ON ASYMPTOTICALLY EQUIVALENT LINEAR DIFFERENTIAL SYSTEMS

Yu. S. Bogdanov

We shall call two linear differential systems asymptotically equivalent if there exists a Lyapunov transformation taking one of these systems into the other. It is shown below that every system with bounded coefficients is asymptotically equivalent to a piecewise constant system whose coefficients take only two values [1].

§ 1. Basic notation and definitions

All quantities considered are assumed to be real functions of a real variable \(\tau\), taking nonnegative values.

Capital Latin letters denote square matrices of order \(\nu\), lowercase Latin letters denote vectors of order \(\nu\) and indices, and Greek letters denote scalars.

We choose and fix some norm of a numerical vector. The norm of a numerical matrix is taken to be that induced by the chosen vector norm [2]. Such a matrix norm is necessarily multiplicative and consistent with the vector norm.

Let us consider \(\mathbf P\)—the collection of all matrix functions \(P\), whose elements \(\pi_{ij}\) are measurable on every bounded interval of the positive half-axis and are equivalent to bounded functions for all \(\tau \ge 0\). Thus

\[ -\infty < \alpha_1 \overset{\mathrm{def}}{=} \inf_{\tau \ge 0}^{*} \pi_{ij}(\tau) \le \alpha_2 \overset{\mathrm{def}}{=} \sup_{\tau \ge 0}^{*} \pi_{ij}(\tau) < +\infty \tag{1} \]

for any \(i, j = 1, 2, \ldots, \nu\). Obviously,

\[ \beta \overset{\mathrm{def}}{=} \sup_{\tau \ge 0}^{*} \|P(\tau)\| < +\infty . \tag{2} \]

Let \(\mathbf L\) denote the collection of absolutely continuous matrix functions \(S\), bounded together with their inverses \(S^{-1}\) for all \(\tau \ge 0\),

\[ \sup_{\tau \ge 0}\|S(\tau)\| < +\infty,\qquad \sup_{\tau \ge 0}\|S^{-1}(\tau)\| < +\infty . \tag{3} \]

The conditions (3) may be replaced by the equivalent requirements

\[ \sup_{\tau \ge 0}\|S(\tau)\| < +\infty,\qquad \inf_{\tau \ge 0}|\det S(\tau)| > 0 . \]

We shall call the substitution \(y=Sx\) a Lyapunov transformation if \(S\in \mathbf{L}\).

Consider the linear differential system

\[ dx/d\tau=Px,\qquad P\in \mathbf{P}. \tag{4} \]

By a solution of (4) we mean any absolutely continuous vector function \(x(\tau)\), defined for all \(\tau\geq 0\) and such that \(dx(\tau)/d\tau\) exists almost everywhere and satisfies (4). We shall call system (4) asymptotically equivalent to the system

\[ dy/d\tau=Qy,\qquad Q\in \mathbf{P}, \tag{5} \]

if there exists a Lyapunov transformation \(y=Sx\), \(S\in \mathbf{L}\), carrying (4) into (5). The relation of asymptotic equivalence is obviously reflexive, symmetric, and transitive; therefore the totality of all systems (4) decomposes into nonintersecting classes of systems pairwise asymptotically equivalent to one another.

§ 2. An Auxiliary Inequality

Denote

\[ R \stackrel{\mathrm{def}}{=} Q-P. \]

Then system (5) can be represented in the form

\[ dy/d\tau=Py+Ry \]

and replaced, in the familiar way, by the system of linear integral equations

\[ y(\tau)=X(\tau)\left(y(0)+\int_0^\tau X^{-1}(\sigma)R(\sigma)y(\sigma)\,d\sigma\right), \]

where \(X\) is a fundamental system of solutions of (4), normalized at \(\tau=0\), i.e.

\[ X(0)=I \]

(\(I\) is the identity matrix). The fundamental system of solutions \(Y\), \(Y(0)=I\), of system (5) satisfies the relation

\[ Y(\tau)=X(\tau)\left(I+\int_0^\tau X^{-1}(\sigma)R(\sigma)Y(\sigma)\,d\sigma\right). \tag{6} \]

