Abstract
Full Text
N. I. Shkil
ON THE ASYMPTOTIC SOLUTION OF A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS CONTAINING A PARAMETER
(Presented by Academician N. N. Bogolyubov on 10 I 1963)
MATHEMATICS
- Numerous works have been devoted to the construction of the asymptotic solution of systems of differential equations containing a parameter \(\left({}^{1-7}\right)\). Fundamental results were obtained by N. N. Bogolyubov \(\left({}^{1}\right)\).
Consider, in \(n\)-dimensional space, the system of differential equations\(^*\)
\[ \frac{dx}{dt}=A(\tau,\varepsilon)x+\varepsilon B(\tau,\varepsilon)e^{i\theta}, \tag{1} \]
where \(A(\tau,\varepsilon)\) is a real matrix of order \(n\); \(B(\tau,\varepsilon)\) is an \(n\)-dimensional vector, admitting formal expansions
\[ A(\tau,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^{s}A^{(s)}(\tau),\qquad B(\tau,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^{s}B^{(s)}(\tau),\qquad 0\leqslant \tau=\varepsilon t\leqslant L; \tag{2} \]
\(\varepsilon\) is a small real parameter.
The asymptotic solution of system (1) is determined by the behavior of the roots of the equation
\[ \det \left\| A^{(0)}(\tau)-\lambda E \right\|=0, \tag{3} \]
where \(E\) is the identity matrix.
In \(\left({}^{2}\right)\) a solution was obtained for the case of simple roots on the segment \([0,L]\) of equation (3). The case of multiple roots is partially considered in \(\left({}^{3,4,7}\right)\). The asymptotic splitting of system (1) is given in \(\left({}^{5,6}\right)\) for arbitrary behavior of the roots of equation (3).
In the present article we consider the case when several multiple elementary divisors correspond to a multiple root.
- Denote by \(\lambda_1(\tau), \lambda_2(\tau), \ldots, \lambda_p(\tau)\) the roots of equation (3), having respectively the constant multiplicities \(k_1,k_2,\ldots,k_p\) \((k_1+k_2+\cdots+k_p=n)\), and suppose that to the root \(\lambda_j(\tau)\) \((j=1,\ldots,p)\) there correspond \(r_j\) elementary divisors
\[ (\lambda-\lambda_j(\tau))^{s_{j1}},\ldots,(\lambda-\lambda_j(\tau))^{s_{jr_j}} \]
\((s_{j1}+\cdots+s_{jr_j}=k_j;\; j=1,\ldots,p)\). Then one can indicate a nonsingular matrix \(T(\tau)\) such that
\[ T^{-1}(\tau)A^{(0)}(\tau)T(\tau)=W(\tau)= \left\| \begin{array}{ccc} W_{k_1}(\tau) & & 0\\ & \ddots & \\ 0 & & W_{k_p}(\tau) \end{array} \right\|, \tag{4} \]
\(^*\) Many differential equations are reducible to a system of the form (1), in particular differential equations with a small parameter at the highest derivatives.
where
\[ W_{k_j}(\tau)= \left\| \begin{array}{cccc} W_{s_{j1}}(\tau) & & & \\ 0 & \ddots & & 0\\ & & W_{s_{jr_j}}(\tau) \end{array} \right\|, \quad W_{s_{jn_j}}(\tau)= \left\| \begin{array}{ccccc} \lambda_j(\tau) & 1 & 0 & \ldots & 0\\ 0 & \lambda_j(\tau) & 1 & 0\ldots 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \ldots & \lambda_j(\tau) \end{array} \right\| \Biggr\}_{s_{jn_j}}, \tag{5} \]
\[ n_j=1,\ldots,r_j;\quad j=1,\ldots,p; \]
\(T^{-1}(\tau)\) is the matrix inverse to the matrix \(T(\tau)\).
Suppose that the function \(i\omega(\tau)\) \((\omega(\tau)=d\theta/dt,\ i=\sqrt{-1})\) at isolated points of the segment \([0,L]\) becomes equal to one of the roots of equation (3), for example \(\lambda_1(\tau)\), but \(i\omega(\tau)\ne \lambda_k(\tau)\) \((k=2,\ldots,p)\) for any \(\tau\in[0,L]\). This case is customarily called “resonant.”
