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ON THE STABILITY OF DISCONTINUOUS SYSTEMS WITH ALMOST REDUCIBLE LINEAR APPROXIMATIONS
I. V. LIVARTOVSKII
For linear differential equations with continuous coefficients, B. F. Bylov in [1] introduced the notion of a class of almost reducible systems. This class is a subclass of the set of regular systems and includes all reducible systems. In the present note the concept of almost reducibility is introduced for systems with discontinuous coefficients and discontinuous solutions. Theorems are proved on the stability and instability of solutions of nonlinear differential equations with discontinuous right-hand sides having almost reducible linear approximations.
§ 1. ALMOST REDUCIBLE DISCONTINUOUS SYSTEMS
Let a system of differential equations be given
\[ \frac{dx}{dt}=P(t)x, \tag{1} \]
where \(x\) is a column vector with coordinates \(x_i\), and \(P(t)=\{p_{ik}(t)\}\) is a real matrix, continuous in each interval \(t_\alpha \le t \le t_{\alpha+1}\) and bounded for \(t\ge 0\).
Here and everywhere below the indices take the values: \(\alpha=1,2,\ldots,\infty\), \(i,k=1,2,\ldots,n\) (\(n\) is a fixed integer); the lengths of the intervals \((t_\alpha,t_{\alpha+1})\) are bounded below by a positive number: \(t_{\alpha+1}-t_\alpha \ge T>0\), and, speaking of intervals \(t_\alpha \le t \le t_{\alpha+1}\), we shall also have in mind the interval \(0\le t\le t_1\).
We shall consider only discontinuous solutions \(x=x(t)\) of system (1) satisfying this system inside each interval \(t_\alpha \le t \le t_{\alpha+1}\) and experiencing, for \(t=t_\alpha\), discontinuities determined by the formulas
\[ x_\alpha^{+}=S_\alpha x_\alpha^{-}, \tag{2} \]
where \(S_\alpha=\{(S_\alpha)_{ik}\}\) are real nonsingular matrices, and the indices “\(+\)” and “\(-\)” refer respectively to the values at
\[ t=t_\alpha+0 \quad \text{and} \quad t=t_\alpha-0 . \]
For systems (1)—(2)\(^*\) we introduce the notion of almost reducibility similarly to the way this was done by B. F. Bylov [1] for differential equations with continuous coefficients.
Definition. System (1)—(2) is called almost reducible to a system with a continuous matrix of coefficients \(B(t)\), bounded for \(t\ge 0\), and continuous solutions
\[ \frac{dz}{dt}=B(t), \qquad z_\alpha^{+}=z_\alpha^{-}, \tag{3} \]
\[ {}^*)\ \text{Systems (1)—(2) are everywhere considered only jointly.} \]
if for any \(\delta>0\) one can specify a Lyapunov transformation discontinuous at \(t=t_\alpha\),
\(x=L_\delta(t)u\), reducing system (1)—(2) to the equations
\[ \frac{du}{dt}=[B(t)+\Phi(t)]u, \]
in which the modulus of each element of the continuous matrix
\[ \Phi(t)=\{\varphi_{ik}(t)\} \]
does not exceed the prescribed number \(\delta\). Concerning the matrices \(L_\delta(t)\), we note that, except for discontinuities at \(t=t_\alpha\), they have the same properties as in the classical case, i.e., on each interval \(t_\alpha \leqslant t \leqslant t_{\alpha+1}\) there exist continuous matrices \(L^{-1}\) and \(dL/dt\), bounded for \(t\geqslant 0\) together with \(L\).
In papers [4] and [5] the author proved that if the matrices \(S_\alpha\) are bounded in their totality and \(\det S_\alpha \geqslant \Gamma>0\), then the following lemma holds.
Lemma. For every system (1)—(2) one can construct a Lyapunov transformation discontinuous at \(t=t_\alpha\), \(x=L(t)y\), reducing it to a system with a continuous coefficient matrix bounded for \(t\geqslant 0\) and continuous solutions,
\[ \frac{dy}{dt}=A(t)y,\qquad y_\alpha^+=y_\alpha^- . \tag{4} \]
Moreover, in order to construct the transformation \(x=L(t)y\), it is sufficient to prescribe only the matrices \(S_\alpha\) and \(P(t_\alpha\pm 0)\).
