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NON-STABILITY OF SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES IN THE CRITICAL CASE
V. Ya. Turin
In the works of M. A. Aizerman, F. R. Gantmakher [1, 2] and Ya. Z. Tsypkin [3], a linear approximation was determined for a system of differential equations
\[ \dot z=f(z,t) \]
with discontinuous right-hand sides.
In works [1, 2, 4], theorems on the stability of discontinuous solutions of this system were proved.
In the present article the question of the instability of periodic solutions of this system in critical cases is resolved.
Let a system of differential equations be given in vector form
\[ \dot z=f(z,t). \tag{1} \]
The function \(f(z,t)\) is defined in an \(n+1\)-dimensional curvilinear cylinder \(C\), whose axis is a continuous integral curve \(z=z^0(t)\) of system (1). In addition, the function \(f(z,t)\) is periodic with period \(\tau\): \(f(z,t+\tau)=f(z,t)\).
The hypersurfaces (surfaces of discontinuity)
\[ F_{\alpha}(z,t)=0 \quad [F_{\alpha}(z,t+\tau)=F_{\alpha}(z,t)] \tag{2} \]
cut the cylinder \(C\) into domains \(H_{\alpha}\), intersecting the curve \(z=z^0(t)\) at the points \(M_{\alpha}\) for \(t=t_{\alpha}\).
The right-hand sides of system (1) satisfy the following conditions.
-
The functions \(f_i(z,t)\) are continuous in each domain \(H_{\alpha}\) (including the boundaries) and continuously differentiable up to order \(N\), and, on passing through the surfaces (2), they and all their partial derivatives up to order \(N\) experience only discontinuities of the first kind.
-
In the angular domains enclosed between the surfaces (2) and the planes \(t=t_{\alpha}\), the conditions
\[ \frac{\partial^m f_i(z,t)} {\partial t^{m_0}\partial z_1^{m_1}\ldots\partial z_n^{m_n}} - \frac{\partial^m f_i(z^0,t)} {\partial t^{m_0}\partial z_1^{m_1}\ldots\partial z_n^{m_n}} \to \xi_i^{m_0\ldots m_n} \tag{3} \]
hold as \((z,t)\to M_{\alpha}\).
(The plus and minus signs correspond to the values \(t=t_{\alpha}+0\) and \(t=t_{\alpha}-0\).)
- The surfaces (2) are continuous, and at the points \(M_{\alpha}\) are smooth up to order \(N\), and along the integral curve \(z=z^0(t)\)
\[ [\dot F_a]_{\overline{M}_a}\ne 0,\qquad \frac{[\dot F_a]^+_{M_a}}{[\dot F_a]^-_{M_a}}>\Gamma>0, \tag{4} \]
where
\[ \dot F_a=\left\{\sum_{i=1}^{n}\frac{\partial F_a}{\partial z_i}\,f_i+\frac{\partial F_a}{\partial t}\right\}_{z=z^0(t)}. \]
Introduce the variable \(x=z-z^0(t)\). System (1) takes the form
\[ \dot x=p(x,t),\quad p(x,t)=f(z^0+x,t)-f(z^0,t), \tag{5} \]
and the equations of the discontinuity surfaces will be
\[ \Phi_a(x,t)=0,\qquad \Phi_a(x,t)\equiv F_a(z^0+x,t). \tag{6} \]
It is obvious that the stability of the solution \(z=z^0(t)\) of system (1) is equivalent to the stability of the zero solution of system (5). The surfaces (6), in contrast to (2), will no longer be smooth at the points \(M_a\), but have corners. In the intersection of the domains \(H_a\) and \(t_a\leqslant t\leqslant t_{a+1}\) (in what follows we shall call these domains central), system (1) can be written in the form
\[ \dot z_i=f_i(z^0,t)+\sum \frac{1}{m!}\, \frac{\partial^m f_i(z^0,t)} {\partial z_1^{m_1}\ldots \partial z_n^{m_n}}\, (z_1-z_1^0)^{m_1}\ldots (z_n-z_n^0)^{m_n}+R_i(z,t)^*, \tag{7} \]
