Full Text
V. A. Pliss
ON THE STABILITY OF MOTION IN THE DOUBTFUL CASE OF TWO ZERO ROOTS
(Presented by Academician V. I. Smirnov on 26 VII 1963)
In paper \((^1)\), Lyapunov studied the question of stability in the doubtful case of two zero roots with a nonsimple elementary divisor, when the number of equations of the system is greater than two. In this connection, only in one case was Lyapunov’s analysis not exhaustive. In this case the problem reduces to the study of the following system of differential equations:
\[ \begin{aligned} \frac{dr}{dt} &= r^{q+1} R_1(r,\vartheta) + r R_2(r,z,\vartheta),\\ \frac{d\vartheta}{dt} &= r^{q-1} + r^q \Theta_1(r,\vartheta) + \Theta_2(r,z,\vartheta), \tag{1}\\ \frac{dz}{dt} &= Az + Z(r,z,\vartheta), \end{aligned} \]
where \(q\) is a natural number greater than one; \(r\) and \(\vartheta\) are scalar variables; \(z\) is an \(n\)-dimensional vector with components \(z_1,\ldots,z_n\); \(A=\{a_{ij}\}\) \((i,j=1,\ldots,n)\) is a constant square matrix with negative real parts of the characteristic roots. The functions \(R_1\) and \(\Theta_1\) are power series in \(r\) with coefficients that are \(\omega\)-periodic with respect to \(\vartheta\), converging absolutely and uniformly for sufficiently small \(r\) and all \(\vartheta\). The functions \(R_2\) and \(\Theta_2\) are analogous series in powers of \(r,z_1,\ldots,z_n\), with \(R_2\) and \(\Theta_2\) vanishing when \(z=0\). The components \(Z_s\) of the vector-function \(Z\) are also series in powers of \(r,z_1,\ldots,z_n\) with coefficients \(\omega\)-periodic in \(\vartheta\); these series contain no terms of dimension lower than the second in \(r,z_s\), and \(r\) enters terms independent of \(z_s\) only in powers not lower than \(2q\). In this case one may assume that \(r \ge 0\).
To study system (1), Lyapunov introduces new variables by the following formulas:
\[ r=c(1+\rho), \qquad z=c^\beta \zeta, \tag{2} \]
where \(c\) is a sufficiently small positive constant, and \(\beta\) is a natural number satisfying the inequalities
\[ q \le \beta < 2q-1. \tag{3} \]
Substituting (2) into system (1) and eliminating \(t\), we obtain
\[ \begin{aligned} \frac{d\rho}{d\vartheta} &= cP(c,\rho,\zeta,\vartheta),\\ c^{q-1}\frac{d\zeta}{d\vartheta} &= A\zeta + cF(c,\rho,\zeta,\vartheta), \end{aligned} \tag{4} \]
where \(P\) is a scalar and \(F\) a vector function, expandable in series in powers of \(c,\rho,\zeta_s\) with coefficients \(\omega\)-periodic in \(\vartheta\); these series converge absolutely and uniformly for all \(\vartheta\) and sufficiently small \(c,|\rho|\), and \(\|\zeta\|\) (where \(\|\zeta\|\) denotes the Euclidean norm of the vector \(\zeta\)).
Along with system (4), Lyapunov considered the system
\[ \begin{gathered} \frac{d\rho}{d\vartheta}=CP(c,\rho,\xi,\vartheta)+\varphi(c),\\ c^{q-1}\frac{d\xi}{d\vartheta}=A\xi+cF(c,\rho,\xi,\vartheta), \end{gathered} \tag{5} \]
where the function \(\varphi(c)\), defined for sufficiently small nonnegative \(c\), is chosen in such a way that system (5), for all sufficiently small \(c\geqslant 0\), has an \(\omega\)-periodic solution \(\rho=\rho_0(c,\vartheta)\), \(\xi=\xi_0(c,\vartheta)\) with the following properties:
\[ \rho_0(c,0)=0,\qquad |\rho_0(c,\vartheta)|<Dc,\qquad \|\xi_0(c,\vartheta)\|<Dc^{2q-\beta}, \tag{6} \]
where \(D\) is a certain positive constant.
Lyapunov proved the existence of the function \(\varphi(c)\) and investigated its properties in detail; in particular, he established that the function \(\varphi(c)\) is continuously differentiable.
From the form of systems (4) and (5) it follows that for those \(c\) for which \(\varphi(c)=0\), system (4) has an \(\omega\)-periodic solution \(\rho=\rho_0(c,\vartheta)\), \(\xi=\xi_0(c,\vartheta)\) with properties (6).
In paper [1], the only case left unresolved was when the function \(\varphi(c)\) can take both zero and negative values in any neighborhood of the point \(c=0\). However, one can prove a statement from which it follows that in this case stability holds. Namely, the following theorem is valid.
Theorem. If there exists a set \(C\) of positive numbers, having zero as its limit point, such that \(\varphi(c)=0\) for \(c\in C\), then the zero solution of system (1) is stable with respect to the variables \(r\) and \(z\).
The proof of this theorem is essentially based on the following considerations. Choose a sufficiently small \(c\in C\); to it there corresponds the periodic solution \(\rho=\rho_0(c,\vartheta)\), \(\xi=\xi_0(c,\vartheta)\) of system (4).
