UDC 517.949.22

On a Question of Rough Systems with Aftereffect

N. G. BULGAKOV

In this paper, N. N. Krasovskii’s proposition, proved for systems of ordinary differential equations, on the decrease according to an exponential law of the norm of every solution, provided that the simplified system has this property, is generalized to systems with aftereffect [1].

Consider the system of equations with aftereffect

\[ \frac{dx_i}{dt} = F_i\bigl(x_1(t+\theta),\ldots,x_n(t+\theta),t\bigr) + \]

\[ + R_i\bigl(x_1(t+\theta),\ldots,x_n(t+\theta),t\bigr) \tag{1} \]

\[ (i=1,\ldots,n), \]

where \(dx_i/dt\) denotes the right derivatives; \(F_i\) and \(R_i\) are functionals defined on piecewise-continuous functions \(x_i(\theta)\) of the argument \(\theta\), which varies within the limits \(-h\leqslant \theta\leqslant 0\) \((h=\mathrm{const}>0)\).

With respect to the functionals \(F_i\) and \(R_i\) we shall assume the following:

1) the functionals \(F_i\) and \(R_i\) are piecewise-continuous in \(t\) in the domain

\[ \|x(\theta)\|^h < H,\qquad t\geqslant 0,\qquad H=\mathrm{const}>0. \tag{2} \]

Here and below

\[ \|x(\theta)\|^h=\sup\bigl(|x_1(\theta)|,\ldots,|x_n(\theta)|\bigr),\qquad -h\leqslant \theta\leqslant 0. \]

By the symbol \(\|x\|\), without the index \(h\), we shall denote the norm of the numerical vector \(\{x_i\}\):

\[ \|x\|=\sup\bigl(|x_1|,\ldots,|x_n|\bigr); \]

2) the functionals \(F_i\) in the domain (2) satisfy the Lipschitz conditions with respect to \(x(\theta)\), i.e.,

\[ \bigl|F_i(x''(\theta),t)-F_i(x'(\theta),t)\bigr| < L\|x''(\theta)-x'(\theta)\|^h \tag{3} \]

\[ (L=\mathrm{const}>0,\quad i=1,\ldots,n); \]

3) \(F_i(0,t)\equiv 0\);

\[ \tag{4} \]

4) the functionals \(R_i\) in the domain (2) are continuous with respect to \(x(\theta)\) and uniformly bounded in \(t\):

\[ |R_i(x(\theta),t)|<\gamma\|x(\theta)\|^h,\qquad \gamma=\mathrm{const}>0. \tag{5} \]

From conditions (3) and (4) it follows that the functionals \(F_i\) in the domain (2) also satisfy the conditions

\[ |F_i(x(\theta),\, t)| < L\|x(\theta)\|^h . \tag{6} \]

We shall understand a solution of system (1) in the sense given by N. N. Krasovskii [1].

Under the assumptions made concerning the functionals \(F_i\) and \(R_i\), for each piecewise-continuous curve \(x_0(\theta_0)\) and for \(t \geq t_0\) there exists ([1], p. 152) at least one solution of system (1); moreover this solution, as in the case of ordinary equations, is continuable to all those values \(t \geq t_0\) for which the continuous curve \(x(t,x_0(\theta_0),t_0)\) still remains in the domain (2).

Along with the complete system (1), we shall consider the system of first approximation

\[ \frac{dy_i}{dt}=F_i(y_i(t+\theta),\,\ldots,\,y_n(t+\theta),\,t) \quad (i=1,\ldots,n). \tag{7} \]

We shall say that the solutions \(y(t,y_0(\theta_0),t_0)\) of the system of first approximation (7) satisfy condition \((\mathrm{A})\) if, for any number \(0<q<1\), there is a number \(T(q)>0\) such that, for all \(y_0(\theta_0)\) and \(t_0\) from the domain (2), the inequality

\[ \|y(t_0+T,\,y_0(\theta_0),\,t_0)\| \leq \frac{1}{2}q\|y_0(\theta_0)\|^h \tag{8} \]

holds.

