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UDC 517.925.2
ON THE CONVERGENCE OF THE OPTIMAL CONTROL OF A COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS
E. M. MARKUSHIN, S. N. SHIMANOV
In the work [1], N. N. Krasovskii showed that the optimal control of A. M. Letov’s problem for a system with delay can be represented in the form of a linear functional. Since, in constructing the optimal control, difficulties of a technical nature arise, it is therefore expedient to consider approximate methods for determining it [1].
In particular, for this purpose one may use the method of splitting the functional space, developed in [2, 3]. This method reduces the solution of the problem to the solution of A. M. Letov’s problem for a sequence of finite-dimensional systems of ordinary differential equations. To justify such an approach it is necessary, in particular, to show the uniform boundedness of the resulting optimal controls in the form of linear forms of the distinguished coordinates, which is shown in the present article.
1. PRELIMINARY REMARKS
Let the motion of some automatic-control system be described by the equation
\[ \frac{dx(t)}{dt}=ax(t-\tau)+u, \tag{1.1} \]
where \(a,\tau\) (\(\tau=\operatorname{const}>0\)) are constants; \(u\) is the action of the regulator. It is required to find a control \(u[x(\vartheta)]\) minimizing the integral
\[ I(u)=\frac{1}{2}\int_{0}^{\infty}\left[x^{2}(t)+u^{2}\right]dt. \tag{1.2} \]
N. N. Krasovskii showed that such an optimal control \(u\) exists and is a linear functional defined on continuous curves \(x(\vartheta)\) [1].
The problem of directly calculating the optimal control \(u[x(\vartheta)]\) is very difficult. Therefore it is natural to attempt to expand the control in the series
\[ \sum_{j=1}^{\infty} p_j y_j(\lambda_j;\,x(\vartheta)), \]
where \(p_j\) are constant coefficients, and \(y\) is a linear functional depending on the roots of the characteristic equation of system (1.1) without
control. The question arises of the convergence of this series to the optimal control of the original problem. In the present article estimates are derived that are important for the solution of this problem. It is also shown that the remainder of the series is majorized by a series of the form \(\sum_n c n^{-2+\varepsilon}\).
In accordance with the decomposition of the functional space [2, 3], the following system of ordinary equations can be associated with equation (1.1):
\[ \frac{d y_i(t)}{dt}=\lambda_i y_i+u \qquad (i=1,2,\ldots), \tag{1.3} \]
where \(\lambda_i\) are the roots of the characteristic equation
\[ \lambda=a e^{-\lambda \tau} \tag{1.4} \]
and
\[ y_i=x(t)+a\int_{-\tau}^{0} x(t+\vartheta)e^{-\lambda_i(t+\vartheta)}\,d\vartheta . \]
Then the coordinate \(x(t)\) can be represented in the form of the series [2]
\[ x(t)=\sum_{i=1}^{\infty}\frac{y_i(t)}{\Delta'(\lambda_i)} . \tag{1.5} \]
In (1.5) the sum on the right is real, since the terms are pairwise conjugate. The roots of the characteristic equation (1.4) are assumed to be simple. This condition is satisfied for all \(a\) and \(\tau\), with the exception of certain individual values. Namely, for \(\tau=1\) it fails only for