\[ 1\leqslant m_1+\ldots+m_n=m\leqslant N, \]
where
\[ \frac{\partial^m f_i(z^0,t)} {\partial z_1^{m_1}\ldots \partial z_n^{m_n}} \]
are functions continuous in each interval \(t_a\leqslant t\leqslant t_{a+1}\), and \(R_i(z,t)\) satisfy the conditions
\[ |R_i(z,t)|<a|x|^{N+1},\qquad |x|=\sqrt{(z_1-z_1^0)^2+\ldots+(z_n-z_n^0)^2}. \tag{8} \]
The discontinuity surfaces, accurate up to infinitesimals of order higher than \(N\), can be written in the form (in a neighborhood of the point \(M_a\) in the lower angular domain):
\[ t_a-t=\sum h_a^{m_1\ldots m_n}x_1^{m_1}\ldots x_n^{m_n}=G(x). \tag{9} \]
We shall seek an approximate solution of equation (5) in a neighborhood of the point \(P(\beta,t_1)\), situated on the discontinuity surface in a neighborhood of the point \(M_a\) in the lower angular domain,
\[ x_i=\beta_i+\sum_{m=1}^{N}\frac{d^{m-1}p_i(\beta,t_1)}{dt^{m-1}}\cdot \frac{(t-t_1)^m}{m!} +O(|t-t_1|^N), \tag{10} \]
where
\[ \frac{dp_i}{dt}=\sum_{j=1}^{n}\frac{\partial p_i}{\partial x_j}p_j+\frac{\partial p_i}{\partial t}. \tag{11} \]
\[ \text{* In what follows, unless specially stated otherwise, we shall assume that the indices } m_1\ldots m_n \text{ run through all possible values satisfying the inequality } 1\leqslant m_1+\ldots+m_n=m\leqslant N. \]
Since the point \(P(\beta,t_1)\) lies on the discontinuity surfaces, with accuracy up to infinitesimals of order higher than \(N\),
\[ t_\alpha-t_1=G_\alpha(\beta). \tag{12} \]
If now in (10) we put \(t=t_\alpha\) and replace \(t_\alpha-t_1\) by formulas (12), then we obtain
\[ x_i^+=\beta_i+\sum_{m=1}^{N}\frac{[G_\alpha(\beta)]^m}{m!}\, \frac{d^{m-1}}{dt^{m-1}}\,p_i[\beta,t_\alpha-G_\alpha(\beta)]+O(|\beta|^N). \tag{13} \]
We expand the last equality in increasing powers and restrict ourselves to terms of order not higher than \(N\) with respect to \(\beta_1,\ldots,\beta_n\):
\[ x_{i\alpha}^{+}=\sum b_{i\alpha}^{m_1\cdots m_n}\beta_1^{m_1}\cdots \beta_n^{m_n}. \tag{14} \]
The coefficients \(b_{i\alpha}^{m_1\cdots m_n}\) depend only on those \(h_\alpha^{s_1\cdots s_n}, \xi_\alpha^{s_1\cdots s_n}\) for which \(s_1+\cdots+s_n<m\), \(m=m_1+\cdots+m_n\). In particular, if \(m_1+\cdots+m_n=1\), we obtain
\[ b_{i\alpha}^{j}=\delta_{ij}+\xi_{i\alpha}h_{j\alpha}, \tag{15} \]
where \(\delta_{ij}\) is the Kronecker symbol. Consider the system
\[ \dot{x}_i=\sum \frac{1}{m!}\, \frac{\partial^m f_i(z^0,t)}{\partial z_1^{m_1}\cdots \partial z_n^{m_n}}\, x_1^{m_1}\cdots x_n^{m_n}. \tag{16} \]
The right-hand sides of this system have discontinuities only on the planes \(t=t_\alpha\) (unlike the right-hand sides of system (5), which undergo discontinuities not only on the planes \(t=t_\alpha\), but also on the surfaces (6)).
Therefore system (16) in the central regions will coincide (with accuracy up to infinitesimals of order \(N\)) with system (5). In the angular regions the discrepancy between systems (5) and (16) will be substantial.
In order to estimate this discrepancy, let us find an approximate solution of system (16) in a neighborhood of the point \(P(\beta,t_1)\), situated on the discontinuity surface, analogously to how this was done for system (5). We denote the corresponding value \(x(t_\alpha)\) by \(x^{-}\).
