Full Text
UDC 517.94
REDUCTION OF SYSTEMS
OF LINEAR DIFFERENTIAL EQUATIONS
TO THE GENERALIZED \(L\)-DIAGONAL FORM
N. I. Shkil
§ 1. INTRODUCTION
- In [1] the asymptotic behavior of solutions of a system of linear differential equations of \(L\)-diagonal form was investigated. In the same work, as well as in [2, 3], methods are given for obtaining substitutions by means of which, in the case of simple roots of the characteristic equation, a certain class of linear differential equations can be reduced to a system of \(L\)-diagonal form.
In the present paper substitutions are constructed for the case when, among the roots of the characteristic equation, multiple roots with multiple elementary divisors appear. By means of the substitutions obtained here, the original system of linear differential equations is reduced to the generalized \(L\)-diagonal form, when in the reduced system the principal coefficient matrix consists of Jordan blocks.
- Consider a system of linear differential equations of the form
\[ \frac{dx}{dt}=A(t)x,\qquad t\geq t_0, \tag{1} \]
where \(x\) is an \(n\)-dimensional vector, and \(A(t)\) is a real square matrix of order \(n\).
Denote the roots of the characteristic equation
\[ \det \|A(t)-\lambda E\|=0 \tag{2} \]
(\(E\) is the identity matrix) by \(\lambda_1(t),\ldots,\lambda_n(t)\). And suppose that on the segment \([t_0,T]\) (\(T\) is any finite number) the roots \(\lambda_1(t),\ldots,\lambda_p(t)\) \((1\leq p<n)\) and their corresponding elementary divisors have respectively constant multiplicities \(k_1,\ldots,k_p\) \((k_1+\cdots+k_p=n)\). Then there exists a nonsingular matrix \(B(t)\) such that
\[ W(t)=B^{-1}(t)A(t)B(t)= \left\| \begin{array}{cccc} W_1(t) & 0 & & \\ \cdot & \cdot & \cdot & \cdot \\ 0 & & & W_p(t) \end{array} \right\|, \tag{3} \]
where
\[ W_j(t)= \left\| \begin{array}{cccccc} \lambda_j(t) & 1 & 0 & \cdots & 0 \\ 0 & \lambda_j(t) & 1 & \cdots & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & \cdots & \cdots & \lambda_j(t) \end{array} \right\|, \qquad j=1,\ldots,p, \tag{4} \]
and \(B^{-1}(t)\) is the matrix inverse to the matrix \(B(t)\).
We shall assume that \(A(t)\) has continuous derivatives on the segment \([t_0,T]\) up to order \(m+k-1\) inclusive (\(m\) is any natural number, \(k=\max\{k_1,\ldots,k_p\}\)). Then, as follows from [4], the matrix \(B(t)\) will also possess this property.
§ 2. THE CASE OF ONE ROOT
- In the present section we shall proceed from the assumption that the characteristic equation (2) has one root, whose multiplicity is equal to \(n\). Then, along with system (1), we consider the system
\[ \varepsilon \frac{dx}{dt}=A(t)x, \tag{5} \]
where \(\varepsilon\) is a parameter (system (5) for \(\varepsilon=1\) coincides with the original system (1)).
Putting
\[ t=\varepsilon t', \tag{6} \]
we arrive at the system
\[ \frac{dx}{dt'}=A(t)x, \tag{7} \]
which has been investigated repeatedly in works [4—10].
Carrying out in (7) the linear substitution
\[ x=U_m(t,\mu)y, \tag{8} \]
where
\[ U_m(t,\mu)=\sum_{s=0}^{n+m-1}\mu^s U^{(s)}(t), \qquad \mu=\sqrt[n]{\varepsilon}, \tag{9} \]
(\(U^{(s)}(t)\), \(s=0,1,\ldots,n+m-1\), are square matrices of order \(n\)), we obtain
\[ U_m(t,\mu)\frac{dy}{dt'} = \left[ A(t)U_m(t,\mu)-\varepsilon\frac{dU_m(t,\mu)}{dt} \right]y. \tag{10} \]
We shall construct the matrix \(U_m(t,\mu)\) starting from the equality
\[ A(t)U_m(t,\mu)-\varepsilon\frac{dU_m(t,\mu)}{dt} = U_m(t,\mu)\left[\Lambda_m(t,\mu)+\mu^{m+n}C_m(t,\mu)\right], \tag{11} \]
in which
\[ \Lambda_m(t,\mu)=\sum_{s=0}^{m}\mu^s\Lambda^{(s)}(t), \tag{12} \]
where, by assumption, \(\Lambda^{(s)}(t)\) \((s=1,\ldots,m)\) are diagonal matrices
\[ \Lambda^{(s)}(t)= \left\| \begin{array}{cccc} \lambda_1^{(s)}(t) & 0 & \cdots & 0\\ 0 & \lambda_2^{(s)}(t) & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & \lambda_n^{(s)}(t) \end{array} \right\|, \tag{13} \]
and \(C_m(t,\mu)\) is a square matrix subject, like \(\Lambda_m(t,\mu)\), to definition.
