m+1)} + \ldots,\qquad Y_2^0 = Y_2^{(m+k)} + Y_2^{(m+k+1)} + \ldots\quad (k \geqslant 1), \tag{5.1} \]
where \(Y_1^{(m)}\) and \(Y_2^{(m+k)}\) are forms of orders \(m\) and \(m+k\). Take the functional \(v(x_t(\theta),t)\):
\[ v(x_t(\theta),t)=v_2^2(z_t(\theta),t)+ \frac{1}{m+2k+1}y_1^{m+2k+1} +\frac{1}{m+k+1}y_2^{m+k+1}, \tag{5.2} \]
and then \(P(y_1,y_2)\) will have the form
\[ P^*(y_1,y_2)=y_1^{m+2k}Y_1^{(m)}+y_2^{m+k}Y_2^{(m+k)}. \tag{5.3} \]
Let us see what can now change in the proof of Theorem 3.1. It turns out that, for the proof of the first two cases, it is only necessary to assume additionally that \(s>m+k\). The third case remains unchanged, while the last can occur only in a special case. Indeed,
\[ P^*(y_1,y_2)=Ay_1^{2m+2k}+\ldots+A_m y_1^{m+2k}y_2^m +By_1^{m+k}y_2^{m+k}+\ldots+B_{m+k}y_2^{2m+2k} \]
can be identically equal to zero only when all the coefficients of the forms \(Y_1^{(m)}\) and \(Y_2^{(m)}\) are equal to zero. Since this is true for forms of any order, it is obvious that \(Y_1^0\equiv 0\) and \(Y_2^0\equiv 0\) must hold.
Thus, for system (2.13), under condition (5.1), Theorem 3.1 is valid, with the exception of case d), which is altogether impossible under condition (5.1).
References
- Krasovskii N. N. Some problems in the theory of stability of motion. Fizmatgiz, 1959.
- Shimanov S. N. PMM, vol. XXIV, issue 3, 1960.
- Shimanov S. N. On the theory of linear differential equations with aftereffect. Differential Equations, vol. I, No. 1, 102–116, 1965.
- Malkin I. G. Theory of Stability of Motion, 1952.
- Shimanov S. N. PMM, vol. XXIV, issue 1, 1960. No. 9, 1939.
- Kamenkov G. V. On the stability of motion. Collection of works of the Kazan Aviation Institute, No. 9, 1939.
- Shimanov S. N. On stability in the critical case of one zero root for systems with aftereffect (special case). Izvestiya Vuzov, Matematika, 1(20), 1961.
Received by the editors
May 29, 1965
Ural State University
named after A. M. Gorky