It is clear that \(x^{-}\) can be obtained from formula (14), in which all
\[ \xi_\alpha^{m_1\cdots m_n}=0. \]
Taking (15) into account and denoting
\[ \left. b_{i\alpha}^{m_1\cdots m_n}\right|_{\xi_\alpha=0}=c_{i\alpha}^{m_1\cdots m_n}, \]
we obtain
\[ x_i^-=\beta_i+\sum c_{i\alpha}^{m_1\cdots m_n}\beta_1^{m_1}\cdots \beta_n^{m_n},\qquad 2\le m_1+\cdots+m_n\le N. \tag{17} \]
The equalities (17) are invertible, with accuracy up to infinitesimals of order not higher than \(N\), in a neighborhood of the origin of coordinates:
\[ \beta_i=\sum d_{i\alpha}^{m_1\cdots m_n}(x_1^-)^{m_1}\cdots (x_n^-)^{m_n}. \tag{18} \]
If now we substitute \(\beta_i\) from (18) into (14) and restrict ourselves to infinitesimals of order not higher than \(N\), then formula (14) takes the form
\[ x_i^+=\sum e_{i\alpha}^{m_1\cdots m_n}(x_1^-)^{m_1}\cdots (x_n^-)^{m_n}, \]
or, in vector form,
\[ x_\alpha^+=S_\alpha(x_\alpha^-). \tag{19} \]
Formula (19) takes into account the divergence between the solutions of systems (5) and (16) in the angular regions. Therefore the stability of the zero solution of system (5) (for which adjacent motions are considered continuous) will be equivalent to the stability of the zero solution of system (16) and of the discontinuity conditions (19). A rigorous proof of this fact will be given in § 2, where certain restrictions will be formulated under which this proof can be carried out.
The totality of system (16) and the discontinuity conditions (19) will be called the system of the first approximation of order \(N\).
In particular, for \(N=1\), system (16), (19), by virtue of (15), takes the form
\[ \dot{x}_i=\sum_{j=1}^{n}\frac{\partial f_i(z^0,t)}{\partial z_j}\,x_j, \tag{20} \]
\[ x_i^+=x_i^-+\sum_{j=1}^{n}\xi_{i\alpha}h_{j\alpha}x_j^- . \tag{21} \]
This system of the first approximation was considered in [2].
§ 1. If \(X(t)\) denotes the fundamental matrix of system (20)—(21), then the equality
\[ X(t+\tau)=X(t)U, \tag{1,1} \]
will hold, where \(U\) is a constant nonsingular matrix.
We now apply to system (5) the transformation \(x=L(t)y\), where
\[ L(t)=X(t)e^{-At},\qquad A=\frac{1}{\tau}\ln U . \]
System (5) takes the form
\[ \dot{y}=q(y,t),\qquad q(y,t)=L^{-1}p-L^{-1}\frac{dL}{dt}\,y . \tag{1,2} \]
If the matrix \(A\) is chosen in Jordan form, then system (1,2) in the central regions will have the form
\[ \dot{y}_i=\lambda_i y_i+\alpha_{i-1}y_{i-1} +\sum_{m=2}^{N}Y_i^{(m)}(y,t)+R_i^*(y,t), \tag{1,3} \]
where
\[ \left|R_i^*(y,t)\right|<a_1|y|^{N+1}, \]
and \(Y_i^{(m)}(y,t)\) are forms of order \(m\) with respect to the variables \(y_1,\ldots,y_n\), with coefficients periodic and discontinuous at \(t=t_\alpha\).
We now transform system (1,3) by means of a nonlinear transformation to such a form that, in it, the terms up to order \(N\) have constant coefficients equal everywhere.
Such a transformation can be performed if between the characteristic numbers of the matrix \(A\) and the period \(\tau\) there exist no relations of the form
\[ m_1\lambda_1+\cdots+m_n\lambda_n =\pm\frac{2\pi}{\tau}\sqrt{-1} \tag{1,4} \]
for nonnegative integers \(m_1,m_2,\ldots,m_n\).