To determine the matrices \(U_m(t,\mu)\), \(\Lambda_m(t,\mu)\), we shall apply the method set forth by the author in [4], requiring, moreover, that for any \(t \in [t_0,T]\)
\[ \{c(t)\}_{n,1}\ne 0, \tag{14} \]
where \(\{c(t)\}_{n,1}\) is the corresponding element of the matrix
\[ C(t)=B^{-1}(t)\frac{dB(t)}{dt}. \tag{15} \]
For this purpose, equating the coefficients in equality (11) at \(\mu^s\) \((s=0,1,\ldots,n+m-1)\) and introducing the matrices
\[ Q^{(s)}(t)=B^{-1}(t)U^{(s)}(t),\qquad F^{(s)}(t)=B^{-1}(t)\frac{dU^{(s)}(t)}{dt} \tag{16} \]
\[ (s=0,1,\ldots,n+m-1), \]
we have
\[ W(t)Q^{(0)}(t)-Q^{(0)}(t)\Lambda^{(0)}(t)=0, \tag{17} \]
\[ W(t)Q^{(s)}(t)-Q^{(s)}(t)\Lambda^{(0)}(t) = \sum_{j=0}^{s-1} Q^{(j)}(t)\Lambda^{(s-j)}(t)+F^{(s-n)}(t) \tag{18} \]
\[ (s=1,2,\ldots,n+m-1), \]
where \(\Lambda^{(m+1)}(t),\ldots,\Lambda^{(m+n-1)}(t)\), by virtue of (12), are equal to the zero matrix. Then, putting in equation (17)
\[ Q^{(0)}(t)=E, \tag{19} \]
we obtain
\[ \Lambda^{(0)}(t)=W(t). \tag{20} \]
Knowing \(Q^{(0)}(t)\), from (16) we find the matrix
\[ U^{(0)}(t)=B(t). \tag{21} \]
We now pass to the determination of the matrices \(Q^{(s)}(t)\), \(\Lambda^{(s)}(t)\) \((s=1,\ldots,n+m-1)\). To this end we shall use equation (18), which, by virtue of (20), can be represented in the form
\[ Hq_i^{(s)}(t) = \sum_{j=0}^{s-1} q_i^{(j)}(t)\lambda_i^{(s-j)}(t) + q_{i-1}^{(s)}(t) + f_i^{(s)}(t) \tag{22} \]
\[ (i=1,\ldots,n), \]
where
\[ H= \left\| \begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdots & \cdot\\ 0 & 0 & 0 & \cdots & 1\\ 0 & 0 & 0 & \cdots & 0 \end{array} \right\|, \tag{23} \]
\(q_i^{(s)}(t)\), \(f_i^{(s)}(t)\) are the corresponding columns of the matrices \(Q^{(s)}(t)\), \(F^{(s)}(t)\).
In system (22), the first components of the vectors \(q_i^{(s)}(t)\), according to (23), remain arbitrary. We shall take them equal to zero,
\[ \{q_i^{(s)}(t)\}_1=0. \tag{24} \]
The remaining components, as follows from (22), are expressed in terms of the unknown functions \(\lambda_i^{(s)}(t)\), for whose determination we shall apply the following method.