Using this lemma, it is not difficult to verify that almost reducibility of system (4) to equations (3) is a necessary and sufficient condition for almost reducibility, to the same equations, of the discontinuous system (1)—(2).
We now introduce the second definition.
Definition. System (1)—(2) is called almost reducible if it is almost reducible to differential equations with constant coefficients.
Let us note that the necessary and sufficient conditions, established by B. F. Bylov [1], for almost reducibility of linear differential equations with continuous coefficient matrices bounded for \(t\geqslant 0\) are readily extended to discontinuous systems (1)—(2) in which the matrices \(S_\alpha\) are bounded in their totality and satisfy the inequalities
\[ \det S_\alpha \geqslant \Gamma>0. \]
Let us proceed to the calculation of the Lyapunov characteristic exponents \(\lambda[x]=\lambda[\|x(t)\|]\), \(\|x\|=[x_1^2+\ldots+x_n^2]^{1/2}\), of the solutions of system (1)—(2).
Consider under what conditions every nontrivial solution \(x=x(t)\) of system (1)—(2) has a finite characteristic exponent. Let
\[ |p_{ik}(t)|\leqslant a \quad (t\geqslant 0). \]
Within each interval \(t_\alpha \leqslant t \leqslant t_{\alpha+1}\), every solution of system (1)—(2) satisfies equations (1). Therefore, restricting ourselves first to nontrivial real solutions, we obtain, for \(t_\alpha \leqslant t \leqslant t_{\alpha+1}\),
\[ -2an \leq \frac{d}{dt}\ln \|x\|^2 \leq 2an \]
and
\[ \|x_\alpha^{+}\| e^{-an(t-t_\alpha)} \leq \|x(t)\| \leq \|x_\alpha^{+}\| e^{an(t-t_\alpha)} . \tag{5} \]
Moreover, it follows from (2) that
\[ \|x_\alpha^{+}\|^2=\sum_{i,k}(Q_\alpha)_{ik}x_i^{-}x_k^{-}, \]
where the matrix \(Q_\alpha=S_\alpha' S_\alpha\), and the prime denotes transposition of the matrix. Denoting the extremal values of the characteristic roots of the matrices \(Q_\alpha\) by \(\mu_{\alpha_1}^2\) and \(\mu_{\alpha_2}^2\), we obtain
\[ |\mu_{\alpha_1}|\,\|x_\alpha^{-}\| \leq \|x_\alpha^{+}\| \leq |\mu_{\alpha_2}|\,\|x_\alpha^{-}\|,\quad 0<\mu_{\alpha_1}\leq \mu_{\alpha_2}. \tag{6} \]
Applying inequalities (5) to each of the intervals \(t_\alpha \leq t \leq t_{\alpha+1}\) and using relations (6), we obtain, for any \(t\geq 0\),
\[ M_1(t)e^{-ant}\|x(0)\|\leq \|x(t)\|\leq M_2(t)e^{ant}\|x(0)\|, \tag{7} \]
where \(M_1(t)\) and \(M_2(t)\) are piecewise-constant functions whose values are determined by the formulas
\[ M_1(t)=M_2(t)=1 \quad \text{for } 0\leq t\leq t_1-0, \]
\[ \begin{aligned} M_1(t)&=\mu_{11}\mu_{21}\cdots \mu_{\alpha 1},\\ M_2(t)&=\mu_{12}\mu_{22}\cdots \mu_{\alpha 2} \end{aligned} \quad \text{for } t_\alpha+0\leq t\leq t_{\alpha+1}-0 . \tag{8} \]
Using inequalities (7), we obtain
\[ -an+\varliminf_{t\to\infty}\frac{1}{t}\ln M_1(t) \leq \lambda[x]=\varlimsup_{t\to\infty}\frac{1}{t}\ln\|x\|\leq an+ \]
\[ +\varlimsup_{t\to\infty}\frac{1}{t}\ln M_2(t). \tag{9} \]
Since any complex solution is composed of two real ones, relation (9) proves the validity of the following theorem, which is a generalization of the well-known theorem of A. M. Lyapunov for the “continuous case.”
Theorem. Every nontrivial solution of system (1)—(2) has a finite characteristic exponent, provided that the functions \(M_1(t)\) and \(M_2(t)\) constructed from the discontinuity matrices \(S_\alpha\) have finite characteristic exponents.