Suppose that these conditions are satisfied. We shall seek a transformation in the form
\[ y_i=u_i+\sum A_i^{m_1\cdots m_n}(t)u_1^{m_1}\cdots u_n^{m_n}, \qquad 2\leq m_1+\cdots+m_n\leq N, \tag{1,5} \]
where \(A_i^{m_1\cdots m_2}(t)\) are periodic functions of period \(\tau\), discontinuous for \(t=t_\alpha\).
In the new variables the system (1,3) takes the form
\[ \dot u_i=\lambda_i u_i+\alpha_{i-1}u_{i-1} +\sum a_i^{m_1\cdots m_n}u_1^{m_1}\cdots u_n^{m_n} +U_i(u,t). \tag{1,6} \]
The discontinuity conditions for the first-approximation system of order \(N\) of the system (1,3) have the form
\[ y_i^+=y_i^-+\sum g_i^{m_1\cdots m_n}(y_1^-)^{m_1}\cdots (y_n^-)^{m_n}, \]
\[ y^+=J(y^-). \tag{1,7} \]
We define the discontinuities of the functions \(A_i^{m_1\cdots m_n}(t)\) so that the discontinuity conditions of the first-approximation system of order \(N\) of (1,6) have the form
\[ u_i^+=u_i^- \tag{1,8} \]
up to terms of order higher than \(N\).
We shall show that this can be done.
Indeed, up to terms of order \(N\), the equality
\[ A(u,t_\alpha+0)=J[A(u,t_\alpha-0)] \tag{1,9} \]
must hold.
Equating the coefficients of like powers \(u_1\cdots u_n\), we obtain the required discontinuity conditions.
Since in equations (1,9) \(A_i^{m_1\cdots m_n}(t_\alpha-0)\) are regarded as known, \(A_i^{m_1\cdots m_n}(t_\alpha+0)\) is determined uniquely.
Equality (1,9), like (1,8), should be understood not in the exact sense, but up to infinitesimals of orders higher than \(N\).
For the possibility of reducing system (1,3) to the form (1,6), it is necessary [5] that the coefficients \(A_i^{m_1\cdots m_n}\) of the transformation (1,5) satisfy the equations
\[ \dot A_i^{m_1\cdots m_n} +\left(m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_i\right) A_i^{m_1\cdots m_n} = -a_i^{m_1\cdots m_n} +B_i^{m_1\cdots m_n}, \tag{1,10} \]
where \(B_i^{m_1\cdots m_n}(t)\) are linear functions of the already computed quantities \(A_i^{m_1\cdots m_n}(t)\), with coefficients periodic of period \(\tau\) and discontinuous for \(t=t_\alpha\).
We shall seek a periodic solution of the system (1,10), having discontinuities at \(t=t_\alpha\) determined from (1,9).
Suppose that all the coefficients of the transformation (1,5) entering into \(B_i^{m_1\cdots m_n}(t)\) have turned out to be periodic; we show that then \(A_i^{m_1\cdots m_n}(t)\) can also be chosen periodic.
Indeed, if \(m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_i=0\), then, in order that \(A_i^{m_1\cdots m_n}(t)\) be periodic, it is necessary and sufficient that
\[ a_i^{m_1\cdots m_n} = \frac{1}{\tau}\int_0^\tau B_i^{m_1\cdots m_n}\,dt + \frac{1}{\tau}\sum_{\alpha=1}^{h}\Delta_\alpha A_i^{m_1\cdots m_n}, \]
where \(h\) is the number of discontinuity surfaces, \(\Delta_\alpha A_i=A_i(t_\alpha+0)-A_i(t_\alpha-0)\).
All the integrals exist, since the discontinuities of the integrands are of the first kind.
If, however, \(m_1\lambda_1+\ldots+m_n\lambda_n-\lambda_i\ne0\), then equation (1.10) has a periodic solution for any constant \(a_i^{m_1\ldots m_n}\). Therefore one may set \(a_i^{m_1\ldots m_n}=0\).
Thus one can determine successively the coefficients of the transformation (1.5). This transformation is such that stability with respect to the variables \((x,t)\) is equivalent to stability with respect to the variables \((u,t)\).