Multiply both sides of equation (22) on the left by the matrix \(H^{n-1}\). Then, taking into account that \(H^n=0\), we have
\[ \sum_{j=0}^{s-1} H^{n-1} q_i^{(j)}(t)\lambda_i^{(s-j)}(t) + H^{n-1}q_{i-1}^{(s)}(t)+H^{n-1}f_i^{(s)}(t)=0 \tag{25} \]
\[ (s=1,2,\ldots,n+m-1;\quad i=1,\ldots,n). \]
Since the matrix \(H^{n-1}\) is equal to
\[ H^{n-1}= \left\| \begin{array}{ccccc} 0&0&\cdot&\cdot&1\\ 0&0&\cdot&\cdot&0\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\cdot&\cdot&0 \end{array} \right\|, \tag{26} \]
then in system (25) the first equation coincides with the last equation of system (22). Therefore, omitting the vectors whose first components are zero, system (25), in accordance with (22), can be put in the form
\[ \sum_{h=0}^{n}\sum_{j_{h-1}=h-1}^{s-1}\cdots \sum_{j_1=1}^{j_2-1}\sum_{i_0=0}^{n-h}\sum_{i_1=0}^{i_0}\cdots \sum_{i_{h-1}=0}^{i_{h-2}} q_{i+h-n}^{(0)}(t)\lambda_{i-i_0}^{(j_1)}(t)\times \]
\[ \times \lambda_{i-i_1}^{(j_2-j_1)}(t)\ldots \lambda_{i-i_{h-1}}^{(s-j_{h-1})}(t)+\omega_i^{(s)}(t)=0 \tag{27} \]
\[ (s=1,\ldots,n+m-1;\quad i=1,\ldots,n), \]
where
\[ \omega_i^{(s)}(t)= \sum_{l=0}^{n-1}\sum_{h=0}^{n-1-l}\cdots \sum_{j_{n-1-l-h}=2n-2-l-h}^{s-1}\cdots \sum_{j_1=n}^{j_2-1} H^l f_{\,n-h}^{(j_1)}(t)\times \]
\[ \times \sum_{i_0=0}^{h}\sum_{i_1=0}^{i_0}\cdots \sum_{i_{n-2-l-h}=0}^{i_{n-3-l-h}} \lambda_{i-i_0}^{(j_2-j_1)}(t)\ldots \lambda_{i-i_{n-2-l-h}}^{(s-j_{n-1-l-h})}(t) \tag{28} \]
\[ (s=1,\ldots,n+m-1;\quad i=1,\ldots,n) \]
(here it is set that \(f_{i-n}^{(j)}(t)=f_{i-n+1+l}^{(s)}(t)\)).
Then, taking into account that the first components of the vectors \(q_2^{(0)}(t), q_3^{(0)}(t), \ldots, q_n^{(0)}(t)\), by virtue of (19), are equal to zero, while
\[ \{q_1^{(0)}(t)\}_1=1, \tag{29} \]
from (27) we obtain
\[ \sum_{j_{n-r}=n-r}^{s-1} \sum_{j_{n-r-1}=n-r-1}^{j_{n-r}-1} \cdots \sum_{j_1=1}^{j_2-1} \sum_{i_0=0}^{r-1}\sum_{i_1=0}^{i_0}\cdots \sum_{i_{n-r}=0}^{i_{n-r-1}} \lambda_{r-i_0}^{(j_1)}(t)\times \]
\[ \times \lambda_{r-i_1}^{(j_2-j_1)}(t)\ldots \lambda_{r-i_{n-r}}^{(s-j_{n-r})}(t) +\{\omega_r^{(s)}(t)\}_1=0 \tag{30} \]
\[ (s=n+1-r,\ n+2-r,\ldots,n+m-r;\quad r=1,\ldots,n), \]
where \(\{\omega_r^{(s)}(t)\}_1\) is the first component of the vector \(\omega_r^{(s)}(t)\).
The scalar equation (30), being equivalent to the last equation in system (22), makes it possible, without particular difficulty, to find the functions \(\lambda_r^{(s)}(t)\) \((r=1,\ldots,n)\). Knowing the functions \(\lambda_r^{(s)}(t)\), the elements of the matrices \(U_m(t,\mu)\) are determined from the system of equations (22), taking (16) into account (for details on this point see [4]).
In what follows we shall assume that the matrix \(U_m(t,\mu)\) constructed by us is nonsingular for \(\mu=1\) and \(t \subseteq [t_0,T]\), i.e.
\[ \det \|U_m(t,1)\| \ne 0. \tag{31} \]
Then, on the basis of (11), the differential system (10) can be written in the form
\[ \frac{dy}{dt} = [\Lambda_m(t,1)+C_m(t,1)]y, \tag{32} \]
where
\[ \Lambda_m(t,1)=\sum_{s=0}^{m}\Lambda^{(s)}(t), \]
\[ C_m(t,1)=-U_m^{-1}(t,1)\left[\sum_{i=0}^{n-1}\frac{dU^{(m+i)}(t)}{dt}+\right. \]
\[ \left. +\sum_{i=0}^{m-1}\sum_{r=1}^{m-i}U^{(m+n-r)}(t)\Lambda^{(i+r)}(t)\right], \tag{33} \]
\(U_m^{-1}(t,1)\) is the matrix inverse to the matrix
\[ U_m(t,1)=\sum_{s=0}^{n+m-1}U^{(s)}(t). \]
If in the transformed system (32) the matrices \(\Lambda_m(t,1)\), \(C_m(t,1)\) satisfy the requirements of theorem (1.6) from [1], then theorem (1.6) makes it possible to investigate the asymptotic behavior of the solutions of system (32), and hence of the original system (1).