\[ a=-\frac{1}{e} \]
[4]. It is known [4] that the asymptotic roots are determined by the equalities
\[ \lambda_n=-\frac{1}{\tau}\ln\frac{2\pi n}{|a|} +\left(2n+\frac{1}{2}\operatorname{sign}a\right)\frac{\pi}{\tau}i +O\left(\frac{\ln n}{n}\right), \]
\[ \lambda_{n+1}=\overline{\lambda}_n =-\frac{1}{\tau}\ln\frac{2\pi n}{|a|} -\left(2n+\frac{1}{2}\operatorname{sign}a\right)\frac{\pi}{\tau}i +O\left(\frac{\ln n}{n}\right). \tag{1.6} \]
Consider the system of ordinary differential equations truncated with respect to (1.3):
\[ \frac{d y_i(t)}{dt}=\lambda_i y_i+u \qquad (i=1,2,\ldots,N). \]
Then, according to [5], the optimal control \(u\) will have the form
\[ u=\sum_{i=1}^{N}p_i y_i, \tag{1.7} \]
and the roots \(\mu_i\) \((i=1,2,\ldots,N)\) of the determinant
\[ \left| \begin{array}{ccccc} \lambda_1+p_1-\mu & p_2 & \cdot & \cdot & p_N\\ p_1 & \lambda_2+p_2-\mu & \cdot & \cdot & p_N\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ p_1 & p_2 & \cdot & \cdot & \lambda_N+p_N-\mu \end{array} \right|=0 \tag{1.8} \]
will satisfy the equation1
\[ 1+\sum_{i=1}^{N}\frac{1}{\Delta'(\lambda_i)(\lambda_i-\mu)} \sum_{j=1}^{N}\frac{1}{\Delta'(\lambda_j)(\lambda_j+\mu)}=0. \tag{1.9} \]
We write the determinant (1.8) in the following form:
\[ \prod_{i=1}^{N}(\lambda_i-\mu)+ \sum_{i=1}^{N}p_i\prod_{j=1}^{N}{}'(\lambda_j-\mu)=0. \tag{1.10} \]
The symbol \(({}')\) means that in the product \(\prod\) the factor with number \(j=i\) is absent. From equality (1.9) it follows that the roots \(\mu_i\) \((i=1,2,\ldots,N)\) of the characteristic determinant (1.8), for sufficiently large \(n\) \((n\leq N)\), have the form
\[ \mu_n=\lambda_n+\theta_1\left(-\frac{1}{\tau}\ln\frac{n+1}{n}\right) +\theta_2\left(\frac{2\pi}{\tau}\right)i, \tag{1.11} \]
where \(|\theta_i|\leq 1/2\) and \(\lambda_n\) is taken according to (1.6)2. From relation (1.9) we have
\[ -|\lambda_n-\mu_n|\,n^s = \sum_{i=1}^{N}\frac{|\lambda_n-\mu_n|\,n^{s/2}}{\Delta'(\lambda_i)(\lambda_i-\mu_n)} \sum_{i=1}^{N}\frac{n^{s/2}}{\Delta'(\lambda_i)(\lambda_i+\mu_n)}. \tag{1.12} \]
Let, without loss of generality, \(\tau=1\). We shall show that the series
\[ \sum_{k=1}^{N} \frac{|\lambda_n-\mu_n|\,n^{s/2}} {\Delta'(\lambda_k)(\lambda_k-\mu_n)} \tag{1.13} \]
is bounded for any arbitrarily large \(n\) \((n\leq N)\) and \(s<2\) by some finite number \(d_1\), independent of the choice of \(N\).
Indeed,
\[ \begin{aligned} \left| \sum_{k=1}^{N}\frac{1}{\Delta'(\lambda_k)} \frac{|\lambda_n-\mu_n|\,n^{s/2}}{\lambda_k-\mu_n} \right| &\leq \sum_{k=1}^{n-1} \left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{|\lambda_n-\mu_n|\,n^{s/2}}{|\lambda_k-\mu_n|} \\ &\quad +2\left|\frac{n^{s/2}}{\Delta'(\lambda_n)}\right| +\sum_{k=n+2}^{N} \left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{|\lambda_n-\mu_n|\,n^{s/2}}{|\lambda_k-\mu_n|}, \end{aligned} \tag{1.14} \]
where \(\overline{\lambda}_k\), conjugate to \(\lambda_k\), is an eigenvalue of the free system \((\overline{\lambda}_k+\lambda_{k+1})\).