It is also evident that, for the terms \(U_i(u,t)\) in system (1.6), the estimate
\[ |U_i(u,t)|<a_2|u|^{N+1}, \]
holds, where
\[ |u|=\sqrt{|u_1|^2+\ldots+|u_n|^2}. \]
§ 2. We now turn to the investigation of the stability of the solution. Suppose that \(l\) characteristic roots of the matrix \(A\) are equal to zero, and \(2r\) characteristic roots are purely imaginary,
\[ \lambda_j=0,\quad j=1,2,\ldots,l, \]
\[ \lambda_{k+l}=\omega_k\sqrt{-1},\quad \lambda_{r+k+l}=-\omega_k\sqrt{-1},\quad k=1,2,\ldots,r. \]
Assume that there are no relations between the proper frequencies of the form
\[ m_1\omega_1+\ldots+m_r\omega_r=0 \tag{2.1} \]
for any integers \(m_1,\ldots,m_r\).
In addition, suppose that to the \(l\) zero characteristic roots of the matrix \(A\) there correspond simple elementary divisors. The remaining characteristic roots of the matrix \(A\) have negative real parts. (If the real part of at least one of the characteristic roots of the matrix \(A\) is positive, then, as shown in [2], the solution \(z=z^0(t)\) will be unstable.)
A characteristic feature of system (1.6), to which we have reduced system (1) in the central regions, is that the variables \(u_{l+2r+1},\ldots,u_n\) from the stable part of the system enter the critical part of the system only in the terms \(U_i(u,t)\), \(i=1,2,\ldots,l+2r\).
Therefore the critical part of system (1.6) can be written more precisely in the form
\[ \dot u_i=\lambda_i u_i+\sum d_i^{m_1\ldots m_{l+2r}\,0\ldots0} u_1^{m_1}\ldots u_{l+2r}^{m_{l+2r}}+U_i(u,t), \tag{2.2} \]
where \(i=1,2,\ldots,l+2r\).
We now replace in system (2.2) the variables corresponding to the purely imaginary characteristic roots of the matrix \(A\) according to the formulas
\[ u_{l+k}=\rho_k e^{i\vartheta_k},\quad u_{l+r+k}=\rho_k e^{-i\vartheta_k}. \tag{2.3} \]
Then system (2.2) assumes the form:
\[ \dot u_i=\sum_{m=\nu}^{N} V_i^{(m)}(u_1\ldots u_l,\rho_1\ldots\rho_r) +V_i(u,\rho,\vartheta,t), \]
\[ \dot\rho_j=\sum_{m=\nu}^{N} R_j^{(m)}(u_1\ldots u_l,\rho_1\ldots\rho_r) +R_j(u,\rho,\vartheta,t), \tag{2.4} \]
\[ \dot\vartheta_j=\omega_j+\Theta_j(u,\rho,\vartheta,t). \]
Here \(V_i^{(m)}(u_1\ldots u_l,\rho_1\ldots \rho_r)\), \(R_j^{(m)}(u_1\ldots u_l,\rho_1\ldots \rho_r)\) are forms of order \(m\) with constant real coefficients (by virtue of the reality of the original system (1)).
The stable part of the system after the transformation (2.3) will take the form
\[ \dot u_k=\lambda_k u_k+a_{k-1}u_{k-1}+\sum_{m=2}^{N}U_k^{(m)}(\rho,\vartheta,u)+U_k(\rho,\vartheta,u,t), \tag{2.5} \]
\[ k=l+2r+1,\ldots,n, \]
where \(U_k^{(m)}(\rho,\vartheta,u)\) are forms of order \(m\) with respect to the variables \(\rho_1,\ldots,\rho_r,\allowbreak u_1,\ldots,u_n\), with coefficients periodic and continuous in \(\vartheta\).
As shown in [5, 6], by changing only the stable variables according to the formulas
\[ u_k=\xi_k+\varphi_k(u_1\ldots u_l,\rho_1\ldots \rho_r,\vartheta_1\ldots \vartheta_r), \tag{2.6} \]
one can arrange that, in the expansion of the right-hand side of the stable part of the system for \(\xi_{l+2r+1}=\cdots=\xi_n=0\), terms from the critical part of the system occur only in powers higher than \(N\).