Suppose that \(R\) is summable on the entire half-line \([0,+\infty)\), and transform (6) by integration by parts, taking into account (4) and (5):

\[ Y(\tau)=X(\tau)\left(I+X^{-1}(\tau)\int_\infty^\tau R(\sigma)\,d\sigma\cdot Y(\tau) +\int_0^\infty R(\sigma)\,d\sigma+\right. \]

\[ \left. +\int_0^\tau X^{-1}(\sigma)P(\sigma)\left(\int_\infty^\sigma R(\omega)\,d\omega\right)Y(\sigma)\,d\sigma -\int_0^\tau X^{-1}(\sigma)\left(\int_\infty^\sigma R(\omega)\,d\omega\right)Q(\sigma)Y(\sigma)\,d\sigma \right), \]

or

\[ \left(I+\int_\tau^\infty R(\sigma)\,d\sigma\right)Y(\tau) = X(\tau)\left(I+\int_0^\infty R(\sigma)\,d\sigma+\right. \]

\[ \left. +\int_0^\tau X^{-1}(\sigma)\left(P(\sigma)\int_\infty^\sigma R(\omega)\,d\omega -\int_\infty^\sigma R(\omega)\,d\omega\cdot Q(\sigma)\right)Y(\sigma)\,d\sigma \right). \]

or, finally,

\[ \left(I+\int_{\tau}^{\infty} R(\sigma)\,d\sigma\right)Y(\tau)X^{-1}(\tau)= \]

\[ = I+\int_{0}^{\infty} R(\sigma)\,d\sigma -\int_{0}^{\tau} X^{-1}(\sigma)\left(P(\sigma)\int_{\sigma}^{\infty} R(\omega)\,d\omega-\right. \]

\[ \left.-\int_{\sigma}^{\infty} R(\omega)\,d\omega\cdot Q(\sigma)\right)Y(\sigma)X^{-1}(\sigma)X(\sigma)\,d\sigma . \]

Put

\[ Z \overset{\mathrm{def}}{=} YX^{-1}. \]

Then the matrix-function \(Z\) satisfies the integral equation

\[ A(\tau)Z(\tau)=B-\int_{0}^{\tau} C(\sigma)Z(\sigma)X(\sigma)\,d\sigma, \tag{7} \]

where

\[ A(\tau)\overset{\mathrm{def}}{=} I+\int_{\tau}^{\infty} R(\sigma)\,d\sigma, \]

\[ B \overset{\mathrm{def}}{=} I+\int_{0}^{\infty} R(\sigma)\,d\sigma, \]

\[ C(\sigma)\overset{\mathrm{def}}{=}X^{-1}(\sigma)\left(P(\sigma)\int_{\sigma}^{\infty}R(\omega)\,d\omega -\int_{\sigma}^{\infty}R(\omega)\,d\omega\cdot Q(\sigma)\right). \]

Denote

\[ a(t)\overset{\mathrm{def}}{=}\|P(t)\|+\|Q(t)\|, \]

\[ \rho(\tau)\overset{\mathrm{def}}{=}\left\|\int_{\tau}^{\infty}R(\sigma)\,d\sigma\right\|, \]

\[ \xi(\tau)\overset{\mathrm{def}}{=}\|X^{-1}(\tau)\|, \]

\[ \xi^{*}(\tau)\overset{\mathrm{def}}{=}\|X(\tau)\|, \]

\[ \zeta(\tau)\overset{\mathrm{def}}{=}\|Z(\tau)\|. \]

Then from (7) we obtain

\[ (1-\rho(\tau))\zeta(\tau)\leq 1+\rho(0)+\int_{0}^{\tau}\xi(\sigma)\xi^{*}(\sigma)\rho(\sigma)a(\sigma)\zeta(\sigma)\,d\sigma . \]

Suppose that

\[ \rho(\tau)\leq \rho_{0}=\mathrm{const}<1 \]

for all \(\tau\geq 0\). Then

\[ \zeta(\tau)\leq \frac{1+\rho_{0}}{1-\rho_{0}} +\frac{1}{1-\rho_{0}}\int_{0}^{\tau}\xi(\sigma)\xi^{*}(\sigma)\rho(\sigma)a(\sigma)\zeta(\sigma)\,d\sigma \]

and, on the basis of Gronwall’s lemma (see, for example, [3], p. 19),

\[ \zeta(\tau)\leq \frac{1+\rho_{0}}{1-\rho_{0}} \exp\left[ (1+\rho_{0})(1-\rho_{0})^{-2} \int_{0}^{\tau}\xi(\sigma)\xi^{*}(\sigma)\rho(\sigma)a(\sigma)\,d\sigma \right]. \]