On the basis of equalities (1.4) and (1.6), the second term on the right-hand side of (1.14) is finite for every \(n\) \((n\leq N)\), since
\[ |\Delta'(\lambda_n)| \geqslant n. \tag{1.15} \]
Further, from equality (1.11) it follows that the modulus \(|\lambda_n-\mu_n|\) is bounded for every \(n\), i.e.
\[ |\lambda_n-\mu_n| \leqslant \gamma \quad (\gamma=\mathrm{const}) \quad (n=1,2,\ldots,N), \tag{1.16} \]
then the series on the right-hand side of (1.14), starting from some value \(k_1\) \((k_1=\mathrm{const})\), are majorized, according to (1.6) and (1.15), by the series
\[ 2\sum_k \frac{n^{s/2}}{k|n-k|}, \tag{1.17} \]
where \(n\ne k\) by virtue of the decomposition (1.14).
The series (1.17) converges for every integer \(n\) and \(s<2\) (Maclaurin–Cauchy test), and its sum is uniformly bounded in \(n\), and, consequently, the series (1.13) converges, and its sum is also uniformly bounded in \(n\) by some finite positive number \(d_1\), independent of the choice of the number \(N\).
The convergence of the series
\[ \sum_{k=1}^{N}\frac{1}{\Delta'(\lambda_k)}\,\frac{n^{s/2}}{\lambda_k+\mu_n} \tag{1.18} \]
is shown analogously. Indeed,
\[ \begin{aligned} \left| \sum_{k=1}^{N}\frac{1}{\Delta'(\lambda_k)}\,\frac{n^{s/2}}{\lambda_k+\mu_n} \right| &\leqslant \sum_{k=1}^{n-1}\left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{n^{s/2}}{|\lambda_k+\mu_n|} \\ &\quad +2\left|\frac{1}{\Delta'(\lambda_n)}\right| \frac{n^{s/2}}{|\lambda_n+\mu_n|} +\sum_{k=n+2}^{N}\left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{n^{s/2}}{|\lambda_k+\mu_n|}. \tag{1.19} \end{aligned} \]
Since \(s<2\) and \(|\lambda_n+\mu_n|\geqslant \ln \dfrac{\pi n}{|a|}\), the second term on the right-hand side of (1.19) is bounded for every \(n\).
The series
\[ \sum_{k=1}^{n-1}\left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{n^{s/2}}{|\lambda_k+\mu_n|}, \qquad \sum_{k=n+2}^{N}\left|\frac{1}{\Delta'(\lambda_k)}\right| \frac{n^{s/2}}{|\lambda_k+\mu_n|} \]
are majorized by the convergent series
\[ 2\sum_k \frac{n^{s/2}}{k|n-k|}, \]
whose sum is uniformly bounded in \(n\). Therefore the series (1.18) also converges for every \(n\) and \(s<2\), and its sum is uniformly bounded in \(n\) by some positive finite number \(d_2\), independent of the choice of the number \(N\).
From the convergence of the series (1.13) and (1.18) and the equality (1.12) it follows that
\[ |\lambda_n-\mu_n|\leq \frac{C_1}{n^s}\quad (C_1=\operatorname{const}<\infty), \tag{1.20} \]
where \(s<2,\ C_1=d_1d_2\).