The following is proved below.

Theorem. If the solutions of the system of first approximation (7) satisfy condition \((\mathrm{A})\) in the domain (2), then the solutions of the complete system (1), in some domain lying inside the domain (2), satisfy condition \((\mathrm{B})\), i.e.

\[ \|x(t,x_0(\theta_0),t_0)\| \leq a\|x_0(\theta_0)\|^h \exp[-\beta(t-t_0)], \quad t\geq t_0 \tag{9} \]

for any choice of the functionals \(R_i\), provided only that the constant \(\gamma\) in inequalities (5) is sufficiently small. Here \(a\) and \(\beta\) are positive constants independent of \(x_0(\theta_0)\) and \(t_0\).

We shall preface the proof of the theorem with two lemmas.

Lemma 1. For any trajectory \(x(t,x_0(\theta_0),t_0)\) of the complete system (1), issuing from an arbitrary point \((x_0(\theta_0),t_0)\) of the domain (2), for all those values of time \(t\geq t_0\) for which this trajectory remains entirely in the domain (2), the estimate

\[ \|x(t,x_0(\theta_0),t_0)\| \leq \|x_0(\theta_0)\|^h \exp[(\gamma+L)(t-t_0)], \quad t\geq t_0. \tag{10} \]

is valid.

The proof of this lemma for the case where the functionals \(R_i\) satisfy a Lipschitz condition is given in [1], p. 153. It is clear that the lemma remains valid also under the assumption made above on the boundedness of the functionals.

Lemma 2. For any trajectories of systems (1) and (7), issuing from the same point \((x_0(\theta_0),t_0)\) of the domain (2), and for any finite interval of time \(t_0\leq t\leq t_0+\tau\) on which these trajectories remain entirely in the domain (2), the estimate

\[ \|x(t,x_0(\theta_0),t_0)-y(t,x_0(\theta_0),t_0)\| \leq \]

\[ \leq \gamma\tau\|x_0(\theta_0)\|^h \exp[(\gamma+2L)\tau]. \tag{11} \]

Proof. We first show that for \(t_0\leq t\leq t_0+\tau\) the inequality

\[ \|z(t)\|\leq \gamma\|x_0(\theta_0)\|^h(t-t_0)\times \]

\[ \times \exp [(\gamma+L)\tau]+L\int_{t_0}^{t}\|z(t)\|\,dt . \tag{12} \]

Here

\[ z(t)=\{x_i(t,x_0(\theta_0),t_0)-y_i(t,x_0(\theta_0),t_0)\}\quad (i=1,\ldots,n). \]

The proof of inequality (12) is constructively no different from the proof of Lemma 27.1 ([1], p. 153). Indeed, inequality (12) is obviously satisfied for \(t=t_0\). Suppose now the contrary, i.e., that inequality (12) is violated. Let \(t=t_1\) \((t_0\leq t_1\leq t_0+\tau)\) be the point in time bounding from above the interval \([t_0,t_1]\) on which inequality (12) is still valid, but for \(t>t_1\) it is already violated, i.e., for any \(\varepsilon>0\) one can specify a number \(t_\varepsilon\) \((t_1<t_\varepsilon<t_1+\varepsilon)\) such that

\[ \|z(t_\varepsilon)\|>\gamma\|x_0(\theta_0)\|^h(t_\varepsilon-t_0)\exp[(\gamma+L)\tau]+L\int_{t_0}^{t_\varepsilon}\|z(t)\|\,dt . \tag{13} \]

Then, obviously, we shall have

\[ \|z(t_1+\theta)\|^h=\|z(t_1)\|, \tag{14} \]

\[ \limsup_{\Delta t\to +0} \frac{\|z(t_1+\Delta t)\|-\|z(t_1)\|}{\Delta t} \geq \gamma\|x_0(\theta_0)\|^h\exp[(\gamma+L)\tau]+L\|z(t_1)\|. \tag{15} \]