§ 2. STABILITY THEOREMS FOR SYSTEMS WITH ALMOST REDUCIBLE LINEAR APPROXIMATIONS
Consider the system of differential equations
\[ \frac{dz}{dt}=f(z,t), \tag{10} \]
where the real vector-function \(f(z,t)\) is given in the \((n+1)\)-dimensional space \(z,t\) inside a curvilinear cylinder \(C\), whose axis is the integral curve \(z=z^0(t)\) of system (10). Let an infinite
the sequence of surfaces \(\Psi_\alpha(z,t)=0\) cuts the cylinder \(C\) into the domains \(H_\alpha\), intersecting the curve \(z=z^0(t)\) for \(t=t_\alpha\) at the points \(M_\alpha\). Here \(t_{\alpha+1}-t_\alpha \geq T>0\). The planes \(t=t_\alpha\) cut the domains \(H_\alpha\) into angular domains, enclosed between these planes and the corresponding surfaces \(\Psi_\alpha=0\), and into central domains, containing segments of the integral curve \(z=z^0(t)\). The angular domain lying below (above) the plane \(t=t_\alpha\) will hereafter be called the lower (upper) one. Concerning the function \(f\) and the surfaces \(\Psi_\alpha=0\) we make the following assumptions:
I. The function \(f\) is continuous in each domain \(H_\alpha\) (including the boundaries \(\Psi_\alpha=0\) and \(\Psi_{\alpha+1}=0\)), and upon passing through the surfaces \(\Psi_\alpha=0\) may undergo only discontinuities of the first kind, whose magnitudes \(\xi_\alpha\) at the points \(M_\alpha\) are bounded in modulus by
\[ \|\xi_\alpha\|\leq m_1\quad (m_1>0). \tag{11} \]
II. Conditions are satisfied which ensure in each domain \(H_\alpha\) the existence and uniqueness of the solution of system (10) for given initial values and its continuous dependence on the initial conditions, as well as conditions for the unobstructed continuation, by continuity, of integral curves from any domain \(H_\alpha\) into the adjacent \(H_{\alpha+1}\).
III. In each central domain
\[ f(z,t)-f(z^0,t)=P(t)(z-z^0)+R(z,t), \tag{12} \]
where \(P(t)\) is a matrix continuous in each interval \(t_\alpha \leq t \leq t_{\alpha+1}\) and bounded for \(t\geq 0\), while \(R(z,t)\) is a nonlinear remainder satisfying the inequality
\[ \|R(z,t)\|\leq a\|z-z^0\|;\quad a=\mathrm{const},\ t\geq 0. \tag{13} \]
IV. The limiting relation, holding in any lower (upper) angular domain,
\(f(z,t)-f(z^0,t)\to +\xi_\alpha\) (respectively, \(-\xi_\alpha\)) as \((z,t)\to M_\alpha\), is fulfilled uniformly with respect to \(\alpha\).
V. The surfaces \(\Psi_\alpha=0\) are continuous, and at the points \(M_\alpha\) are smooth.
VI. Along the integral curve \(z=z^0(t)\)
\[ \left(\frac{d\Psi_\alpha}{dt}\right)_{M_\alpha}^{-}\neq 0 \quad\text{and}\quad \frac{(d\Psi_\alpha/dt)_{M_\alpha}^{+}}{(d\Psi_\alpha/dt)_{M_\alpha}^{-}}\geq \Gamma>0, \tag{14} \]
where
\[ \frac{d\Psi_\alpha}{dt} = \left( \frac{\partial\Psi_\alpha}{\partial z}f + \frac{\partial\Psi_\alpha}{dt} \right)_{z=z^0(t)}; \quad \frac{\partial\Psi_\alpha}{\partial z} \text{ is the gradient vector (row).} \]
Introduce the deviation \(x=z-z^0(t)\) and rewrite system (10) in deviations:
\[ \frac{dx}{dt}=\varphi(x,t);\quad \varphi(x,t)=f(z^0+x,t)-f(z^0,t). \tag{15} \]
In view of (14), the equation of the part of the surface
\[ F_\alpha(x,t)=0,\quad [F_\alpha(x,t)\equiv \Psi_\alpha(z^0+x,t)], \tag{16} \]
lying below (above) the plane \(t=t_\alpha\), can be written in the form
\[ t_\alpha-t=h_\alpha^{-}x+0(\|x\|),\quad [t_\alpha-t=h_\alpha^{+}x+0(\|x\|)], \tag{17} \]
where the row vector
\[ h_\alpha^{\pm} = \left[ \frac{\partial\Psi_\alpha}{\partial z} /\left(\frac{d\Psi_\alpha}{dt}\right)^{\pm} \right]_{M_\alpha}. \]
It is assumed that the quantities \(h_\alpha^{-}\) are bounded in modulus,
\[ \|h_\alpha^{-}\|\leqslant m_2\quad (m_2>0), \tag{18} \]
and the limiting relation occurring in (17)
\[ \frac{o(\|x\|)}{\|x\|}\to 0 \quad \text{as } \|x\|\to 0 \tag{19} \]
holds uniformly with respect to \(\alpha\).
