Abstract Generated abstract
This paper develops an analogue of the Andronov-Witt stability theorem for autonomous ordinary differential equations with a bounded, not necessarily periodic, reference solution. It introduces standard and completely bounded solutions for associated linear systems, proves a representation lemma for the Cauchy matrix under Lyapunov regularity when one solution is completely bounded and the remaining characteristic exponents are negative, and applies this structure to the variational equation along the bounded trajectory. The main result gives sufficient conditions under which such a trajectory, with nonvanishing velocity and exactly one zero characteristic exponent while all others are negative, is asymptotically orbitally stable as time tends to infinity and possesses an asymptotic phase. The proof reduces the nonlinear system near the trajectory to a perturbed linear equation and constructs decaying nearby solutions through a singular integral equation solved by successive approximations.
Full Text
UDC 517.919
MATHEMATICS
B. P. DEMIDOVICH
ON AN ANALOGUE OF THE ANDRONOV–WITT THEOREM
(Presented by Academician A. N. Kolmogorov, 22 XII 1966)
1°. In this paper sufficient conditions are obtained for the asymptotic orbital stability of a completely bounded solution of an autonomous system of ordinary differential equations, analogous to the well-known Andronov–Witt theorem \((^1)\) on the stability of a periodic solution.
2°. Consider the real linear system
\[ dx/dt=A(t)x, \tag{1} \]
where
\[ \mathbf{x}= \begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix} \equiv \operatorname{colon}(x_1,\ldots,x_n), \]
\[ A(t)\in C(I_t^+) \quad (I_t^+=(t_0,+\infty)) \text{ is an } n\times n\text{-matrix};\quad \sup_t \|A(t)\|<\infty . \]
By
\[ \chi[\mathbf{x}(t)]=\lim_{t\to\infty}\left\{\frac{1}{t}\ln\|\mathbf{x}(t)\|\right\} \]
we shall denote the characteristic exponent of the solution \(\mathbf{x}(t)\).
Definition 1. A bounded solution \(\mathbf{x}(t)\) of system (1) will be called standard if
\[ 0<\inf_t\|\mathbf{x}(t)\|\leq \sup_t\|\mathbf{x}(t)\|<\infty . \]
Definition 2. A bounded solution \(\mathbf{x}(t)\) of system (1) will be called completely bounded if: 1) it is standard and 2) the adjoint system
\[ d\vec{\xi}/dt=A^T(t)\vec{\xi} \tag{2} \]
(\(A^T(t)\) is the transposed matrix of \(A(t)\)) has a bounded solution \(\vec{\xi}(t)\), with
\[ \mathbf{x}^T(t)\vec{\xi}(t)=\text{const}\ne 0. \]
Definition 3. A square matrix \(L(t)\) will be called a Lyapunov matrix on the set \(Z\subset I_t^+\) if, for \(t\in Z\), the following conditions are satisfied:
a) \(\quad L(t)\in C^{(1)};\)
b) \(\quad \sup_t\|L(t)\|<\infty,\quad \sup_t\|\dot L(t)\|<\infty;\)
c) \(\quad \inf_t|\det L(t)|>0.\)
Lemma. Let system (1) be regular in the sense of Lyapunov \((^2)\), and let its normal fundamental matrix \(X(t)=(x_{ij}(t))\) consist of solutions
\[ \mathbf{x}^{(j)}(t)=\operatorname{colon}[x_{1j}(t),\ldots,x_{nj}(t)] \quad (j=1,\ldots,n), \]
of which one, \(\mathbf{x}^{(1)}(t)\), is completely bounded, while all the remaining \(\mathbf{x}^{(j)}(t)\) \((j>1)\) have negative characteristic exponents.
Then the half-plane \(\{t_0 \le t < +\infty,\ t_0 \le \tau < +\infty\}\) can be represented as a finite sum of open sets \(O_{pq}=\{t,\tau: t\in O_p,\ \tau\in O_q\}\), on each of which the Cauchy matrix
\[ K(t,\tau)=X(t)X^{-1}(\tau) \]
admits the representation
\[ K(t,\tau)=U_p(t)\operatorname{diag}[E_1,Y_{pq}(t,\tau)]V_q^T(\tau), \tag{3} \]
where \(E_1\) is the identity matrix of order \(1\), \(U_p(t)\) and \(V_q(\tau)\) are Lyapunov matrices on \(O_p\) and \(O_q\), respectively, \(Y_{pq}(t,\tau)\in C^{(1)}(O_{pq})\), and
\[ \|Y_{pq}(t,\tau)\|\le ce^{-\alpha(t-\tau)}e^{\varepsilon t} \tag{4} \]
for \(t_0\le \tau\le t\), where \(\varepsilon>0\) is arbitrary,
\[ -\alpha=\max_{j>1}\chi[\mathbf{x}^{(j)}(t)]<0 \]
and \(c=c(\varepsilon,t_0)\) is a positive constant.