In system (4), make the change of variables:
\[ \rho=\rho_0(c,\vartheta)+\alpha,\qquad \xi=\xi_0(c,\vartheta)+\psi . \tag{7} \]
The equations for \(\alpha\) and \(\psi\) have the form
\[ \frac{d\alpha}{d\vartheta}=cQ(c,\alpha,\psi,\vartheta),\qquad c^{q-1}\frac{d\psi}{d\vartheta}=A\psi+cG(c,\alpha,\psi,\vartheta), \tag{8} \]
where the functions \(Q\) and \(G\) satisfy the relations
\[ Q(c,0,0,\vartheta)\equiv 0,\qquad G(c,0,0,\vartheta)\equiv 0; \tag{9} \]
\[ \begin{gathered} |Q(c,\alpha_1,\psi_1,\vartheta)-Q(c,\alpha_2,\psi_2,\vartheta)| <L\bigl(|\alpha_1-\alpha_2|+\|\psi_1-\psi_2\|\bigr),\\ \|G(c,\alpha_1,\psi_1,\vartheta)-G(c,\alpha_2,\psi_2,\vartheta)\| <L\bigl(|\alpha_1-\alpha_2|+\|\psi_1-\psi_2\|\bigr). \end{gathered} \tag{10} \]
The Lipschitz conditions (10) are fulfilled for all sufficiently small \(c\in C\), \(|\alpha|\), \(\|\psi\|\), and for all \(\vartheta\); moreover, the constant \(L\) can be chosen the same for all sufficiently small \(c\in C\) (and, correspondingly, for all periodic solutions \(\rho=\rho_0(c,\vartheta)\), \(\xi=\xi_0(c,\vartheta)\) of system (4)).
Relying on the fact that all eigenvalues of the matrix \(A\) have negative real parts and that, in conditions (10), the Lipschitz constant \(L\) does not depend on \(c\), one can prove the following assertion.
Lemma. There exist constants \(\lambda>0\), \(c_0>0\), \(a_0>0\) and a scalar function \(f(c,a,\theta)\), where \(a\) is an \(n\)-dimensional vector, such that:
1) the function \(f\) is defined for any \(c\in C\), \(c\leqslant c_0\), \(\|a\|\leqslant a_0\), and for all \(\theta\);
2) \(f\) is continuous with respect to \(a\) and \(\theta\) for any fixed \(c\) in its domain of definition;
3) \[ f(c,a,\theta+\omega)=f(c,a,\theta); \]
4) \[ |f(c,a,\theta)|\leqslant \|a\|\,c^{q-1}; \]
5) for any solution of system (8) with initial data
\[ \vartheta=\theta,\qquad \psi=a,\qquad \alpha=f(c,a,\theta),\qquad \|a\|\leqslant a_0 \]
the inequalities
\[ \|\psi(\vartheta)\|<E\|a\|\exp\left[-\frac{\lambda}{c^{q-1}}(\vartheta-\theta)\right], \]
\[ |\alpha(\vartheta)|<\|a\|c^{q-1}\exp\left[-\frac{\lambda}{c^{q-1}}(\vartheta-\theta)\right] \]
hold for \(\vartheta\geqslant\theta\), where \(E\) is some positive number.
Let us indicate the method for constructing the function \(f\). Consider the system of integral equations:
\[ \alpha=-c\int_{\vartheta}^{+\infty} Q(c,\alpha,\psi,\tau)\,d\tau, \]
\[ \psi=\exp\left[\frac{1}{c^{q-1}}(\vartheta-\theta)\right]a+ \frac{1}{c^{q-2}}\int_{\theta}^{\vartheta} \exp\left[\frac{A}{c^{q-1}}(\vartheta-\tau)\right] G(c,\alpha,\psi,\tau)\,d\tau, \tag{11} \]
where \(a\) is an \(n\)-dimensional constant vector.
Using the fact that all eigenvalues of the matrix \(A\) have negative real parts and that the functions \(\theta\) and \(G\) satisfy conditions (10), one can show, by applying the method of successive approximations with initial approximation
\[ \alpha_0=0,\qquad \psi_0=\exp\left[\frac{A}{c^{q-1}}(\vartheta-\theta)\right]a, \]
that system (11) has a solution \(\alpha(c,a,\vartheta,\theta)\), \(\psi(c,a,\vartheta,\theta)\), \(\omega\)-periodic in \(\vartheta\) and \(\theta\), and one must set
\[ f(c,a,\theta)=\alpha(c,a,\theta,\theta). \]
The formulated lemma makes it possible to prove the following assertion. Let the solution \(r(t)\), \(\vartheta(t)\), \(z(t)\) of system (1) have initial data \(r=\bar r\), \(\vartheta=\bar\vartheta\), \(z=\bar z\) at \(t=\bar t\), and let \(\varepsilon>0\) be an arbitrary number; then, for all \(t\geqslant \bar t\), the relations
\[ r(t)<\varepsilon,\qquad \|z(t)\|<\varepsilon \]
hold, provided only that
\[ \|\bar z\|<\delta,\qquad \bar r\leqslant c\,[1+\rho_0(c,\bar\vartheta)+f(c,c^{-p}\bar z-\zeta_0(c,\bar\vartheta),\bar\vartheta)], \]
where \(c\in C\) and \(\delta\) are sufficiently small. This proves the assertion of the theorem.
Leningrad State University
named after A. A. Zhdanov
Received
22 VII 1963
REFERENCES
- A. M. Lyapunov, “Investigation of one of the special cases of the problem of stability of motion,” L., 1963.