Abstract
Full Text
B. P. DEMIDOVICH
ON BOUNDED SOLUTIONS OF A CERTAIN QUASILINEAR SYSTEM
(Presented by Academician A. N. Kolmogorov on 15 II 1961)
1°. In this paper we generalize the theorem of Farnell, Langhenhop, and Levinson (¹) on the existence and stability of forced harmonic oscillations of weakly nonlinear systems of differential equations (see Theorems 1, 2, and 3).
2°. Consider the linear system
\[ \frac{dx}{dt}=P(t)x, \tag{1} \]
where \(x\) is an \(n\)-dimensional vector (an \(n\times 1\) matrix) and \(P(t)\in C(-\infty,+\infty)\) is an \(n\times n\) matrix.
Definition. We shall say that system (1) (or the matrix \(P(t)\)) has the property \((\alpha,\beta)\) if its Cauchy matrix \(X(t,t_0)=X(t)X^{-1}(t_0)\) decomposes into two subsystems:
\[ X_\alpha(t,t_0)=X(t)AX^{-1}(t_0),\qquad X_\beta(t,t_0)=X(t)BX^{-1}(t_0) \]
(\(A\) and \(B\) are constant \(n\times n\) matrices, with \(A+B=I\)) such that
\[ \begin{aligned} \|X_\alpha(t,t_0)\| &\leq ae^{-\alpha(t-t_0)} &&\text{for } t\geq t_0; \tag{2'}\\ \|X_\beta(t,t_0)\| &\leq be^{\beta(t-t_0)} &&\text{for } t\leq t_0. \tag{2''} \end{aligned} \]
Here \(a,b,\alpha,\beta\) are positive constants.
Conditions \((2')\) and \((2'')\) are a generalization of the known conditions of K. P. Persidskii (²) and are satisfied, for example, for a system reducible to a constant matrix whose characteristic numbers have nonzero real parts.
Lemma 1. Suppose that the matrix \(P(t)\) has the property \((\alpha,\beta)\). Then there exists an \(n\times n\) matrix \(G(t,t_1)\in C^1\) for \(t\ne t_1\) \((-\infty<t<+\infty,\ -\infty<t_1<+\infty)\) such that:
1) \(G(t,t-0)-G(t,t+0)=I\);
2) \(G'_t(t,t_1)=P(t)G(t,t_1)\) for \(t\ne t_1\),
\[ G'_{t_1}(t,t_1)=-G(t,t_1)P(t_1)\quad \text{for } t\ne t_1; \]
3) \(\|G(t,t_1)\|\leq ce^{-\gamma|t-t_1|}\) (\(c\) and \(\gamma\) are positive constants);
4) any nonhomogeneous system
\[ \frac{dy}{dt}=P(t)y+f(t), \tag{3} \]
where \(f(t)\in C(-\infty,+\infty)\), has a solution bounded on \((-\infty,+\infty)\),
\[ \hat y(t)=\int_{-\infty}^{+\infty} G(t,t_1)f(t_1)\,dt_1 . \tag{4} \]
under the condition that the integral (4) converges uniformly in \(t\) on every finite interval and is bounded on \((-\infty,+\infty)\).
For the proof we set
\[ G(t,t_1)= \begin{cases} X_\alpha(t,t_1), & \text{for } t>t_1,\\ -X_\beta(t,t_1), & \text{for } t<t_1. \end{cases} \]
Lemma 2. If \(P(t)\) and \(f(t)\) are almost periodic and system (1) has no nontrivial bounded solutions, then the solution \(\hat y(t)\) (4) is also almost periodic.
The lemma is a modified Favard theorem \((^3)\).
\(3^\circ\). Consider the quasilinear system
\[ \frac{dy}{dt}=P(t)y+f(\omega t,y)+e(\omega t), \tag{5} \]
where \(P(t)\in C(-\infty,+\infty)\), is bounded\(^*\) and has the property \((\alpha,\beta)\); \(\omega\) is a large parameter \((\omega\ge \omega_0>0)\); \(f(t,y)\in C(|t|<+\infty,\|y\|<+\infty)\), satisfies a Lipschitz condition in \(y\) with constant \(L\), and
\[ f(t,0)\equiv 0, \]
and \(e(t)\) is a continuous function with bounded integral
\[ E(t)=\int_0^t e(t_1)\,dt_1. \]
Theorem 1. If the constant \(L\) is sufficiently small, then system (5) has at least one solution \(y=\hat y(t)\), bounded on \((-\infty,+\infty)\), such that
\[ \|\hat y(t)\|<\frac{\Gamma}{\omega}\sup_t \|E(t)\|, \]
where \(\Gamma\) is a constant depending only on the matrix \(P(t)\).
This result is analogous to Perron’s theorem \((^4)\), but is not its consequence, since \(e(t)\) may be unbounded.
The proof is carried out by considering the integral equation
\[ y(t)=\int_{-\infty}^{+\infty} G(t,t_1)\,[f(\omega t_1,y(t_1))+e(\omega t_1)]\,dt_1, \]
which is solved by the method of successive approximations.
Theorem 2. If the matrix \(P(t)\) is almost periodic (periodic with period \(T/\omega\)), and also \(f(t,y)\), \(e(t)\), and \(E(t)\) are almost periodic in \(t\) (periodic with period \(T\)), and the homogeneous system (1) with matrix \(P(t)\) has no bounded nontrivial solutions, then the bounded solution \(\hat y(t)\) of system (5) is almost periodic (periodic with period \(T/\omega\)).
For the case of a constant matrix \(P(t)=\mathrm{const}\), the existence of an almost periodic solution for quasilinear systems analogous to (5) was established by G. I. Biryuk \((^5)\), by the author \((^6)\), and by Langenhop \((^7)\).
\(^*\) One may allow a weak unboundedness of the matrix \(P(t)\), namely, it suffices to assume that \(P(t)=P_0(t)+P_1(t)\), where \(P_0(t)\) is a bounded matrix and \(P_1(t)\) is absolutely integrable on \((-\infty,+\infty)\).
Theorem 3. If
\[ \operatorname{rank} X_a(t_0,t_0)=n, \]
where \(n\) is the order of system (1), then the bounded solution \(\hat y(t)\) is exponentially stable in the sense of Lyapunov as \(t\to +\infty\).
I express my gratitude to V. V. Nemytskii, in whose seminar the preliminary results of this work were presented and discussed.
Moscow State University
named after M. V. Lomonosov
Received
4 I 1961
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