Suppose that

\[ \int_0^\infty \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\alpha(\sigma)\,d\sigma \]

converges. Then for all \(\tau \ge 0\) we obtain

\[ \left\|Y(\tau)X^{-1}(\tau)\right\| \le \gamma, \tag{8} \]

where

\[ \gamma \stackrel{\mathrm{def}}{=} \frac{1+\rho_0}{1-\rho_0} \exp\left[ (1+\rho_0)(1-\rho_0)^{-2} \int_0^\infty \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\alpha(\sigma)\,d\sigma \right] = \]

\[ = \mathrm{const}<+\infty . \]

If one carries out analogous arguments for the systems that are adjoint to systems (3) and (4), then we obtain

\[ \left\|X(\tau)Y^{-1}(\tau)\right\|\le \gamma . \tag{9} \]

§ 3. A SUFFICIENT CONDITION FOR ASYMPTOTIC STABILITY

By means of a well-known device [4], pp. 31—32, using relation (1), one can prove the existence of constants \(\lambda\) and \(\mu\) such that

\[ \|X(\tau)\|\le e^{\lambda\tau},\quad \|X^{-1}(\tau)\|\le e^{\mu\tau} \]

and, consequently,

\[ \xi^*(\tau)\le e^{\lambda\tau},\quad \xi(\tau)\le e^{\mu\tau}. \tag{10} \]

Theorem. If

\[ \left\|\int_\tau^\infty R(\sigma)\,d\sigma\right\|\le \rho_0 e^{-\beta_0\tau}, \tag{*} \]

where \(\rho_0,\ \beta_0\) are positive constants, with \(\rho_0<1,\ \beta_0>\max\{\lambda,\mu\}\), then system (4) is asymptotically equivalent to system (5).

Proof. System (4) can be transformed into system (5) by the substitution \(y=Sx\), where

\[ S(\tau)\stackrel{\mathrm{def}}{=}Y(\tau)X^{-1}(\tau). \]

On the basis of (2), the matrix-functions \(P\) and \(Q\) are equivalent to matrices having bounded norm, and therefore

\[ \beta^*=\sup_{\tau\ge0}^{*} a(\tau)=\sup_{\tau\ge0}^{*}(\|P(\tau)\|+\|Q(\tau)\|)<+\infty \]

and, consequently,

\[ \int_0^\tau \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\alpha(\sigma)\,d\sigma \le \beta^* \int_0^\tau \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\,d\sigma . \]

On the basis of (10) and (*)

\[ \xi(\sigma)\xi^*(\sigma)\le e^{(\lambda+\mu)\tau}, \]

\[ \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\le \rho_0 e^{(\lambda+\mu-\beta_0)\tau}, \]

but \(\lambda+\mu-\beta_0<0\), and therefore the integral

\[ \int_0^\infty \xi(\sigma)\xi^*(\sigma)\rho(\sigma)\alpha(\sigma)\,d\sigma \]

converges. But then from (8) the boundedness of \(S\) follows. Similarly, from (9) the boundedness of \(S^{-1}\) follows. The theorem is proved.

Remark. The integral identity (7) can be used to construct other sufficient conditions for the asymptotic equivalence (3) and (4), for example, to derive conditions that take into account structural features of \(X(\tau)\).

§ 4. APPROXIMATION OF \(P\) BY A MATRIX WHOSE ELEMENTS TAKE ONLY TWO VALUES

Lemma. Let \(P \in \mathbf P\). Then for every function \(\varphi\), given, positive and continuous for \(\tau \ge 0\), there exists a matrix \(Q\) with the properties:

1) \(Q\) is a piecewise continuous matrix whose elements take only two values \(\alpha_1\) or \(\alpha_2\) (see (1)); in particular \(Q \in \mathbf P\);

2) the integral

\[ \int_0^\infty (Q(\sigma)-P(\sigma))\,d\sigma \]

converges, and

\[ \left\|\int_\tau^\infty (Q(\sigma)-P(\sigma))\,d\sigma\right\| \le \varphi(\tau). \]