On the other hand, from equations (1) and (7), taking into account conditions (3) and (5), we have

\[ \sup_{(i=1,\ldots,n)} \left|\frac{dz_i}{dt}\right|_{t=t_1} < \gamma\|x(t_1+\theta)\|^h+L\|z(t_1+\theta)\|^h . \tag{16} \]

Moreover, on the basis of Lemma 1,

\[ \|x(t_1+\theta)\|^h\leq \|x_0(\theta_0)\|^h\exp[(\gamma+L)\tau]. \tag{17} \]

Taking into account relations (14) and (17), inequality (16) can be written as

\[ \sup_{(i=1,\ldots,n)} \left|\frac{dz_i}{dt}\right|_{t=t_1} < \gamma\|x_0(\theta_0)\|^h\exp[(\gamma+L)\tau]+L\|z(t_1)\|. \]

It follows that for any number \(\varepsilon>0\) one can specify a number \(\delta>0\) such that

\[ |z_i(t_1+\Delta t)-z_i(t_1)| < \{\gamma\|x_0(\theta_0)\|^h \]

\[ \times \exp[(\gamma+L)\tau]+L\|z(t_1)\|+\varepsilon\}\Delta t \]

for all \(i=1,\ldots,n\), \(0\leq \Delta t\leq \delta\), or

\[ \|z_i(t_1+\Delta t)\|-\|z_i(t_1)\| < \{\gamma\|x_0(\theta_0)\|^h\times \]

\[ \times \exp[(\gamma+L)\tau]+L\|z(t_1)\|+\varepsilon\}\Delta t . \tag{18} \]

Since, as \(\Delta t\to +0\), \(\delta\to 0\), i.e., \(\varepsilon\to 0\), it follows from inequalities (18) that

\[ \limsup_{\Delta t\to +0} \frac{\|z(t_1+\Delta t)\|-\|z(t_1)\|}{\Delta t} < \gamma\|x_0(\theta_0)\|^h\exp[(\gamma+L)\tau]+L\|z(t_1)\|, \]

which contradicts inequality (15). The contradiction proves the inequality

(12), which for \(t_0 \leq t \leq t_0+\tau\) can be written as follows:

\[ \|z(t)\|\leq \gamma \tau \|x_0(\theta_0)\|^h \exp[(\gamma+L)\tau]+L\int_{t_0}^{t}\|z(t)\|\,dt. \]

Hence, on the basis of Lemma 1 [2], we arrive at (11). This is what was required to prove.

We proceed to the proof of the theorem. We shall carry out the proof by the method used in [3].

Consider any two trajectories \(x(t,x_0(\theta_0),t_0)\) and \(y(t,x_0(\theta_0),t_0)\), respectively, of equations (1) and (7), issuing from the same point and the same instant \((x_0(\theta_0),t_0)\) of the domain (2). On the basis of Lemma 1, for all values of the time \(t\geq t_0\) for which these trajectories remain entirely in the domain (2), we have

\[ \begin{gathered} \|x(t,x_0(\theta_0),t_0)\|\leq \|x_0(\theta_0)\|^h \exp[(\gamma+L)(t-t_0)],\\ \|y(t,x_0(\theta_0),t_0)\|\leq \|x_0(\theta_0)\|^h \exp[L(t-t_0)]. \end{gathered} \tag{19} \]

It follows from these estimates that, for \(t_0\leq t\leq t_0+T\), for any points \((x_0(\theta_0),t_0)\) lying in the domain

\[ \|x(\theta)\|^h < H/\omega,\quad t\geq 0, \tag{20} \]

where \(\omega=\exp[(\gamma+L)T]>1\), the trajectories under consideration will not leave the domain (2), and consequently, on the basis of Lemma 2, we shall have

\[ \|x(t,x_0(\theta_0),t_0)-y(t,x_0(\theta_0),t_0)\|< \]

\[ <\gamma T\|x_0(\theta_0)\|^h \exp[(\gamma+2L)T]. \tag{21} \]