The linear approximation of system (10) is defined (see [3] and [4]) as the totality: 1) of the differential equations
\[ \frac{dx}{dt}=P(t)x, \tag{20} \]
which are satisfied by the solution inside each interval \(t_\alpha \leqslant t < t_{\alpha+1}\), and 2) of the jump conditions, for \(t=t_\alpha\), of the integral curves \(x=x(t)\), determined by the formulas
\[ x_\alpha^{+}=S_\alpha x_\alpha^{-}, \tag{21} \]
where the matrix
\[ S_\alpha=\{(S_\alpha)_{ik}\},\quad (S_\alpha)_{ik}=\delta_{ik}+\xi_{\alpha i}h_{\alpha k}^{-}, \]
\(\delta_{ik}\) is the Kronecker symbol, and \(\xi_{\alpha i}\) and \(h_{\alpha k}^{-}\) are the coordinates of the vectors \(\xi_\alpha\) and \(h_\alpha^{-}\).
For the system of the linear approximation the following theorem is valid.
Theorem. Every nontrivial solution of the system of the linear approximation (20)—(21) has a finite characteristic exponent.
Indeed, from inequalities (11) and (18) it follows that the matrices \(S_\alpha\) are bounded as a whole, and from the structure of the matrices \(S_\alpha\) and relations (14) we obtain that
\[ \det S_\alpha= \frac{(d\Psi_\alpha/dt)_{\overline{M}_\alpha}^{+}} {(d\Psi_\alpha/dt)_{\overline{M}_\alpha}^{-}} \geqslant \Gamma>0. \]
Since the matrices \(Q_\alpha=S_\alpha' S_\alpha\) are also bounded as a whole, and \(\det Q_\alpha \geqslant \Gamma^2>0\), there exist positive constants \(K_1\) and \(K_2\) such that, for all \(\alpha\),
\[ 0<K_1^2<\mu_{\alpha 1}^2\leqslant \mu_{\alpha 2}^2<K_2^2, \]
where \(\mu_{\alpha 1}^2\) and \(\mu_{\alpha 2}^2\), as in formulas (6), denote the extreme values of the characteristic numbers of the positive definite matrices \(Q_\alpha\).
Consequently, for any interval \(t_\alpha \leqslant t < t_{\alpha+1}\), the values of the piecewise-constant functions \(M_1(t)\) and \(M_2(t)\), constructed from the jump matrices \(S_\alpha\) according to formulas (8), satisfy the inequalities
\[ 0<K_1^\alpha<M_1(t)\leqslant M_2(t)<K_2^\alpha<K_2^{\frac{t}{T}+1}, \]
whence
\[ K_1' \leqslant \lim_{t\to\infty}\frac{1}{t}\ln M_1(t) \leqslant \lim_{t\to\infty}\frac{1}{t}\ln M_2(t) \leqslant \frac{\ln K_2}{T}, \]
i.e.,
\[ K_1' \leqslant \lambda[M_1]\leqslant \lambda[M_2]\leqslant \frac{\ln K_2}{T}, \]
where \(K'_1=0\) for \(K_1 \geqslant 1\) and \(K'_1=\dfrac{1}{T}\ln K_1\) for \(K_1<1\). Since the matrix \(P(t)\) of the coefficients of equations (20) is continuous in each interval \(t_a \leqslant t < t_{a+1}\) and is bounded for \(t \geqslant 0\), it follows from the finiteness of the characteristic exponents of the functions \(M_1(t)\) and \(M_2(t)\), according to the theorem proved earlier, that every nontrivial solution of the system of the linear approximation (20)—(21) also has a finite characteristic exponent.