After applying the transformation (2.6), system (2.5) takes the form
\[ \dot \xi_k=\lambda_k\xi_k+\alpha_{k-1}\xi_{k-1}+Z_k(\rho,\vartheta,\xi,u,t), \tag{2.7} \]
where \(Z_k(\rho,\vartheta,\xi,u,t)\), for small \(|\rho+u+\xi|\), satisfy the inequality
\[ |Z_k(\rho,\vartheta,\xi,u,t)|<\gamma\left(|\rho+u|^{N+1}+|\rho+u||\xi|+|\xi|^2\right), \tag{2.8} \]
where \(\gamma\) is a positive constant.
As a result of all transformations, system (1) in the central domains takes the form:
\[ \begin{aligned} \dot u_i&=V_i^{(\nu)}(u,\rho)+V_i(u,\rho,\vartheta,\xi,t),\\ \dot \rho_j&=R_j^{(\nu)}(u,\rho)+R_j(u,\rho,\vartheta,\xi,t),\\ \dot \vartheta_j&=\omega_j+\theta(u,\rho,\vartheta,\xi,t),\\ \dot \xi_k&=\lambda_k\xi_k+\alpha_{k-1}\xi_{k-1}+Z_k(u,\rho,\vartheta,\xi,t), \end{aligned} \tag{2.9} \]
where \(V_i(u,\rho,\vartheta,\xi,t)\), \(R_j(u,\rho,\vartheta,\xi,t)\) are periodic functions of \(\vartheta\) and \(t\) and, for sufficiently small \(|\rho+u+\xi|\), satisfy the inequalities
\[ |R_i|<\alpha\left(|\rho+u|^{\nu+1}+|\xi|^2\right), \]
\[ |V_i|<\beta\left(|\rho+u|^{\nu+1}+|\xi|^2\right), \tag{2.10} \]
where \(\alpha\) and \(\beta\) are positive constants.
We now prove the following theorem. If the zero solution of the system
\[ \dot u_i=V_i^{(\nu)}(u_1\ldots u_l,\rho_1\ldots \rho_r), \]
\[ \dot \rho_j=R_j^{(\nu)}(u_1\ldots u_l,\rho_1\ldots \rho_r) \tag{2.11} \]
is unstable, then the zero solution of system (2.9) is also unstable.
Indeed, in this case there exists [7] a continuously differentiable function \(V(u,\rho)\) satisfying the following inequalities:
\[ |V(u,\rho)|\leq b_1|u+\rho|^B. \]
\[ \left(\frac{dV}{dt}\right)_{(2,11)} \geq b_2 |u+\rho|^{B+\nu-1}, \qquad \left|\frac{\partial V}{\partial \rho_j}\right| \leq b_3 |u+\rho|^{B-1}, \tag{2,12} \]
\[ \left|\frac{\partial V}{\partial u_i}\right| \leq b_3 |u+\rho|^{B-1}, \]
and in an arbitrarily small neighborhood of the point \(u=0,\ \rho=0\) one can specify points \(u_0\ne0,\ \rho_0\ne0\), where
\[ V(u_0,\rho_0)>0. \tag{2,13} \]
In order to detect instability, it is sufficient to specify at least one trajectory of system (5) that leaves the cylinder \(V=\delta\) under arbitrarily small initial perturbations.