For a trajectory of system (1) issuing from an arbitrary point of the domain (20), moreover, on the basis of (19) we have

\[ \|x(t,x_0(\theta_0),t_0)\|\leq \omega \|x_0(\theta_0)\|^h,\quad t_0\leq t\leq t_0+T. \tag{22} \]

From inequality (21), for sufficiently small \(\gamma\), for any points \((x_0(\theta_0),t_0)\) from the domain (20), we have

\[ \|x(t,x_0(\theta_0),t_0)-y(t,x_0(\theta_0),t_0)\|\leq \]

\[ \leq \frac{1}{2}\,q\|x_0(\theta_0)\|^h,\quad t_0\leq t\leq t_0+T. \tag{23} \]

For this it is sufficient that \(\gamma\) satisfy the inequality

\[ \gamma T\exp[(\gamma+2L)T]\leq \frac{1}{2}\,q. \tag{24} \]

Adding inequalities (8) and (23), and taking into account that \(x_0(\theta_0)=y_0(\theta_0)\), for any \(x_0(\theta_0)\) and \(t_0\) from the domain (20) we obtain

\[ \|x(t_0+T,x_0(\theta_0),t_0)\|\leq q\|x_0(\theta_0)\|^h. \tag{25} \]

Applying estimates (22) and (25) to the interval \(t_0+T\leq t\leq t_0+2T\), we obtain

\[ \|x(t,x_0(\theta_0),t_0)\|\leq \omega \|x(t_0+T,x_0(\theta_0),t_0)\|\leq \omega q\|x_0(\theta_0)\|^h, \]

\[ \|x(t_0+2T,x_0(\theta_0),t_0)\|\leq q\|x(t_0+T,x_0(\theta_0),t_0)\|\leq q^2\|x_0(\theta_0)\|^h. \]

Continuing this process further, one can verify that

\[ \|x(t, x_0(\theta_0), t_0)\| \leq \omega \|x_0(\theta_0)\|^h q^{\,n-1}, \qquad t_0+(n-1)T \leq t \leq t_0+nT, \]

\[ \|x(t_0+nT, x_0(\theta_0), t_0)\| \leq q^n \|x_0(\theta_0)\|^h . \]

Hence the inequality follows

\[ \|x(t, x_0(\theta_0), t_0)\| \leq \frac{\omega}{q}\,\|x_0(\theta_0)\|^h q^{\frac{t-t_0}{T}}, \qquad t \geq t_0 . \]

Now put

\[ q=\exp(-\beta T), \qquad \frac{\omega}{q}=\alpha . \]

The quantities \(\alpha\) and \(\beta\) obviously do not depend on \(x_0(\theta_0)\) and \(t_0\). Then, finally, for any \(x_0(\theta_0)\) and \(t_0\) from the domain (20), we shall have

\[ \|x(t, x_0(\theta_0), t_0)\| \leq \alpha \|x_0(\theta_0)\|^h \exp[-\beta(t-t_0)], \qquad t \geq t_0 . \]

This is what was required to prove.

Let us indicate some consequences of the theorem.

Corollary 1. Condition (A) is equivalent to condition (B).

Corollary 2. If the solutions of the system of first approximation (7) in the domain (2) satisfy condition (B), then the solutions of the complete system (1) in some domain lying inside the domain (2) also satisfy condition (B) with changed constants for any choice of the functionals \(R_i\), provided only that the constant \(\gamma\) in the inequalities (5) is sufficiently small. In the absence of delay, this assertion was first proved by N. N. Krasovskii [1] by the method of Lyapunov functions.

References

1. N. N. Krasovskii, Some Problems in the Theory of Stability of Motion. Fizmatgiz, 1959.

2. V. V. Nemitskii, V. V. Stepanov, Qualitative Theory of Differential Equations. GITL, Moscow—Leningrad, 1949, p. 19.

3. E. A. Barbashin, M. A. Skalkina, PMM, 19, no. 5, 623—624, 1955.

Received by the editors
January 7, 1966

Belorussian State University
named after V. I. Lenin