Abstract Generated abstract
This paper studies the accumulation of perturbations in nonstationary linear oscillatory systems described by systems of differential equations with time dependent polynomial operator coefficients. The author reduces the general system to an equivalent first order linear matrix system, introduces matrix weight functions, and interprets them as responses to unit impulse disturbances, including a procedure suitable for electronic modeling devices. Using the fundamental matrix and the associated adjoint system, the paper derives estimates for the maximum possible deviation of a chosen coordinate at a fixed time under bounded external forces. A further extension treats systems in which the forcing terms include both the perturbations and their derivatives, yielding analogous representation formulas and bounds.
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MATHEMATICS
Ya. N. ROITENBERG
ON THE ACCUMULATION OF PERTURBATIONS IN NONSTATIONARY LINEAR SYSTEMS
(Presented by Academician S. L. Sobolev on 10 III 1958)
The problem of the accumulation of perturbations in linear oscillatory systems with constant parameters was posed and solved by B. V. Bulgakov \((^{1-3})\). The accumulation of perturbations in nonstationary linear systems described by differential equations of the form
\[ \dot y_j=\sum_{k=1}^{n} a_{jk}(t)y_k=x_j(t)\qquad (j=1,\ldots,n), \]
was studied in the work \((^4)\) of B. V. Bulgakov and N. T. Kuzovkov. The aim of the present work is to study systems of a more general form.
Consider a linear oscillatory system described by the differential equations
\[ \sum_{k=1}^{n} f_{jk}(D)y_k=x_j(t)\qquad (j=1,\ldots,n), \tag{1} \]
where \(f_{jk}(D)\) are polynomials in \(D\), whose coefficients are prescribed functions of time, and \(D=d/dt\) is the operator of differentiation with respect to time. By \(m_k\) \((k=1,\ldots,n)\) we denote the order of the highest derivative
\[ \overset{m_k}{y_k} \]
with respect to time occurring in equations (1). We shall assume that \(m_k\geq 1\) \((k=1,\ldots,n)\). The polynomial matrix \(f(D)=\|f_{jk}(D)\|\) is assumed to be nonsingular; the determinant of this matrix \(\Delta(D)\not\equiv 0\).
The system of equations (1) can be represented as follows:
\[ b_{j1}(t)\overset{m_1}{y_1}+b_{j2}(t)\overset{m_2}{y_2}+\ldots+b_{jn}(t)\overset{m_n}{y_n} = \Psi_j\left(\overset{m_1-1}{y_1},\ldots,y_1,\ldots,\overset{m_n-1}{y_n},\ldots,y_n\right)+x_j(t) \tag{2} \]
\[ (j=1,\ldots,n), \]
where
\[ \Psi_j\left(\overset{m_1-1}{y_1},\ldots,y_1,\ldots,\overset{m_n-1}{y_n},\ldots,y_n\right) \]
are linear functions of their arguments.
By \(\Delta^*\) we denote the determinant formed from the coefficients of \(\overset{m_k}{y_k}\) in equations (2):
\[ \Delta^*=\left|b_{jk}(t)\right| \tag{3} \]
and assume that this determinant is not identically equal to zero. Solving equations (2) with respect to \(\overset{m_k}{y_k}\) \((k=1,\ldots,n)\), we obtain
\[ \overset{m_j}{y_j} = \Phi_j\left(\overset{m_1-1}{y_1},\ldots,y_1,\ldots,\overset{m_n-1}{y_n},\ldots,y_n\right) + \frac{B_{1j}(t)}{\Delta^*(t)}x_1(t)+\ldots+ \frac{B_{nj}(t)}{\Delta^*(t)}x_n(t) \tag{4} \]
\[ (j=1,\ldots,n). \]
Here \(\Phi_j\bigl(\overset{m_1-1}{y_1}, \ldots, y_1, \ldots, \overset{m_n-1}{y_n}, \ldots, y_n\bigr)\) are linear functions of their arguments, and \(B_{ij}\) are the algebraic cofactors of the elements \(b_{ij}\) in the determinant (3).
We now introduce new variables \(z_i\) by means of the relations
\[ z_1=y_1,\quad z_2=\dot y_1,\ldots,\quad z_{m_1}=\overset{m_1-1}{y_1},\ldots,\quad z_r=\overset{m_n-1}{y_n}, \tag{5} \]
where
\[ r=m_1+m_2+\cdots+m_n. \tag{6} \]
We shall denote the linear combinations of the external forces \(x_\mu(t)\) in the right-hand sides of equations (4) by
\[ X_{\sigma_j}(t)=\frac{B_{1j}(t)}{\Delta^*(t)}x_1(t)+\cdots+\frac{B_{nj}(t)}{\Delta^*(t)}x_n(t) \quad (\sigma_j=\sigma_1,\ldots,\sigma_n), \tag{7} \]
where
\[ \sigma_1=m_1,\quad \sigma_2=m_1+m_2,\ldots,\quad \sigma_n=r. \tag{8} \]
Equations (4) can now be rewritten as
\[ \dot z_1-z_2=0,\ldots,\dot z_{m_1}-\Phi_1(z_1,z_2,\ldots,z_r)=X_{\sigma_1}(t),\ldots \]
\[ \ldots,\dot z_r-\Phi_n(z_1,z_2,\ldots,z_r)=X_{\sigma_n}(t). \tag{9} \]
In view of the linearity of the functions \(\Phi_j(z_1,z_2,\ldots,z_r)\), equations (9) may be represented in the form
\[ \dot z_j+\sum_{k=1}^{r} a_{jk}(t)z_k=X_j(t)\quad (j=1,\ldots,r). \tag{10} \]
In equations (10), the functions \(X_\mu(t)\) for which \(\mu\ne\sigma_l\) \((l=1,\ldots,n)\) are identically equal to zero.