Proof. For fixed \(\nu\), for any \(\varepsilon>0\) there exists a \(\delta>0\) such that for every \(\nu \times \nu\) matrix \(A\) with elements \(a_{ij}\), from \(|a_{ij}|<\delta\) it always follows that \(\|A\|<\varepsilon\). The function \(\delta_\varepsilon\) is defined for all \(\varepsilon>0\) and can always be chosen continuous. Consequently, the function

\[ \delta(\tau)\stackrel{\mathrm{def}}{=}\delta_{\varphi(\tau)}>0,\qquad \tau \ge 0. \]

will also be continuous.

Take an arbitrary natural number \(\chi\). Denote by \(\chi\) the smallest value of \(\delta(\tau)\) on the interval \([\chi-1,\chi]\). Obviously, \(\chi>0\).

Choose and fix any values \(i,j\). Put

\[ \pi^*(\tau)\stackrel{\mathrm{def}}{=}\int_0^\tau \pi_{ij}(\sigma)\,d\sigma. \]

It follows from (1) that for any \(\tau',\tau''\) \((0\le \tau'<\tau'')\) one has

\[ \alpha_1(\tau''-\tau')\le \pi^*(\tau'')-\pi^*(\tau')\le \alpha_2(\tau''-\tau'). \tag{11} \]

On the interval \([\chi-1,\chi]\) we construct an auxiliary function \(\tilde\rho\). If \(\alpha_1=\alpha_2\), then put \(\tilde\rho(\tau)=\alpha_1\). If \(\alpha_1<\alpha_2\), then we construct \(\tilde\rho\) in the form of a piecewise linear function.

The continuous function

\[ \Psi_1(\tau)\stackrel{\mathrm{def}}{=} \chi+\pi^*(\tau)-\pi^*(\chi-1)-\alpha_2(\tau-\chi+1) \]

at \(\tau=\chi-1\) takes the value \(\chi>0\). For all \(\tau\ge \chi-1\), on the basis of (11), it is true that

\[ \Psi_1(\tau)=\chi+[\pi^*(\tau)-\pi^*(\chi-1)]-\alpha_2(\tau-\chi+1)\le \]

\[ \le \chi+\alpha_2(\tau-\chi+1)-\alpha_2(\tau-\chi+1)=\chi, \]

\[ \Psi_1(\tau)\le \chi,\qquad \tau\ge \chi-1. \tag{12} \]

If \(\Psi_1(\tau)>0\) for all \(\tau\in[\chi-1,\chi]\), then put \(\tau_1\stackrel{\mathrm{def}}{=}\chi\). Otherwise denote by \(\tau_1\) the smallest of the roots of \(\Psi_1\) on \([\chi-1,\chi]\). By the definition of \(\tau_1\) and on the basis of (12),

\[ 0\le \Psi_1(\tau)\le \chi,\qquad \tau\in[\chi-1,\tau_1]. \tag{13} \]

Estimate the difference \(\tau_1-\varkappa+1\). On the one hand,

\[ \Psi_1(\tau_1)-\Psi_1(\varkappa-1)=0-\chi=-\chi . \]

On the other hand,

\[ \Psi_1(\tau_1)-\Psi_1(\varkappa-1) =\pi^*(\tau_1)-\chi-\pi^*(\varkappa-1)-\alpha_2(\tau_1-\varkappa+1)-\chi \]

\[ =\pi^*(\tau_1)-\pi^*(\varkappa-1)-\alpha_2(\tau_1-\varkappa+1). \]

Therefore

\[ \pi^*(\tau_1)-\pi^*(\varkappa-1)=\alpha_2(\tau_1-\varkappa+1)-\chi, \]

but on the basis of (1)

\[ \alpha_1(\tau_1-\varkappa+1)\leqslant \alpha_2(\tau_1-\varkappa+1)-\chi \]

or

\[ \chi \leqslant (\alpha_2-\alpha_1)(\tau_1-\varkappa+1) \]

and, consequently,

\[ \tau_1-\varkappa+1 \geqslant \frac{\chi}{\alpha_2-\alpha_1}>0 . \tag{14} \]