- As an example, consider the equation
\[ \frac{d^2x}{dt^2}+2p(t)\frac{dx}{dt}+p^2(t)x=0. \tag{34} \]
Then, as is known, equation (34) can be represented in the form of system (1), in which
\[ A(t)= \left\| \begin{array}{cc} 0 & 1\\ -p^2(t) & -2p(t) \end{array} \right\|. \tag{35} \]
The characteristic equation constructed for the matrix \(A(t)\) has the roots
\[ \lambda_1(t)\equiv\lambda_2(t)=-p(t). \tag{36} \]
Assuming that the elementary divisor has multiplicity two for every \(t \subseteq [t_0,T]\), as the transformation matrix one may take the matrix
\[ B(t)= \left\| \begin{array}{cc} 1 & 1\\ -p(t) & 1-p(t) \end{array} \right\|. \tag{37} \]
Then, putting, for example, \(m=1\) in relations (9), (32) and requiring the fulfillment of conditions (14), (31), which in the present case respectively have the form
\[ \frac{dp(t)}{dt}=p'(t)\ne 0, \tag{38} \]
\[ \beta(t)=[1+p(t)(1-p'(t))](1-\sqrt{-p'(t)})\ne 0, \]
equation (34), by virtue of the preceding arguments, with the aid of the substitution
\[ x=\sum_{s=0}^{2} U^{(s)}(t)y,\qquad x=\left[x,\frac{dx}{dt}\right] \tag{39} \]
is reduced to the system (32), where
\[ U^{(0)}(t)=B(t),\qquad U^{(1)}(t)=\sqrt{-p'(t)} \left\|\begin{matrix} 1 & 0\\ 1-p(t) & 0 \end{matrix}\right\|, \]
\[ U^{(2)}(t)=p'(t) \left\|\begin{matrix} 1 & 1\\ 1-p(t) & 1-p(t) \end{matrix}\right\|, \]
\[ \Lambda^{(0)}(t)= \left\|\begin{matrix} -p(t) & 1\\ 0 & -p(t) \end{matrix}\right\|,\qquad \Lambda^{(1)}(t)= \left\|\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix}\right\|\sqrt{-p'(t)}, \tag{40} \]
and the elements of the matrix \(C_1(t,1)\) are respectively equal to
\[ \{c_1(t,1)\}_{11} =\frac{1}{\beta(t)} \left[p'(t)\sqrt{-p'(t)}+(p'(t))^2\sqrt{-p'(t)}+(p'(t))^2+(p'(t))^3\right], \]
\[ \{c_1(t,1)\}_{12}=\frac{(p'(t))^3}{\beta(t)}, \]
\[ \{c_1(t,1)\}_{21}= \]
\[ =-\frac{1}{\beta(t)} \left[ 2(p'(t))^3+(p'(t))^2\sqrt{-p'(t)}-p''(t)+(p'(t))^2+ \frac{p''(t)}{2\sqrt{-p'(t)}} \right], \]
\[ \{c_1(t,1)\}_{22} =\frac{1}{\beta(t)} \left[p''(t)-(p'(t))^2-p'(t)\sqrt{-p'(t)}-\right. \]
\[ \left. -(p'(t))^2\sqrt{-p'(t)}-(p'(t))^3 \right], \]
\[ p''(t)=\frac{d^2p(t)}{dt^2}. \tag{41} \]
Putting in the relations (9), (12) \(m=2,3,\ldots\), we shall each time obtain new substitutions reducing equation (34) to a system of the form (32).