Theorem 1. If \(A(\tau,\varepsilon)\), \(B(\tau,\varepsilon)\), \(\omega(\tau)\) have derivatives of all orders with respect to \(\tau\), and for all \(\tau\in[0,L]\) the \(\nu_{jn_j}\)-component of the vector
\[ C(\tau)=T^{-1}(\tau)\left[ A^{(1)}(\tau)T_{\nu_{jn_j-1}+1}(\tau) -\frac{dT_{\nu_{jn_j-1}+1}(\tau)}{d\tau} \right], \tag{6} \]
where \(T_{\nu_{jn_j-1}+1}(\tau)\) is the column of the matrix \(T(\tau)\) numbered \(\nu_{jn_j-1}+1\) \((\nu_{jn_j}=k_1+\ldots+k_{j-1}+s_{j1}+\ldots+s_{jn_j};\ n_j=1,\ldots,r_j;\ j=1,\ldots,p)\), is not equal to zero, then the asymptotic solution of system (1) in the “resonant” case can be represented in the form
\[ x=\sum_{n_1=1}^{r_1}\left[U_{1n_1}(\tau,\mu_{1n_1})h_{1n_1} +P_{n_1}(\tau,\mu_{1n_1})\right]e^{i\theta} + \]
\[ +\sum_{k=2}^{p}\sum_{n_k=1}^{r_k} U_{kn_k}(\tau,\mu_{kn_k})h_{kn_k}; \tag{7} \]
\[ \frac{dh_{1n_1}}{dt} = [\lambda_{1n_1}(\tau,\mu_{1n_1})-i\omega(\tau)]h_{1n_1} +z_{n_1}(\tau,\mu_{1n_1}), \quad n_1=1,\ldots,r_1; \tag{8} \]
\[ \frac{dh_{kn_k}}{dt} = \lambda_{kn_k}(\tau,\mu_{kn_k})h_{kn_k}, \quad n_k=1,\ldots,r_k;\quad k=2,\ldots,p, \tag{9} \]
where \(U_{jn_j}(\tau,\mu_{jn_j})\), \(P_{n_1}(\tau,\mu_{1n_1})\) are \(n\)-dimensional vectors; \(h_{jn_j}\), \(z_{n_1}(\tau,\mu_{1n_1})\) are scalar functions \((n_j=1,\ldots,r_j;\ j=1,\ldots,p)\), admitting formal expansions
\[ U_{jn_j}(\tau,\mu_{jn_j}) = \sum_{s=0}^{\infty}\mu_{jn_j}^{s}U_{jn_j}^{(s)}(\tau), \quad \lambda_{jn_j}(\tau,\mu_{jn_j}) = \lambda_j(\tau)+ \sum_{s=1}^{\infty}\mu_{jn_j}^{s}\lambda_{jn_j}^{(s)}(\tau); \tag{10} \]
\[ P_{n_1}(\tau,\mu_{1n_1}) = \sum_{s=0}^{\infty}\mu_{1n_1}^{s}P_{n_1}^{(s)}(\tau), \quad z_{n_1}(\tau,\mu_{1n_1}) = \sum_{s=0}^{\infty}\mu_{1n_1}^{s}z_{n_1}^{(s)}(\tau); \tag{11} \]
\[ \mu_{jn_j}=\sqrt[s_{jn_j}]{\varepsilon}, \quad n_j=1,\ldots,r_j;\quad j=1,\ldots,p. \tag{12} \]
Substituting the vector \(x\), determined by relations (7)—(9), into system (1) and, in the identity obtained, equating the coefficients of the functions \(h_{jn_j}\) and the free terms, we have
\[ (A-\lambda_{jn_j}E)U_{jn_j} = \varepsilon U'_{jn_j}, \quad n_j=1,\ldots,r_j;\quad j=1,\ldots,p^*, \tag{13} \]
\[ (A-i\omega E)P_{n_1} = U_{1n_1}z_{n_1} +\varepsilon(P'_{n_1}-B), \quad n_1=1,\ldots,r_1. \tag{14} \]
\[ \text{* Here and below we omit the arguments of the quantities.} \]
To determine the coefficients of the series (10), we shall use relation (13). Equating in it the coefficient of \(\mu_{j n_j}^{0}\), we have
\[ (A^{(0)}-\lambda_j E)U_{i n_j}^{(0)}=0,\qquad n_j=1,\ldots,r_j;\ j=1,\ldots,p. \tag{15} \]
Introducing the vector
\[ Q_{j n_j}^{(s)}=T^{-1}U_{j n_j}^{(s)},\qquad s=0,1,\ldots, \tag{16} \]
equation (15) may be put in the form
\[ (W-\lambda_j E)Q_{j n_j}^{(0)}=0, \tag{17} \]
or, according to (4),
\[ (W_{k_r}-\lambda_j E)Q_{j n_j k_r}^{(0)}=0,\qquad n_j=1,\ldots,r_j;\ r,j=1,\ldots,p, \tag{18} \]
where \(Q_{j n_j k_r}^{(0)}\) is the vector with components
\[ Q_{j n_j k_r}^{(0)}= \bigl(\{q_{j n_j}^{(0)}\}_{l_{r-1}+1},\ \{q_{j n_j}^{(0)}\}_{l_{r-1}+2},\ldots,\{q_{j n_j}^{(0)}\}_{l_r}\bigr), \tag{19} \]
\[ l_r=k_1+\cdots+k_r;\quad r=1,\ldots,p. \]
From (18) we find
\[ Q_{j n_j k_r}^{(0)}\equiv 0,\qquad j\ne r;\ j,r=1,\ldots,p;\ n_j=1,\ldots,r_j. \tag{20} \]