If \(\tau_1<\varkappa\), then consider the continuous function

\[ \Psi_2(\tau)\stackrel{\mathrm{def}}{=}\pi^*(\tau)-\chi-\pi^*(\tau_1)-\alpha_1(\tau-\tau_1). \]

On the basis of (11)

\[ \Psi_2(\tau)\geqslant \alpha_1(\tau-\tau_1)-\alpha_1(\tau-\tau_1)-\chi=-\chi<0, \]

\[ \Psi_2(\tau)\geqslant -\chi,\quad \tau\geqslant \tau_1 . \tag{15} \]

For \(\tau=\tau_1\) the function \(\Psi_2\) assumes a negative value. If \(\Psi_2(\tau)<0\) for all \(\tau\in[\tau_1,\varkappa]\), then put

\[ \tau_2\stackrel{\mathrm{def}}{=}\varkappa . \]

Otherwise denote by \(\tau_2\) the smallest of the zeros of \(\Psi_2\) on \([\tau_1,\varkappa]\). On the basis of inequality (15) and the definition of \(\tau_2\)

\[ 0\geqslant \Psi_2(\tau)\geqslant -\chi,\quad \tau\in[\tau_1,\tau_2]. \tag{16} \]

Estimate the difference \(\tau_2-\tau_1\). On the one hand,

\[ \Psi_2(\tau_2)-\Psi_2(\tau_1)=0-\Psi_2(\tau_1)=\chi . \]

On the other hand,

\[ \Psi_2(\tau_2)-\Psi_2(\tau_1)=\pi^*(\tau_2)-\pi^*(\tau_1)-\alpha_1(\tau_2-\tau_1). \]

Therefore

\[ \pi^*(\tau_2)-\pi^*(\tau_1)=\alpha_1(\tau_2-\tau_1)+\chi, \]

but on the basis of (1)

\[ \alpha_2(\tau_2-\tau_1)\geqslant \alpha_1(\tau_2-\tau_1)+\chi, \]

\[ (\alpha_2-\alpha_1)(\tau_2-\tau_1)\geqslant \chi \]

and, consequently,

\[ \tau_2-\tau_1\geqslant \frac{\chi}{\alpha_2-\alpha_1}>0 . \tag{17} \]

If \(\tau_2<\varkappa\), then we consider the function \(\Psi_3\), replacing \(\varkappa-1\) by \(\tau_2\) in \(\Psi_1\), and define \(\tau_3\). Then (if necessary) the function \(\Psi_4\), obtained from \(\Psi_2\) by replacing \(\tau_1\) by \(\tau_3\), and so on.

We obtain a collection of intervals

\[ \{[\tau_{k-1},\tau_k]\}, \]

where \(\tau_0=\varkappa-1\). On each of these intervals \([\tau_{k-1},\tau_k]\) the function

\[ \Psi_k(\tau)=\pi^*(\tau)-\pi^*(\tau_{k-1})+(-1)^k\chi-\widetilde a(\tau-\tau_{k-1}) \]

is defined, where \(\widetilde a=a_2\) for odd \(k\) and \(\widetilde a=a_1\) for even \(k\). The functions \(\Psi_k\) satisfy the inequalities

\[ \begin{gathered} 0\leq \Psi_{2m+1}(\tau)\leq \chi,\quad \tau\in[\tau_{2m},\tau_{2m+1}],\\ 0\geq \Psi_{2m+2}(\tau)\geq -\chi,\quad \tau\in[\tau_{2m+1},\tau_{2m+2}]. \end{gathered} \tag{18} \]

Moreover,

\[ \tau_k-\tau_{k-1}\geq \frac{\chi}{a_2-a_1}. \tag{19} \]

It follows from inequality (19) that the number of intervals \([\tau_{k-1},\tau_k]\) is finite, not exceeding

\[ \left[\frac{a_2-a_1}{\chi}\right]+1. \]

Denote this number by \(k_0\). Then \(\tau_{k_0}=\varkappa\).

