D. M. GROBMÁN

ASYMPTOTICS OF SOLUTIONS OF ALMOST LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovskii, 22 IV 1964)

\(1^\circ\). In the present note we state theorems that generalize and refine the author’s results (1) concerning questions of asymptotic equivalence of solutions of the systems

\[ \begin{aligned} x'&=Ax+F(t,x); \tag{1}\\ y'&=Ay. \tag{2} \end{aligned} \]

Here \(A\) is a square matrix of order \(n\) with constant coefficients; \(x,y,F(t,x)\) are \(n\)-dimensional vectors; \(F(t,x)\) is defined for \(t\ge t_0\) and arbitrary \(x\);

\[ F(t,0)=0; \tag{3} \]

\[ |F(t,x_1)-F(t,x_2)|\le g(t)|x_1-x_2|, \tag{4} \]

where \(g(t)\) is a nonnegative function.

\(2^\circ\). Obviously, the fraction \(|x(t)-y(t)|/|y(t)|\) may be taken as a measure of the closeness of the vectors \(x(t)\) and \(y(t)\). We shall call this fraction the deviation of \(x\) from \(y\), or simply the deviation. If the deviation of \(x\) from \(y\) tends to 0 as \(t\to\infty\), then we shall call the vectors \(x(t)\) and \(y(t)\) analogous.

It is easy to see that the ratio of the lengths of analogous vectors tends to unity as \(t\to\infty\), and the differences between the direction cosines (in the case of real \(x\) and \(y\)) tend to zero. Clearly, this occurs the more rapidly, the more rapidly the deviation tends to zero.

\(3^\circ\). Let \(\omega_1<\omega_2<\cdots<\omega_s\) be all the distinct real parts of the eigenvalues of the matrix \(A\). Consider those Jordan blocks in the Jordan form of \(A\) whose diagonal contains an eigenvalue with real part \(\omega_k\). Denote by \(m_{k+1}\) the order of the largest of these blocks.

Take any nonnegative number \(\alpha\), and let \(\omega_0\) denote an arbitrary number smaller than \(\omega_1-\alpha\): \(\omega_0<\omega_1-\alpha\). For each \(k=1,2\ldots,s\), define the index \(\widetilde{k}\) by means of the inequalities

\[ \omega_k-\alpha>\omega_{\widetilde{k}-1}, \qquad \omega_k-\alpha\le \omega_{\widetilde{k}}. \]

Obviously, the index \(\widetilde{k}\) is determined uniquely from this. In particular, if \(\alpha=0\), then \(\widetilde{k}=k\).

Denote by \(m_{\widetilde{k}}^{0}\) the number defined by the conditions

\[ m_{\widetilde{k}}^{0}= \begin{cases} 0, & \text{if } \omega_k-\alpha<\omega_{\widetilde{k}},\\ m_{\widetilde{k}}, & \text{if } \omega_k-\alpha=\omega_{\widetilde{k}}. \end{cases} \]

Since for \(\alpha=0\) one has \(\widetilde{k}=k\) and \(\omega_k-\alpha=\omega_{\widetilde{k}}\), in this case \(m_{\widetilde{k}}^{0}=m_k\).

\(4^\circ\). Theorem 1. Let \(\alpha\) and \(\beta\) be arbitrary real numbers, with \(\alpha>0\). Suppose that

\[ \int_{t_0}^{\infty} e^{\alpha\tau}\tau^\beta g(\tau)\,d\tau<+\infty. \]

Then there exists a topological mapping \(\Phi\) of the space \((x)\) onto the space \((y)\) with the following properties: a) \(\Phi\) and \(\Phi^{-1}\) satisfy the Lipschitz condition; b) through the points corresponding under \(\Phi\) at the instant \(t=t^*\), where \(t^*\) is sufficiently large, there pass solutions of systems (1) and (2) that are analogous and have deviation

\[ o\!\left(e^{-\alpha t}t^{m_{\widetilde{k}}^{0}-\beta}\right). \]

Theorem 2. Let, for some nonnegative number \(\beta\),

\[ \int_{t_0}^{\infty} \tau^\beta g(\tau)\,d\tau < +\infty . \tag{5} \]

Then there exists a homeomorphism \(\Phi\) mapping the space \((x)\) onto the space \((y)\) and having the following properties: a) \(\Phi\) and \(\Phi^{-1}\) satisfy a Lipschitz condition; b) the solutions of systems (1) and (2) passing at \(t=t^*\), where \(t^*\) is sufficiently large, through points corresponding under \(\Phi\), have identical exponents; c) for every index \(k\) for which \(\beta \geq m_k\), the solutions of systems (1) and (2) with exponents \(\omega_k\), passing at the initial instant through points corresponding under \(\Phi\), are analogous, and their deviation is \(o(t^{m_k-\beta})\) as \(t \to \infty\).

\(5^\circ.\) A simple consequence of Theorem 1 is the criterion of V. A. Yakubovich \((^2)\), which guarantees the existence of such a homeomorphic mapping of the space \((x)\) onto the space \((y)\) that, through corresponding points at the initial instant, there pass solutions whose difference tends to zero as \(t \to \infty\). From Theorem 2 it is not difficult to obtain one of the reducibility theorems \((^3)\) of the same author. By the standard method, using the principle of linear inclusion \((^5)\), one can derive from Theorem 2 a proposition very close to the theorem of A. Wintner and P. Hartman \((^4)\).