We now turn to the theorems on stability and instability of solutions of a nonlinear system of differential equations with discontinuous right-hand sides, having almost reducible linear approximations.
Theorem 1. If the system of the linear approximation (20)—(21) is almost reducible and has negative characteristic exponents, then the solution \(z=z^0(t)\) of the original nonlinear system (10) is asymptotically stable, provided only that the constant “\(a\)” in inequality (13) is sufficiently small.
For the proof of the theorem we first use the lemma and apply to the system (20)—(21) the Lyapunov transformation \(x=L(t)y\), reducing this system to the linear differential equations (4) with coefficients continuous and bounded for \(t \geqslant 0\) and with continuous solutions. The system (4) obtained in this way is also almost reducible, and therefore, according to Theorems 5 and 3 of B. F. Bylov [1], the elements of the fundamental matrix normalized at \(t=t_0\)
\[ Y(t,t_0)=\{y_{ik}\}, \qquad [Y(t_0,t_0)=E=\{\delta_{ik}\}] \]
satisfy, for all \(t \geqslant t_0>0\), the inequalities
\[ |y_{ik}(t,t_0)| \leqslant C(\varkappa)e^{(\lambda_k+\varkappa)(t-t_0)}, \]
where \(\lambda_k\) is the characteristic exponent of the solution, \(\varkappa\) is any positive number, and \(C(\varkappa)\) is a constant depending on \(\varkappa\), but not depending on \(t\) and \(t_0\). Choosing a fixed positive \(\varkappa<\min(-\lambda_k)\) and \(0<\beta<\min(-\lambda_k-\varkappa)\), we obtain
\[ |y_{ik}(t,t_0)|<Be^{-\beta(t-t_0)}\quad (t\geqslant t_0>0), \tag{22} \]
where \(B\) and \(\beta\) are positive constants not depending on \(t\) and \(t_0\). Since the fundamental matrix, normalized at \(t=t_0\),
\[ X(t,t_0)=\{x_{ik}(t,t_0)\} \]
of the system (20)—(21) is related to \(Y(t,t_0)\) by the relation
\[ X(t,t_0)=L(t)Y(t,t_0)L^{-1}(t_0), \]
it follows from (22), in view of the boundedness for \(t \geqslant 0\) of the matrices \(L(t)\) and \(L^{-1}(t)\), that for arbitrary
\[ t \geqslant t_0>0,\qquad |x_{ik}(t,t_0)|<B_1 e^{-\beta(t-t_0)}, \]
where \(B_1\) and \(\beta\) are positive constants not depending on \(t\) and \(t_0\). When these latter inequalities are fulfilled, as was shown by the author in [4] and [5], the solution \(z=z^0(t)\) of the original nonlinear system (10) is asymptotically stable, provided only that the constant \(a\) in inequality (13) is sufficiently small. The theorem is proved.
Theorem 2. If the system of the linear approximation (20)—(21) is almost reducible and has at least one positive characteristic
ristic exponent, then the solution \(z=z^0(t)\) of the original nonlinear system (10) is unstable, provided only that the constant “\(a\)” in inequality (13) is sufficiently small.
Indeed, suppose that, by means of the transformation \(x=L_\delta(t)y\), the system of the linear approximation (20)—(21) has been reduced to equations with continuous solutions
\[ \frac{dy}{dt}=[D+\Phi(t)]y,\qquad D=\mathrm{const},\qquad y_\alpha^+=y_\alpha, \tag{23} \]
in which the elements \(\varphi_{ik}\) of the matrix \(\Phi(t)\) satisfy the inequalities
\[ |\varphi_{ik}(t)|<\delta \qquad (t\geq 0). \tag{23a} \]
Then one of the roots of the characteristic equation of the matrix \(D\) has positive real part. In this case, as A. M. Lyapunov [2] showed, there exists a positive number \(\gamma\) and a quadratic form \(V\), taking positive values for some arbitrarily small (in modulus) \(y_1\) and satisfying the relation
\[ \left(\frac{\partial V}{\partial y}\right)Dy=\gamma V+\|y\|^2, \tag{24} \]
where \(\dfrac{\partial V}{\partial y}\) denotes the gradient vector (row). Applying to equations (15) the transformation \(x=L_\delta(t)y\), we obtain the system
\[ \frac{dy}{dt}=q(y,t),\qquad q=L_\delta^{-1}\left(\varphi-\frac{dL_\delta}{dt}y\right). \tag{25} \]
The cylinder \(C\) in the space \(yt\) (as also in the space \(zt\)) is divided by the surfaces
\[ Q_\alpha(y,t)=0,\qquad [Q_\alpha(y,t)\equiv F_\alpha(L_\delta y,t)] \]
and by the planes \(t=t_0\) into angular and central regions.