Put

\[ \rho^*(\tau)\overset{\mathrm{def}}{=} \begin{cases} \pi^*(\tau_{2m})+a_2(\tau-\tau_{2m}), & \tau\in[\tau_{2m},\tau_{2m+1}),\\ \pi^*(\tau_{2m+1})+\chi+a_1(\tau-\tau_{2m+1}), & \tau\in[\tau_{2m+1},\tau_{2m+2}). \end{cases} \]

Consequently,

\[ \begin{gathered} \rho^*(\tau_0)=\pi^*(\varkappa-1),\quad \rho^*(\tau_1)=\chi+\pi^*(\tau_1),\\ \rho^*(\tau_{2m})=\pi^*(\tau_{2m}),\\ \rho^*(\tau_{2m+1})=\chi+\pi^*(\tau_{2m+1}). \end{gathered} \tag{20} \]

From the definition of \(\tau_k\) it follows that

\[ \Psi_k(\tau_k)=0, \]

therefore

\[ \chi+\pi^*(\tau_1)-\pi^*(\varkappa-1)-a_2(\tau_1-\varkappa+1)=0, \]

\[ -\chi+\pi^*(\tau_2)-\pi^*(\tau_1)-a_1(\tau_2-\tau_1)=0, \]

\[ \cdots \]

\[ \chi+\pi^*(\tau_{2m+1})-\pi^*(\tau_{2m})-a_2(\tau_{2m+1}-\tau_{2m})=0, \]

\[ -\chi+\pi^*(\tau_{2m+2})-\pi^*(\tau_{2m+1})-a_1(\tau_{2m+2}-\tau_{2m+1})=0, \]

\[ \cdots \]

or

\[ \lim_{\tau\to\tau_1-0}\rho^*(\tau)=\chi+\pi^*(\tau_1), \]

\[ \lim_{\tau\to\tau_2-0}\rho^*(\tau)=\pi^*(\tau_2), \]

\[ \cdots \]

\[ \lim_{\tau\to\tau_{2m+1}}\rho^*(\tau)=\chi+\pi^*(\tau_{2m+1}), \]

\[ \lim_{\tau\to\tau_{2m+2}}\rho^*(\tau)=\pi^*(\tau_{2m+2}), \]

\[ \cdots \]

The function \(\rho^*(\tau)\) is thus continuous on \([\varkappa-1,\varkappa]\).

From inequalities (18) it follows that

\[ 0 \leq \pi^*(\tau)-\pi^*(\tau_{2m})+\chi-a_2(\tau-\tau_{2m}) \leq \chi, \]

\[ 0 \geq \pi^*(\tau)-\pi^*(\tau_{2m+1})-\chi-a_1(\tau-\tau_{2m+1}) \geq -\chi \]

respectively for

\[ \tau \in [\tau_{2m},\tau_{2m+1}] \quad \text{and} \quad \tau \in [\tau_{2m+1},\tau_{2m+2}]. \]

Consequently,

\[ 0 \leq \pi^*(\tau)+\chi-\rho^*(\tau) \leq \chi,\quad \tau \in [\tau_{2m},\tau_{2m+1}], \]

\[ 0 \geq \pi^*(\tau)-\rho^*(\tau) \geq -\chi,\quad \tau \in [\tau_{2m+1},\tau_{2m+2}], \]

or

\[ \pi^*(\tau)\leq \rho^*(\tau)\leq \pi^*(\tau)+\chi \tag{21} \]

for any \(\tau\in[\varkappa,\varkappa-1]\).

By the construction of \(\rho^*(\tau)\),

\[ \rho^*(\varkappa-1)=\pi^*(\varkappa-1). \tag{22} \]

If, moreover, \(\rho^*(\varkappa)=\pi^*(\varkappa)\), then we put

\[ \tilde{\rho}(\tau)\stackrel{\mathrm{def}}{=}\rho^*(\tau),\quad \tau\in[\varkappa-1,\varkappa]. \]

But if \(\rho^*(\varkappa)\ne\pi^*(\varkappa)\), then on the basis of (21)

\[ \pi^*(\varkappa)<\rho^*(\varkappa)\leq \pi^*(\varkappa)+\chi. \tag{23} \]

In the latter case consider the function

\[ \Psi(\tau)\stackrel{\mathrm{def}}{=}\pi^*(\varkappa)+a_1(\tau-\varkappa). \]