For \(r=j\), equation (18) is representable in the form
\[ (W_{s_j i_j}-\lambda_j E)Q_{j n_j s_j i_j}^{(0)}=0, \tag{21} \]
where
\[ Q_{j n_j s_j i_j}^{(0)} = \bigl(\{q_{j n_j}^{(0)}\}_{v_{i_j j-1}+1},\ \{q_{j n_j}^{(0)}\}_{v_{i_j j-1}+2},\ldots, \{q_{j n_j}^{(0)}\}_{v_{i_j j}}\bigr), \tag{22} \]
\[ v_{i_j j}=k_1+\cdots+k_{j-1}+s_{j1}+\cdots+s_{j i_j}; \]
\[ n_j,i_j=1,\ldots,r_j;\quad j=1,\ldots,p. \]
Taking (5) into account, from (21) we find
\[ \{q_{j n_j}^{(0)}\}_{v_{i_j j-1}+2} =\cdots= \{q_{j n_j}^{(0)}\}_{v_{i_j j}}=0,\qquad n_j,i_j=1,\ldots,r_j;\ j=1,\ldots,p. \tag{23} \]
The component \(\{q_{j n_j}^{(0)}\}_{v_{i_j j-1}+1}\) is arbitrary. Put
\[ \{q_{j n_j}^{(0)}\}_{v_{i_j j-1}+1} = \begin{cases} 1, & i_j=n_j,\\ 0, & i_j\ne n_j;\ i_j,n_j=1,\ldots,r_j;\ j=1,\ldots,p. \end{cases} \tag{24} \]
To determine the vector \(Q_{j n_j}^{(1)}\) and the function \(\lambda_{j n_j}^{(1)}\), compare in (13) the coefficients of \(\mu_{j n_j}^{1}\). Then, repeating the preceding arguments, we find:
\[ Q_{j n_j k_r}^{(1)}\equiv 0,\qquad Q_{j n_j s_j i_j}^{(1)}\equiv 0,\qquad r\ne j;\ i_j\ne n_j;\ i_j,n_j=1,\ldots,r_j; \tag{25} \]
\[ r,j=1,\ldots,p; \]
\[ \{q_{j n_j}^{(1)}\}_{v_{j n_j-1}+2} =\lambda_{j n_j}^{(1)},\qquad \{q_{j n_j}^{(1)}\}_{v_{j n_j-1}+3} =\cdots= \{q_{j n_j}^{(1)}\}_{v_{j n_j}}=0. \tag{26} \]
The component \(\{q_{j n_j}^{(1)}\}_{v_{j n_j-1}+1}\) is arbitrary. We put it equal to zero. Equating now in relation (13) the coefficients successively of \(\mu_{j n_j}^{2},\ldots,\mu_{j n_j}^{s_{j n_j}-1}\) and applying the method of induction, we have
\[ \{q_{j n_j}^{(s_{j n_j}-1)}\}_{v_{j n_j}} = (\lambda_{j n_j}^{(1)})^{s_{j n_j}-1},\qquad \{q_{j n_j}^{(\eta_{j n_j})}\}_{v_{j n_j}}=0,\qquad 0\le \eta_{j n_j}\le s_{j n_j}-2. \tag{27} \]
Then, equating in (13) the coefficients of \(\mu_{j n_j}^{s_{j n_j}}\) and taking (27) into account, we find
\[ \lambda_{i n_j}^{(1)}(\tau) = \sqrt[s_{j n_j}]{ \left\{ T^{-1}(\tau) \left[ A^{(1)}(\tau)T_{\nu_{j n_j}-1+1}(\tau) - \frac{dT_{\nu_{j n_j}-1+1}(\tau)}{dt} \right] \right\}_{\nu_{j n_j}} }, \tag{28} \]
\[ n_j=1,\ldots,r_j;\quad j=1,\ldots,p. \]
All subsequent coefficients of the series (10) are determined in the same way.
The vector \(P_{n_1}(\tau,\mu_{1n_1})\) and the function \(z_{n_1}(\tau,\mu_{1n_1})\) \((n_1=1,\ldots,r_1)\) are determined from relation (14).
Applying the method set forth in \((^1)\), one can prove the following theorem.
Theorem 2. If the conditions of Theorem 1 are satisfied and
\[ x\big|_{t=0}=x_m\big|_{t=0};\qquad \operatorname{Re}\left( \sum_{s=0}^{s_{j n_j}-1} \mu_{j n_j}^{s}\lambda_{j n_j}^{(s)} \right)\leq 0, \quad n_j=1,\ldots,r_j;\quad j=1,\ldots,p, \tag{29} \]
where \(x_m\) is the vector defined by relations (7)—(12); if the series are cut off after the \(m\)-th terms, then for any \(L>0\) and \(0<\mu_{j n_j}\leq \bar{\mu}_{j n_j}\) one can indicate constants \(C_{j n_j}\), independent of the parameter \(\mu_{j n_j}\), such that
\[ \|x-x_m\| \leq \sum_{j=1}^{p}\sum_{n_j=1}^{r_j} \mu_{j n_j}^{\,m+2-2s_{j n_j}}C_{j n_j}. \tag{30} \]
Relation (30) proves the asymptotic character of the solution \(x_m\).
Kyiv State Pedagogical Institute
named after A. M. Gorky
Received
3 I 1963
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