Next, using (1.10), we obtain
\[ \prod_{i=1}^{N}(\lambda_i-\mu)+ \sum_{i=1}^{N}p_i\prod_{j=1}^{N}{}'(\lambda_j-\mu) = \prod_{i=1}^{N}(\mu_i-\mu), \tag{1.21} \]
where \(\mu_i\) \((i=1,2,\ldots,N)\) are the roots of the characteristic determinant (1.8). For \(\mu=\lambda_i\), equality (1.21) takes the form
\[ p_i\prod_{j=1}^{N}{}'(\lambda_j-\lambda_i) = \prod_{k=1}^{N}(\mu_k-\lambda_i). \tag{1.22} \]
Introducing the notation
\[ \mu_k=\lambda_k+\varepsilon_k, \tag{1.23} \]
we write (1.22) in the form
\[ p_i=\varepsilon_i\prod_{j=1}^{N}{}' \left( 1+\frac{\varepsilon_j}{\lambda_j-\lambda_i} \right). \tag{1.24} \]
From equality (1.24) it follows that
\[ |p_i|\leq |\varepsilon_i| \prod_{j=1}^{N}{}' \left( 1+\frac{|\varepsilon_j|}{|\lambda_j-\lambda_i|} \right). \tag{1.25} \]
But the product on the right in (1.25) converges for any \(N\), since, for sufficiently large difference \((i-j)\), by virtue of (1.6),
\[ |\lambda_i-\lambda_j|\geq 1\quad (i\ne j), \tag{1.26} \]
and \(|\varepsilon_j|\) satisfy the inequality (1.20), i.e.
\[ \prod_{j=1}^{N}{}' \left( 1+\frac{|\varepsilon_j|}{|\lambda_j-\lambda_i|} \right) \leq B_1\quad (B_1=\operatorname{const}<\infty), \tag{1.27} \]
where the quantity \(B_1\) does not depend on the choice of \(N\). Therefore, from (1.25) and (1.20),
\[ |p_i|\leq |\varepsilon_i|B_1\leq \frac{B}{n^s}\quad (B=\operatorname{const}<\infty), \tag{1.28} \]
where \(s<2\), and the quantity \(B\) does not depend on the choice of \(N\).
2. BOUNDEDNESS OF THE CONTROL
Consider the system of equations truncated with respect to (1.3),
\[ \frac{dy_i}{dt}=\lambda_i y_i+u\quad (i=1,2,\ldots,N), \tag{2.1} \]
where
\[ u=\sum_{i=1}^{N}p_i y_i. \tag{2.2} \]
The initial values \(y_i(0)\), according to (2), are determined from the equality
\[ y_i(0)=\varphi(0)+a\int_{-\tau}^{0}\varphi(\vartheta)e^{-\lambda_i(\tau+\vartheta)}\,d\vartheta . \]
Let the initial curve \(\varphi(\vartheta)\,[-\tau\leq \vartheta\leq 0]\) be bounded together with its first derivative. Then, first integrating by parts, we obtain
\[ |y_i(0)|\leq 2|\varphi(0)|+|\varphi(-\tau)|e^{\operatorname{Re}\lambda_i\tau}+ \]
\[ +\|\dot{\varphi}(\vartheta)\|^{(h)} \left|\frac{1-e^{\operatorname{Re}\lambda_i\tau}}{\operatorname{Re}\lambda_i\tau}\right| \leq D\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\}, \tag{2.3} \]
where
\[ \|\dot{\varphi}(\vartheta)\|^{(h)}=\sup|\dot{\varphi}(\vartheta)|[-\tau\leq \vartheta\leq 0];\qquad D=\text{const}<\infty . \]
Inequality (2.3) shows that the initial values \(y_i(0)\) \((i=1,2,\ldots)\) are bounded.
Using the transformation of A. I. Lur’e [6], we reduce system (2.1) to the canonical form1
\[ \dot z_\rho=\mu_\rho z_\rho\qquad (\rho=1,2,\ldots,N). \tag{2.4} \]
The transformation is effected by the substitution
\[ y_k=\sum_{\rho=1}^{N}\frac{H_k(\mu_\rho)}{D'(\mu_\rho)}z_\rho \qquad (k=1,2,\ldots,N), \tag{2.5} \]
where \(D'(\mu_\rho)\) is the derivative of the determinant (1.8), and \(H_k(\mu_\rho)\) is the determinant (1.8) in which the \(k\)-th column is replaced by the numbers \(h_i\). In the present case we set
\[ h_1=1,\quad h_i=0\qquad (i=2,3,\ldots,N). \tag{2.6} \]
The variables \(z_\rho\) can be represented in the form [6]
\[ z_\rho=\frac{1}{H_m(\mu_\rho)}\sum_{\alpha=1}^{N}D_{\alpha m}(\mu_\rho)y_\alpha \qquad (\rho=1,2,\ldots,N), \tag{2.7} \]
where \(D_{\alpha m}\) is the algebraic cofactor of the element in the \(\alpha\)-th row and \(m\)-th column of the determinant (1.8).