Let us estimate from below the difference \(\Psi(\tau)-\pi^*(\tau)\), \(\tau\in[\varkappa-1,\varkappa]\):

\[ \Psi(\tau)-\pi^*(\tau)=\pi^*(\varkappa)+a_1(\tau-\varkappa)-\pi^*(\tau)= \]

\[ =\bigl[\pi^*(\varkappa)-\pi^*(\tau)\bigr]+a_1(\tau-\varkappa). \]

On the basis of (11),

\[ a_1(\varkappa-\tau)+a_1(\tau-\varkappa)\leq \Psi(\tau)-\pi^*(\tau), \]

therefore

\[ \Psi(\tau)\geq \pi^*(\tau),\quad \tau\in[\varkappa-1,\varkappa]. \tag{24} \]

In particular,

\[ \Psi(\varkappa-1)\geq \pi^*(\varkappa-1). \tag{25} \]

If \(\Psi(\varkappa-1)=\pi^*(\varkappa-1)\), then we put \(\tau^*\stackrel{\mathrm{def}}{=}\varkappa-1\).

Suppose that \(\Psi(\varkappa-1)\ne\pi^*(\varkappa-1)\); then from (25) it follows that

\[ \Psi(\varkappa-1)>\pi^*(\varkappa-1). \tag{26} \]

The difference
\[ \Psi^*(\tau)\stackrel{\mathrm{def}}{=}\rho^*(\tau)-\Psi(\tau) \]
is a continuous function. On the basis of (22) and (26),

\[ \Psi^*(\varkappa-1)=\rho^*(\varkappa-1)-\Psi(\varkappa-1) =\pi^*(\varkappa-1)-\Psi(\varkappa-1)<0. \]

On the basis of the definition of \(\Psi\) and (21),

\[ \Psi^*(\varkappa)=\rho^*(\varkappa)-\Psi(\varkappa) =\rho^*(\varkappa)-\pi^*(\varkappa)>0. \]

Consequently, \(\Psi^*(\tau)\) must vanish between \(x-1\) and \(x\). Denote by \(\tau^*\) the last of the zeros of the function \(\Psi^*(\tau)\) on the interval \([x-1,x]\). By the definition of \(\tau^*\), for all \(\tau \in [\tau^*,x]\) we have

\[ \Psi^*(\tau^*)=0\leq \Psi^*(\tau), \]

therefore

\[ \rho^*(\tau)\geq \Psi(\tau),\quad \tau\in[\tau^*,x]. \tag{27} \]

Moreover, on the basis of (24),

\[ \Psi(\tau)\geq \pi^*(\tau),\quad \tau\in[\tau^*,x]. \tag{28} \]

Comparing (27), (28), and (21), we obtain

\[ \pi^*(\tau)\leq \Psi(\tau)\leq \pi^*(\tau)+\chi \tag{29} \]

for \(\tau\in[\tau^*,x]\). Finally, set

\[ \widetilde{\rho}(\tau)\stackrel{\mathrm{def}}{=} \begin{cases} \rho^*(\tau), & \tau\in[x-1,\tau^*),\\ \Psi(\tau), & \tau\in[\tau^*,x]. \end{cases} \]

The function \(\widetilde{\rho}(\tau)\), defined on the interval \([x-1,x]\), is piecewise linear. The angular coefficients of the segments of the graph of this function have the values \(\alpha_1\) or \(\alpha_2\). Throughout the interval \([x-1,x]\), on the basis of (21) and (29), the function \(\widetilde{\rho}\) satisfies the inequality

\[ \pi^*(\tau)\leq \widetilde{\rho}(\tau)\leq \pi^*(\tau)+\chi. \tag{30} \]

At the ends of the interval \([x-1,x]\), the constructed function coincides with \(\pi^*\):

\[ \widetilde{\rho}(x-1)=\pi^*(x-1),\quad \widetilde{\rho}(x)=\pi^*(x). \tag{31} \]

Construct, in the indicated manner, the function \(\widetilde{\rho}\) on each of the intervals \([x-1,x]\). On the basis of (31) we obtain a function \(\widetilde{\rho}(\tau)\), defined and continuous for all \(\tau\geq 0\). The function \(\widetilde{\rho}\) is piecewise linear and, consequently, has a derivative almost everywhere. The derivative of \(\widetilde{\rho}\) takes the values \(\alpha_1\) or \(\alpha_2\). On the basis of (30) and the definition of \(\chi\),

\[ 0\leq \widetilde{\rho}(\tau)-\pi^*(\tau)\leq \delta(\tau),\quad \tau\geq 0. \tag{32} \]