The system of scalar differential equations (10) is equivalent to the matrix equation
\[ \dot z+a(t)z=X(t), \tag{11} \]
where \(z\), \(a(t)\), and \(X(t)\) are the following matrices:
\[ z=\|z_j\|,\quad a(t)=\|a_{jk}(t)\|,\quad X(t)=\|X_j(t)\|. \tag{12} \]
The general solution of equation (11) has the form
\[ z(t)=\theta(t)\theta^{-1}(t_0)z(t_0)+\int_{t_0}^{t}\theta(t)\theta^{-1}(\tau)X(\tau)\,d\tau, \tag{13} \]
where \(\theta(t)\) is the fundamental matrix for the homogeneous matrix equation obtained from (11) when \(X(t)=0\), and \(\theta^{-1}(t)\) is the inverse matrix.
Denoting by \(N(t,\tau)\) the matrix weight function
\[ N(t,\tau)=\theta(t)\theta^{-1}(\tau), \tag{14} \]
we represent solution (13) in the form
\[ z(t)=N(t,t_0)z(t_0)+\int_{t_0}^{t}N(t,\tau)X(\tau)\,d\tau. \tag{15} \]
Since the functions \(X_\mu(t)\) for which \(\mu\ne\sigma_l\) \((l=1,\ldots,n)\) are identically equal to zero, the elements of the matrix \(z\) will be
\[ z_j(t)=\sum_{k=1}^{r}N_{jk}(t,t_0)z_k(t_0)+\int_{t_0}^{t}\sum_{i=1}^{n}N_{j\sigma_i}(t,\tau)X_{\sigma_i}(\tau)\,d\tau \quad (j=1,\ldots,r). \tag{16} \]
Taking into account expressions (7), by which \(X_{\sigma_i}(t)\) are defined, we obtain
\[ z_j(t)=\sum_{k=1}^{r}N_{jk}(t,t_0)z_k(t_0)+ \int_{t_0}^{t}\sum_{l=1}^{n}\sum_{i=1}^{n}N_{j\sigma_i}(t,\tau) \frac{B_{li}(\tau)}{\Delta^*(\tau)}x_l(\tau)\,d\tau \quad (j=1,\ldots,r). \tag{17} \]
Denoting
\[ W_{jl}(t,\tau)=\sum_{i=1}^{n}N_{j\sigma_i}(t,\tau)\frac{B_{li}(\tau)}{\Delta^*(\tau)} \quad (j=1,\ldots,r;\ l=1,\ldots,n), \tag{18} \]
we represent the general solution in the form
\[ z_j(t)=\sum_{k=1}^{r}N_{jk}(t,t_0)z_k(t_0) +\sum_{l=1}^{n}\int_{t_0}^{t}W_{jl}(t,\tau)x_l(\tau)\,d\tau \quad (j=1,\ldots,r). \tag{19} \]
The function \(W_{js}(t,\tau)\) may be called the weight function, in the coordinate \(z_j\), for the external force applied in equation number \(s\) of the original system of equations (1). For a fixed value \(\tau_1\) of the argument \(\tau\), it represents the law of motion of the system in the coordinate \(z_j\) under the action of a unit impulse \(x_s(\tau)=\delta(\tau-\tau_1)\), provided that all the other external forces \(x_\mu(\tau)\equiv 0\) \((\mu=1,2,\ldots,s-1,s+1,\ldots,n)\) and that the initial deviations \(z_k(t_0)=0\) \((k=1,\ldots,n)\).
From expression (19) it is clear that the law of motion
\[ z_j(t)=W_{js}(t,\tau_1)\quad (j=1,\ldots,r) \tag{20} \]
can also be obtained in the system free from the action of external forces, i.e., with \(x_k(\tau)\equiv 0\) \((k=1,\ldots,n)\), if the initial time is taken as \(t_0=\tau_1\) and the initial conditions are prescribed as
\[ z_{\sigma_i}(t_0)=\frac{B_{si}(t_0)}{\Delta^*(t_0)} \quad (\sigma_i=\sigma_1,\sigma_2,\ldots,\sigma_n); \qquad z_p(t_0)=0\quad (p\ne\sigma_i). \tag{21} \]
Such a variant of determining the function \(W_{js}(t,\tau_1)\) may prove advisable when electronic modeling devices are used for this purpose.