\(3^\circ\). Let the real autonomous system
\[ dy/dt=\mathbf{f}(\mathbf{y}), \tag{5} \]
where \(\mathbf{f}(\mathbf{y})\in C^{(2)}(R^n)\), \(n\ge2\), admit a solution \(\boldsymbol{\eta}=\boldsymbol{\eta}(t)\) bounded on \(I_t=(-\infty,+\infty)\).
Definition 4. The solution \(\boldsymbol{\eta}(t)\) is called orbitally stable as \(t\to\infty\) if, for some \(t_0\), for every \(\varepsilon>0\) there exists \(\delta>0\) such that, for every solution \(\mathbf{y}(t)\) \((t_0\le t<\infty)\) satisfying the condition
\[ \|\mathbf{y}(t_0)-\boldsymbol{\eta}(t_0)\|<\delta, \]
the inequality
\[ \rho(\mathbf{y}(t),L_{\boldsymbol{\eta}}^+)=\inf_{t_0\le t_1<\infty}\|\mathbf{y}(t)-\boldsymbol{\eta}(t_1)\|<\varepsilon \]
will be ensured for \(t_0\le t<\infty\), where \(L_{\boldsymbol{\eta}}^+\) is the positive semitrajectory of the solution \(\boldsymbol{\eta}(t)\).
If, moreover, for \(\|\mathbf{y}(t_0)-\boldsymbol{\eta}(t_0)\|<\Delta\), where \(\Delta>0\) is sufficiently small, we have \(\rho(\mathbf{y}(t),L_{\boldsymbol{\eta}}^+)\to0\) as \(t\to\infty\), then the solution \(\boldsymbol{\eta}(t)\) is called asymptotically orbitally stable as \(t\to\infty\).
Definition 5. We shall say that the solution \(\boldsymbol{\eta}(t)\) has an asymptotic phase \((^3)\) if, for every solution \(\mathbf{y}(t)\) sufficiently close to it at \(t=t_0\), there exists a constant number \(c\) (the phase) such that
\[ \lim_{t\to\infty}\|\mathbf{y}(t+c)-\boldsymbol{\eta}(t)\|=0. \]
Obviously, an orbitally stable solution that has an asymptotic phase is asymptotically orbitally stable.
Main theorem. Suppose the autonomous system (5) admits a solution \(\boldsymbol{\eta}(t)\), bounded on \(I_t\), such that
\[ \inf_t\|\dot{\boldsymbol{\eta}}(t)\|>0, \]
and suppose that the variational equation for this solution,
\[ dx/dt=\mathbf{f}'(\boldsymbol{\eta}(t))x \tag{6} \]
form a regular linear system, among whose characteristic exponents only one is zero, corresponding to the completely unbounded solution \(\mathbf{x}_0=\vec{\eta}(t)\), while the others are negative. Then the solution \(\vec{\eta}(t)\) is asymptotically orbitally stable as \(t\to\infty\) and has an asymptotic phase.
To prove the main theorem, by means of the change of variables
\[ \mathbf{y}=\vec{\eta}(t)+\mathbf{z} \]
we bring system (5) to the form
\[ dz/dt=f'(\vec{\eta}(t))z+\vec{\varphi}(t,z), \tag{7} \]
where
\[ \|\vec{\varphi}(t,\widetilde{z})-\vec{\varphi}(t,z)\|\leq \]
\[ \leq N\max(\|z\|,\|\widetilde{z}\|)\|\widetilde{z}-z\| \]
for \(\|z\|,\|\widetilde{z}\|<h<\infty\) (\(N\) is a constant). To find a family of solutions \(z(t,a)\) of equation (7) such that \(z(t,a)\to0\) as \(t\to\infty\), where \(a=\operatorname{colon}(0,a_2,\ldots,a_n)\) is a vector parameter, we use the singular integral equation
\[ z(t,a)=X(t)a+\int_{t_0}^{\infty}G(t,\tau)\vec{\varphi}(\tau,z(\tau,a))\,d\tau \quad (t\geq t_0), \tag{8} \]
where \(X(t)\) is a suitable fundamental matrix of the linear system (6) and
\[ G(t,\tau)= \begin{cases} X(t)AX^{-1}(\tau), & \tau>t\geq t_0,\\ X(t)BX^{-1}(\tau), & t_0\leq \tau<t; \end{cases} \]
\[ A=-\operatorname{diag}(E_1,0),\qquad B=\operatorname{diag}(0,E_{n-1}) \]
(\(E_m\) is the identity matrix of order \(m\)). The integral equation (8) is solved by the method of successive approximations analogously to \((4)\).
Moscow State University
named after M. V. Lomonosov
Received
11 VIII 1966
CITED LITERATURE
- A. Andronov, A. Witt, ZhETF, 3, no. 5 (1933).
- A. M. Lyapunov, The General Problem of the Stability of Motion, L.—M., 1935.
- O. Vejvoda, Czechoslovak Mathematical Journal, 9 (84), 390 (1959).
- E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, IL, 1958.