If the vector \(F(t,x)\), instead of satisfying requirement (4), satisfies the inequality \(|F(t,x)| \leq g(t)|x|\) and condition (5) is fulfilled, then every solution of system (1) with exponent \(\omega_k\), where \(k\) is such that \(m_k \leq \beta\), is analogous to some solution of system (2), and their deviation is \(o(t^{m_k-\beta})\) as \(t \to \infty\).

\(6^\circ.\) To illustrate the sharpness of the results obtained, let us consider examples.

Example 1. Let the systems be given by

\[ \dot{x}_1=-x_1+g(t)x_4;\qquad \dot{x}_2=x_1-x_2;\qquad \dot{x}_3=-x_3;\qquad \dot{x}_4=x_3-x_4; \tag{6} \]

\[ \dot{y}_1=-y_1;\qquad \dot{y}_2=y_1-y_2;\qquad \dot{y}_3=-y_3;\qquad \dot{y}_4=y_3-y_4, \tag{7} \]

where \(g(t)\) is some positive function.

If

\[ \int_{t_0}^{\infty} \tau g(\tau)\,d\tau < +\infty , \]

then, by Theorem 2, every solution of (6) is analogous to some solution of system (7). Suppose that

\[ \int_{t_0}^{\infty} \tau g(\tau)\,d\tau = \infty . \]

Consider the solution \(x(t)\) of system (6) with initial conditions at \(t=0\)
\((0;0;1;0)\). Obviously, the coordinates of this solution are given by the formulas

\[ x_1=e^{-t}\int_0^t \xi g(\xi)\,d\xi;\qquad x_2=e^{-t}\int_0^t d\tau \int_0^\tau \xi g(\xi)\,d\xi;\qquad x_3=e^{-t};\qquad x_4=te^{-t}. \]

Put

\[ \int_0^t \xi g(\xi)\,d\xi=\varphi(t). \]

By assumption, \(\varphi(t)\to\infty\) as \(t\to\infty\), and since \(g(\xi)>0\), \(\varphi(t)\) increases monotonically.

For any solution \(y(t)=\{y_1(t);y_2(t);y_3(t);y_4(t)\}\) of system (7), the inequality

\[ |x(t)-y(t)|\geq |x_2(t)-y_2(t)| = e^{-t}\left|\int_0^t \varphi(\tau)\,d\tau-at-b\right|\geq \]

\[ \geq e^{-t}\left\{\int_0^t \varphi(\tau)\,d\tau -t\left|a+\frac{b}{t}\right|\right\}. \]

(Here \(a\) and \(b\) are arbitrary constants.)

Using the monotonicity and positivity of \(\varphi(t)\), we can write

\[ \int_0^t \varphi(\tau)\,d\tau > \int_{t/2}^t \varphi(\tau)\,d\tau > \frac{t}{2}\,\varphi\!\left(\frac{t}{2}\right). \]

Hence

\[ \frac{|x(t)-y(t)|}{|y(t)|} \ge \frac{1}{|y(t)|}\,e^{-tt} \left\{ \frac{1}{2}\,\varphi\!\left(\frac{t}{2}\right) - \left|a+\frac{b}{t}\right| \right\}. \]

Since \(|y(t)|\) “grows” no faster than \(e^{-tt}\), and \(\varphi(t)\to\infty\) as \(t\to\infty\), the deviation of the solution \(x(t)\) of system (6) under consideration from any solution \(y(t)\) of system (7) increases without bound as \(t\to\infty\). Consequently, for \(x(t)\) there is no analogous solution of system (7).

Thus, without imposing some additional restrictions on the matrix \(A\) or the vector \(F(t,x)\), the conditions of Theorem 2 cannot be weakened.

Example 2. Consider the systems

\[ \dot{x}_1=-2x_1+g(t)x_4;\qquad \dot{x}_2=x_1-2x_2;\qquad \dot{x}_3=-x_3;\qquad \dot{x}_4=x_3-x_4; \tag{8} \]

\[ \dot{y}_1=-2y_1;\qquad \dot{y}_2=y_1-2y_2;\qquad \dot{y}_3=-y_3;\qquad \dot{y}_4=y_3-y_4. \tag{9} \]

If

\[ \int_{t_0}^{\infty} e^\tau g(\tau)\,d\tau < +\infty, \]

then, by Theorem 1, any solution of system (8) is analogous to some solution of system (9), and their deviation is \(o(e^{-t})\). Let

\[ \int_{t_0}^{\infty} e^\tau g(\tau)\,d\tau=\infty. \]

Then it can be shown, just as was done in Example 1, that the deviation of the solution \(x(t)\) of system (8), passing at \(t=0\) through the point \((0;\,0;\,1;\,0)\), from any solution of system (9) is a quantity infinitely large in comparison with \(e^{-t}\).

Institute of Electronic
Control Machines

Received
15 IV 1964

CITED LITERATURE

1. D. M. Grobman, DAN, 86, No. 1, 19 (1952).
2. V. A. Yakubovich, DAN, 63, No. 4 (1948).
3. V. A. Yakubovich, DAN, 66, 577 (1949).
4. P. Hartman, A. Wintner, Am. J. Math., 77, No. 4, 692 (1955).
5. B. F. Bylov, D. M. Grobman, UMN, 17, issue 3, 159 (1962).