Consider the change in the values of the form \(V(y)\) along the integral curves \(y=y(t)\) of system (25).
In the central region, in accordance with (12), (15), and (25), we have
\[ q(y,t)=Dy+\Phi(t)y+R_\delta^*(y,t),\qquad R_\delta^*=L_\delta^{-1}(t)R[L_\delta y+z^0(t),t]. \tag{26} \]
Therefore, according to (13), we obtain
\[ \|R_\delta^*(y,t)\|<a_1\|y\|, \tag{27} \]
where \(a_1=am_\delta\), \(m_\delta>0\). From the boundedness for \(t\geq 0\) of the matrices \(L_\delta(t)\) and \(L_\delta^{-1}(t)\) it follows that \(m_\delta\) is a finite quantity. Denoting by \(V'\) the total derivative of \(V\) with respect to \(t\), computed by means of equation (25), we obtain
\[ V'=\left(\frac{\partial V}{\partial y}\right)Dy+ \left(\frac{\partial V}{\partial y}\right)\Phi y+ \frac{\partial V}{\partial y}R_\delta^*. \]
According to (24), we have
\[ V'=\gamma V+\|y\|^2+ \frac{\partial V}{\partial y}(\Phi y+R_\delta^*)\geq \gamma V, \tag{28} \]
provided only that
\[ \left\|\frac{\partial V}{\partial y}\right\|\|\Phi y+R_\delta^*\|\leq \|y\|^2. \]
From (23a) and (27) it follows that
\[ \left\|\frac{\partial V}{\partial y}\right\|\,\|\Phi y+R^*\|< \left(\sqrt n\,\delta+am_\delta\right)\left\|\frac{\partial V}{\partial y}\right\|\,\|y\|. \]
Therefore, if \(\delta>0\) is chosen so small that
\[
\sqrt n\,\delta\left\|\frac{\partial V}{\partial y}\right\|<
\frac12\|y\|,
\]
and if we assume that the transformation \(x=L_\delta y\) used corresponds to the chosen value of \(\delta\), then relation (28) is valid, provided only that the constant “\(a\)” in (13) is so small that
\[ am_\delta\left\|\frac{\partial V}{\partial y}\right\|< \frac12\|y\|. \]
From (28) it follows that the values of \(V\) at the times \(t^*\) and \(t>t^*\), when the point of the integral curve lies in one and the same central region, satisfy the relation
\[ V\geq V^*e^{\gamma(t-t^*)}. \tag{29} \]
Passing to the consideration of the behavior of integral curves in an angular region, let us assume, for definiteness, that the surface bounding it, \(\psi_\alpha=0\), is situated below the plane \(t=t_\alpha\). The equation of this surface, according to (20) and (22), can be written in the form
\[ t_\alpha-t=h_\alpha x+(*). \]
Here and everywhere below, by \((*)\) and \((**)\) we denote quantities whose order of smallness with respect to \(x\) (uniformly with respect to \(t\) and with respect to the number \(\alpha\) of the angular region) is, respectively, higher than the first or the second.