Choose the quantities \(u_0\ne0,\ \rho_0\ne0,\ \zeta_0=0\) inside the cylinder \(V=\delta\) in such a way that
\[ V_0=V(u_0,\rho_0)>0. \]
By virtue of the estimates (2,10), in which \(\zeta=0\) is set, the function \(V(u,\rho)\) will increase along an integral curve of system (2,9), and the inequality
\[ \left(\frac{dV}{dt}\right)_{(2,9)} > \mu^2 |u+\rho|^{B+\nu-1} \tag{2,14} \]
will hold. By virtue of (2,12),
\[ \frac{dV}{dt} > \frac{\mu^2}{b_1^\sigma} V^\sigma, \tag{2,15} \]
where
\[ \sigma=\frac{B+\nu-1}{B}, \]
whence
\[ V>\frac{V_0}{\left[1-\beta V_0^{\sigma-1}(t-t_0)\right]^{\frac{1}{\sigma-1}}}. \tag{2,16} \]
Let us now trace the change of the function \(V(u,\rho)\) in the corner regions. By the continuity of this function, up to infinitesimals of order higher than \(N\),
\[ V^{-K}(x_-,t_\alpha)-V^{-K}(x_+,t_\alpha)>-\varkappa, \tag{2,17} \]
where \(\varkappa\) is an arbitrarily small positive number. Increasing \(N\), one may choose \(K\) sufficiently large. Let
\[ K=\sigma-1. \tag{2,18} \]
We now choose the cylinder \(V=\delta\) so that, in addition,
\[ \varkappa<\beta T, \tag{2,19} \]
where
\[ T=\min_\alpha |t_{\alpha+1}-t_\alpha|. \]
The radius of the cylinder can be chosen so small that the planes
\[ t=t_\alpha^*=\frac{1}{2}(t_\alpha+t_{\alpha+1}) \tag{2,20} \]
inside the cylinder \(C\) do not intersect the corner regions. As \(t\) varies within \(t_1^*\leq t\leq t_2^*\), the function \(V\) along an integral curve of the system
(5) increases in the central region up to the point \((x_1,t_1)\) of intersection of the curve with the surface of discontinuity. According to \((2,16)\),
\[ V(x_1,t_1)>\frac{V_1}{[1-\beta V_1^K(t_1-t_1^*)]^{1/K}},\qquad V_1=V(x_1^*,t_1^*). \tag{2,21} \]
Further, in the corner region the function \(V\) along a trajectory of the shortened system will increase, and
\[ V(x_-,t_\alpha)>\frac{V_1}{[1-\beta V_1^K(t_\alpha-t_1^*)]^{1/K}}. \tag{2,22} \]
Then, according to \((2,17)\),
\[ V(x,t_\alpha)>\frac{V_1}{\{1-V_1^K[\beta(t_\alpha-t_1^*)-x]\}^{1/K}}, \tag{2,23} \]
where \(x=x(t_\alpha)\) is the point on the trajectory of system (5) at the moment of its meeting with the plane \(t=t_\alpha\). In the next central region the function \(V\) will again increase, and
\[ V_2>\frac{V_1}{\{1-V_1^K[\beta(t_2^*-t_1^*)-\varkappa]\}^{1/K}} >\frac{V_1}{\{1-V_1^K(\beta T-\varkappa)\}^{1/K}}. \tag{2,24} \]
If during the time \(t_1^*\le t\le t_2^*\) the trajectory of system (5) has not gone outside the cylinder \(V=\delta\), then by virtue of \((2,19)\)
\[ V_2=V(x_2^*,t_2^*)>V_1. \]
Further:
\[ V_3>\frac{V_1}{[1-2V_1^K(\beta T-\varkappa)]^{1/K}}, \]
and, by induction,
\[ V_{i+1}>\frac{V_1}{[1-iV_1^K(\beta T-\varkappa)]^{1/K}}, \]
from which the instability of the solution \(z=z^0(t)\) of system (1) follows.
For example, in the case when the characteristic equation has one root with modulus equal to one, while the remaining roots have moduli less than one, system \((2,11)\) reduces to a single equation
\[ \dot u=gu^\nu. \]
Then, if \(\nu\) is even, or if \(\nu\) is odd and \(g>0\), the unperturbed motion \(z=z^0(t)\) is unstable.
References
- Aizerman M. A., Gantmakher F. R. Dokl. Akad. Nauk SSSR, 116, No. 4, 1957.
- Aizerman M. A., Gantmakher F. R. Prikl. Mat. Mekh., 21, issue 5, 1957.
- Tsypkin Ya. Z. Theory of Relay Systems of Automatic Control. Moscow, Gostekhizdat, 1955.
- Livartovskii I. V. Dokl. Akad. Nauk SSSR, 125, No. 4, 1959.
- Malkin I. G. Theory of Stability of Motion. Moscow—Leningrad, Gostekhizdat, 1952.
- Zubov V. I. Methods of A. M. Lyapunov and Their Application. Leningrad State University Press, 1957.
- Krasovskii N. N. Prikl. Mat. Mekh., 19, issue 5, 1955.
Received by the editors
February 4, 1965
Moscow Electrotechnical Institute
of Communications