Let us now turn to the question of the accumulation of perturbations in the system under consideration. The position of the system at a fixed instant of time \(t=t_1\), according to (19), is determined by the expressions
\[ z_j(t_1)=\sum_{k=1}^{r}N_{jk}(t_1,t_0)z_k(t_0) +\sum_{l=1}^{n}\int_{t_0}^{t_1}W_{jl}(t_1,\tau)x_l(\tau)\,d\tau \quad (j=1,\ldots,r). \tag{22} \]
Assuming that the external forces \(x_l(t)\) are bounded in absolute value,
\[ |x_l(t)|\leq K_l \tag{23} \]
and taking (18) into account, we obtain an estimate of the greatest possible deviation in a certain coordinate of the system \(z_j\) at the time \(t_1\):
\[ |z_j(t_1)|\leq \left|\sum_{k=1}^{r}N_{jk}(t_1,t_0)z_k(t_0)\right| +\sum_{l=1}^{n}K_l\int_{t_0}^{t_1} \left|\sum_{i=1}^{n}N_{j\sigma_i}(t_1,\tau) \frac{B_{li}(\tau)}{\Delta^*(\tau)}\right|\,d\tau . \tag{24} \]
Let us note that, as is known \((^{5,6})\),
\[ N_{j\sigma_i}(t_1,\tau)=Z_{\sigma_i}(\tau), \tag{25} \]
where \(Z_\mu(\tau)\) are the integrals of the adjoint system of equations constructed for the system of equations (10),
\[ \frac{dZ_\mu}{d\tau}-\sum_{k=1}^{r} a_{k\mu}(\tau) Z_k = 0 \qquad (\mu=1,\ldots,r), \tag{26} \]
which take, for \(\tau=t_1\), the values
\[ Z_j(t_1)=1;\qquad Z_k(t_1)=0 \quad (k=1,2,\ldots,j-1,j+1,\ldots,r). \tag{27} \]
Therefore expression (24) can be represented in the form
\[ |z_j(t_1)| \leq \left|\sum_{k=1}^{r} Z_k(t_0) z_k(t_0)\right| + \sum_{l=1}^{n} K_l \int_{t_0}^{t_1} \left| \sum_{i=1}^{n} Z_{\sigma i}(\tau)\, \frac{B_{li}(\tau)}{\Delta^*(\tau)} \right|\,d\tau, \tag{28} \]
where \(Z_{\sigma i}(\tau)\) are defined according to (26) and (27). From what has been indicated there also follows a method for determining the quantities \(|z_j(t_1)|\) with the aid of electronic modeling devices.
For nonstationary linear systems described by the differential equations
\[ \sum_{k=1}^{n} f_{jk}(D)y_k = [L_j(t)D+R_j(t)]x_j(t) \qquad (j=1,\ldots,n), \tag{29} \]
analogously to (17), we obtain the general solution in the form
\[ z_j(t)= \sum_{k=1}^{r} N_{jk}(t,t_0) z_k(t_0) + \]
\[ + \int_{t_0}^{t} \sum_{l=1}^{n}\sum_{i=1}^{n} N_{j\sigma i}(t,\tau)\, \frac{B_{li}(\tau)}{\Delta^*(\tau)} [L_l(\tau)\dot{x}_l(\tau)+R_l(\tau)x_l(\tau)]\,d\tau \qquad (j=1,\ldots,r). \tag{30} \]
From (30) we determine the position of the system at the instant \(t=t_1\):
\[ z_j(t_1)= \sum_{k=1}^{r} N_{jk}(t_1,t_0)z_k(t_0) + \sum_{l=1}^{n} M_{jl}(t_1,t_1)x_l(t_1) - \sum_{l=1}^{n} M_{jl}(t_1,t_0)x_l(t_0) - \]
\[ - \sum_{l=1}^{n} \int_{t_0}^{t_1} \left[ \frac{d}{d\tau}M_{jl}(t_1,\tau)-S_{jl}(t_1,\tau) \right]x_l(\tau)\,d\tau \qquad (j=1,\ldots,r), \tag{31} \]
where
\[ M_{jl}(t_1,\tau)= \sum_{i=1}^{n} N_{j\sigma i}(t_1,\tau)\, \frac{B_{li}(\tau)}{\Delta^*(\tau)}\,L_l(\tau), \]
\[ S_{jl}(t_1,\tau)= \sum_{i=1}^{n} N_{j\sigma i}(t_1,\tau)\, \frac{B_{li}(\tau)}{\Delta^*(\tau)}\,R_l(\tau). \tag{32} \]
Hence, for \(|x_l(t)|\leq K_l\), we obtain for \(z_j(t_1)\) the estimate
\[ |z_j(t_1)| \leq \left| \sum_{k=1}^{r} N_{jk}(t_1,t_0) z_k(t_0) \right| + \sum_{l=1}^{n} K_l \left\{ |M_{jl}(t_1,t_1)| + \right. \]
\[ \left. + |M_{jl}(t_1,t_0)| + \int_{t_0}^{t_1} \left| \frac{d}{d\tau}M_{jl}(t_1,\tau)-S_{jl}(t_1,\tau) \right|\,d\tau \right\}. \tag{33} \]
Moscow State University
named after M. V. Lomonosov
Received
8 III 1958
REFERENCES
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