Put

\[ \rho_{ij}(\tau)\stackrel{\mathrm{def}}{=}\frac{d\rho(\tau)}{d\tau}. \]

Then, on the basis of (32) and the definition of \(\pi^*\),

\[ 0\leq \int_0^\tau [\rho_{ij}(\sigma)-\pi_{ij}(\sigma)]\,d\sigma\leq \delta(\tau). \tag{33} \]

Without loss of generality,

\[ \delta(\tau)\downarrow 0\quad \text{as } \tau\to+\infty, \tag{34} \]

since otherwise, in the preceding constructions, instead of the function \(\delta(\tau)\) one could use

\[ \delta^*(\tau)\stackrel{\mathrm{def}}{=} \begin{cases} \min\limits_{0\leq \sigma\leq \tau}\delta(\sigma), & \tau\in[0,1],\\[6pt] \dfrac{1}{\tau}\min\limits_{0\leq \sigma\leq \tau}\delta(\sigma), & \tau\geq 1. \end{cases} \]

From (33) and (34) it follows that the integral

\[ \int_0^\infty |\rho_{ij}(\sigma)-\pi_{ij}(\sigma)|\,d\sigma \]

exists and is equal to zero; therefore

\[ 0 \leq \int_\tau^\infty [\pi_{ij}(\sigma)-\rho_{ij}(\sigma)]\,d\sigma \leq \delta(\tau) \]

and, a fortiori,

\[ \left|\int_\tau^\infty [\pi_{ij}(\sigma)-\rho_{ij}(\sigma)]\,d\sigma\right|\leq \delta(\tau). \tag{35} \]

Denote by \(Q(\tau)\) the matrix with elements \(\rho_{ij}(\tau)\). The matrix \(Q(\tau)\) is piecewise constant. The elements of \(Q(\tau)\) take only the values \(\alpha_1\) or \(\alpha_2\). From the definition of \(\delta(\tau)\) and (35) it follows that

\[ \left\|\int_\tau^\infty [Q(\sigma)-P(\sigma)]\,d\sigma\right\|\leq \varphi(\tau) \]

for all \(\tau\geq 0\). Thus the required matrix has been constructed. The lemma is proved.

§ 5. MAIN RESULT

Theorem. Among the systems

\[ dy/d\tau=Qy,\quad Q\in \mathbf{P}, \]

asymptotically equivalent to the given system

\[ dx/d\tau=Px,\quad P\in \mathbf{P}, \]

there is always a system with a piecewise constant coefficient matrix whose elements take only two values.

The proof follows from the sufficient condition for asymptotic stability in § 3 and the lemma of § 4, in which one should take, for example, as \(\varphi(\tau)\),

\[ \varphi(\tau)=\frac{1}{2}e^{-\beta_0\tau}. \]

Remark. It was proved above that as the set of values of the elements of \(Q\) one may take the set \(\{\alpha_1,\alpha_2\}\), where \(\alpha_1\) and \(\alpha_2\) are defined in (1). However, all the constructions can also be carried out in the case when, as the set of values of the elements of \(Q\), an arbitrary set \(\{\beta_1,\beta_2\}\) is prescribed, where \(\beta_1\leq \alpha_1\), \(\beta_2\geq \alpha_2\).

References

1. Bogdanov Yu. S. Proceedings of the Fourth All-Union Mathematical Congress, Leningrad, July 3–12, 1961, vol. 2, Sectional Reports. Nauka Publishing House, Leningrad, 1964, pp. 424–432; see also the abstract in V. V. Nemytskii, RZhMat 1B162, 1965.
2. Lozinskii S. M. Izv. vuzov, matematika, No. 5 (6), 1958, pp. 52–90.
3. Nemytskii V. V. and Stepanov V. V. Qualitative Theory of Differential Equations. 2nd ed., GITTL, Moscow–Leningrad, 1949, p. 550.
4. Lyapunov A. M. Collected Works, vol. 2. Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1956, p. 473.

Received by the editors
February 13, 1965

Belorussian State University
named after V. I. Lenin.