§ 3. GENERAL CASE
Let us now consider the case when the characteristic equation (2) has \(p\) multiple roots \((1<p<n)\), to each of which there corresponds one elementary divisor. Then the auxiliary system (5), by means of the substitution
\[ x=\sum_{j=1}^{p} U_j(t)y_j, \tag{42} \]
where \(U_j(t)\) are rectangular matrices of dimensions \((n\times k_j)\); \(y_j\) are vectors of dimension \(k_j\) \((j=1,\ldots,p)\), is reduced to the system
\[ \sum_{j=1}^{p} U_j(t)\frac{dy_j}{dt} = \sum_{j=1}^{p}\bigl[A(t)U_j(t)-\varepsilon U'_j(t)\bigr]y_j, \tag{43} \]
\[ U'_j(t)=\frac{dU_j(t)}{dt}. \]
We shall require that
\[ A(t)U_j(t)-\varepsilon U'_j(t) = U_j(t)\bigl[W_j(t)+\varepsilon C_j(t)\bigr] \qquad (j=1,\ldots,p), \tag{44} \]
where \(W_j(t)\) is the Jordan block corresponding to the root \(\lambda_j(t)\), and \(C_j(t)\) is a square matrix of order \(k_j\), to be determined.
Equating in relation (44) the coefficients of \(\varepsilon^s\) \((s=0,1)\), we obtain
\[ A(t)U_j(t)=U_j(t)W_j(t), \tag{45} \]
\[ U_j(t)C_j(t)=-U'_j(t) \qquad (j=1,\ldots,p). \tag{46} \]
Multiplying equation (45) on the left by the matrix \(B^{-1}(t)\), and taking (3) into account, we have
\[ W(t)Q_j(t)=Q_j(t)W_j(t), \tag{47} \]
where
\[ Q_j(t)=B^{-1}(t)U_j(t). \tag{48} \]
Equation (47) splits into \(p\) equations
\[ W_r(t)Q_{jr}(t)=Q_{jr}(t)W_j(t) \qquad (j,r=1,\ldots,p), \tag{49} \]
where \(Q_{jr}(t)\) are rectangular matrices of dimensions \((k_r\times k_j)\).
Since the matrices \(W_r(t)\), \(W_j(t)\) for \(r\ne j\) have different eigenvalues, it follows by [6] that
\[ Q_{jr}(t)=0 \qquad (j\ne r;\quad j,r=1,\ldots,p). \tag{50} \]
For \(r=j\), put
\[ Q_{jj}(t)=E_j, \tag{51} \]
where \(E_j\) is the identity matrix of order \(k_j\) \((j=1,\ldots,p)\).
Knowing the matrix \(Q_j(t)\), from (48) we find
\[ U_j(t)=B(t)Q_j(t). \tag{52} \]
Then equation (46) can be represented in the form
\[ B(t)Q_j(t)C_j(t)=-B'(t)Q_j(t) \]
or
\[ Q_j(t)C_j(t)=D(t)Q_j(t) \qquad (j=1,\ldots,p), \tag{53} \]
where
\[ D(t)=-B^{-1}(t)B'(t). \tag{54} \]
Represent the matrix \(D(t)\) in the form
\[ D(t)= \left\| \begin{array}{cccc} D_{11}(t) & \cdot & \cdot & D_{1p}(t)\\ \cdot & \cdot & \cdot & \cdot\\ D_{p1}(t) & \cdot & \cdot & D_{pp}(t) \end{array} \right\|, \tag{55} \]
where \(D_{rj}(t)\) are rectangular matrices of dimensions \((k_r \times k_j)\) \((r,j=1,\ldots,p)\). Requiring now that
\[ D_{rj}(t)\equiv 0 \tag{56} \]
for \(r\ne j\) \((r,j=1,\ldots,p)\), from (53), taking into account (50), (51), we find
\[ C_j(t)=D_{jj}(t)\quad (j=1,\ldots,p). \tag{57} \]
Then the differential system (43), by virtue of (44), (52), (57), can be represented in the form
\[ B(t)\sum_{j=1}^{p} Q_j(t)\frac{dy_j}{dt'} = B(t)\sum_{j=1}^{p} Q_j(t)[W_j(t)+\varepsilon D_{jj}(t)]y_j . \tag{58} \]
Hence, taking into account (50), (51), we obtain
\[ \frac{dy_j}{dt'}=[W_j(t)+\varepsilon D_{jj}(t)]y_j\quad (j=1,\ldots,p). \tag{59} \]
The characteristic equation
\[ \det\|W_j(t)-\lambda E_j\|=0 \tag{60} \]
of system (59) has a single root \(\lambda_j(t)\) of multiplicity \(k_j\). Consequently, the arguments of § 2 are applicable to system (59).
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Received by the editors
October 5, 1965
Kyiv State Pedagogical Institute
named after A. M. Gorky