Let the integral curve of the nonlinear system (15), (25) pass from the point \(x_1t_1(y_1t_1)\) on the discontinuity surface to the point \(x_\alpha t_\alpha(y_\alpha t_\alpha)\) on the plane \(t=t_\alpha\). According to assumption IV, \(\varphi\Rightarrow \xi\) as \(x\to0\), therefore
\[ x_\alpha-x_1=\int_{t_1}^{t_\alpha}\varphi\,dt =\xi_\alpha(t_\alpha-t_1)+(*)=\xi_\alpha h_\alpha x_1+(*) \]
and
\[ x_\alpha=S_\alpha x_1+(*), \]
where the sign \(\Rightarrow\) means convergence to the limit uniformly with respect to \(\alpha\). Moreover, from the boundedness, for \(t\geq0\), of the matrix \(dL_\delta/dt\), the uniform continuity of \(L_\delta(t)\) follows. Therefore, as \(t\to t_\alpha-0\),
\[ L_\delta(t)\Rightarrow L_\delta(t_\alpha-0)\equiv L_{\delta\alpha}^{-}. \]
By virtue of (11), (18), and the relations
\[ L_{\delta\alpha}^{+}=S_\alpha L_{\delta\alpha}^{-}, \]
which follow from the continuity of the solutions of system (23), \(y_\alpha^{+}=y_\alpha^{-}\), and the equalities
\[ L_{\delta\alpha}^{+}y_\alpha^{+}=x_\alpha^{+}=S_\alpha x_\alpha^{-}=S_\alpha L_{\delta\alpha}^{-}y_\alpha^{-}, \]
we obtain
\[ x_1=L_\delta(t_1)y_1=L_{\delta\alpha}^{-}y_1+ [L_\delta(t_1)-L_\delta(t_\alpha-0)]y_1 =L_{\delta\alpha}^{-}y_1+(*) \]
and
\[ L_{\delta\alpha}^{+} y_{\alpha}^{+}=x_{\alpha}=S_{\alpha}x_1+(*)=S_{\alpha}L_{\bar\delta\alpha}y_1+(*)=L_{\bar\delta\alpha}^{+}y_1+(*), \]
whence1
\[ y_{\alpha}^{+}=y_1+(*) \quad \text{and} \quad V(y_{\alpha}^{+})=V(y_1)+(**). \]
Consequently, for sufficiently small \(y_1\),
\[ e^{-\eta}<\frac{V(y_{\alpha}^{+})}{V(y_1)}<e^{\eta}, \tag{30} \]
where \(\eta\) is an arbitrarily small positive number.
Now let \(\varepsilon>0\) be chosen so small that, for \(V(y)\leq \varepsilon\), inequalities (29), (30) hold, \(\eta<\dfrac{1}{2}T(\gamma-\nu)\), and the residence time \(\Delta t\) in any angular domain is less than \(\dfrac{1}{2\gamma}(\gamma-\nu)\) (which is possible by virtue of relations (17), (18), and (19)), where \(0<\nu<\gamma\). The planes
\[ t=t_{\alpha}^{*}=\frac{t_{\alpha+1}+t_{\alpha}}{2} \]
inside the cylinder \(V(y)\leq \varepsilon\) do not intersect the angular domains. Choose the initial point \(y_1=y(t_1^{*})\) so that \(y_1\) is arbitrarily small in modulus and
\[ V_1=V[y(t_1^{*})]>0. \]
Then, by (29) and (30),
\[ V_2=V[y(t_2^{*})]>V_1 e^{\gamma(T-\Delta t)-\eta}>V_1 e^{\nu T}, \]
provided that on the interval \(t_1^{*}\leq t\leq t_2^{*}\) the integral curve lies inside the cylinder \(V=\varepsilon\). Since \(V_2>V_1\), the arguments given above can be repeated for the interval \(t_2^{*}\leq t\leq t_3^{*}\), and so on. In this case
\[ V_{\alpha}=V[y(t_{\alpha}^{*})]>V_1 e^{(\alpha-1)\nu T}. \]
Consequently, at some moment of time the integral curve will leave the cylinder \(V=\varepsilon\). The theorem is proved.
References
- Bylov B. F. Siberian Mathematical Journal, III, No. 3, 333–360, 1962.
- Lyapunov A. M. The General Problem of the Stability of Motion. Gostekhizdat, Moscow, 1950.
- Aizerman M. A. and Gantmakher F. R. DAN SSSR, 116, No. 4, 1957.
- Livartovskii I. V. DAN SSSR, 125, No. 4, 733–737, 1959.
- Livartovskii I. V. PMM, XXIII, issue 3, 598–604, 1959.
Received by the editors
March 13, 1965
All-Union Scientific Research Institute
of Agricultural Machine Building
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In view of the transformation \(x=L_{\delta}(t)y\), quantities infinitely small relative to \(x\) are infinitely small quantities relative to \(y\) of the same order of smallness, and conversely. Therefore the former notation may be used. ↩