We write the determinant (1.8) in the following form:
\[ \Delta= \left| \begin{array}{ccccc} p_1+\lambda_1-\mu & p_2 & \cdots & p_N\\ -(\lambda_1-\mu) & \lambda_2-\mu & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots\\ -(\lambda_1-\mu) & 0 & \cdots & \lambda_N-\mu \end{array} \right|=0. \tag{2.8} \]
Then, putting \(m=1\) in (2.7), we obtain
\[ z_j=y_1+\sum_{i=2}^{N}\frac{p_i}{\lambda_i-\mu_j}y_i \tag{2.9} \]
and, consequently,
\[ |z_j(0)|\leq |y_1(0)|+\sum_{i=2}^{N}\left|\frac{p_i}{\lambda_i-\mu_j}\right|\,|y_i|. \tag{2.10} \]
We write the series in (2.10) in the form
\[ |z_j|\leq |y_1|+\left|\frac{p_j}{\lambda_j-\mu_j}\right|\,|y_j| +\sum_{i=2}^{N}{}'\left|\frac{p_i}{\lambda_i-\mu_j}\right|\,|y_i|, \]
where \(({}')\) means that in the sum \(\sum\) the term with number \(i=j\) is absent. But
\[ \left|\frac{p_j}{\lambda_j-\mu_j}\right|\leq K_1 \qquad (K_1=\mathrm{const}<\infty). \tag{2.11} \]
Indeed, according to (1.10),
\[ \left|\frac{p_j}{\lambda_j-\mu_j}\right| \leq 1+\sum_{k=1}^{N}{}'\left|\frac{p_k}{\lambda_k-\mu_j}\right|. \tag{2.12} \]
Since, for sufficiently large difference \((k-j)\), in accordance with (1.6) and (1.11),
\[ \frac{1}{|\lambda_k-\lambda_j|}\leq 1, \tag{2.13} \]
the series on the right in (2.12) is majorized by the series convergent on the basis of (1.28),
\[ \sum_k |p_k|, \tag{2.14} \]
and, consequently, inequality (2.11) holds, and the quantity \(K_1\) does not depend on the choice of \(N\).
Further, since \(|y_i(0)|\leq D\) \((i=1,2,\ldots)\), using (2.11) and (2.13) and the convergence of the series (2.14), from (2.10) we obtain
\[ |z_j(0)|\leq C_3\{ \|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)} \} \qquad (C_3=\mathrm{const}<\infty), \]
where \(j=1,2,\ldots,N\), and the quantity \(C_3\) does not depend on the choice of \(N\), i.e. the initial values of the variables \(z_j\) are bounded, and, consequently, using (2.4),
\[ |z_j(t)|\leq |z_j(0)|\leq C_3\{ \|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)} \} \quad (j=1,2,\ldots,N). \tag{2.15} \]
From equality (2.8)
\[ \frac{H_k(\mu_p)}{D'(\mu_p)} = -\frac{1}{(\lambda_k-\mu_p)\left(\displaystyle\sum_{i=1}^{N}\frac{p_i}{(\lambda_i-\mu_p)^2}\right)}. \tag{2.16} \]
Next, since for sufficiently large \(\rho\), on the basis of (1.24) and (1.20),
\[ \left|\frac{p_{\rho}}{(\lambda_{\rho}-\mu_{\rho})^2}\right|>K_2\rho^{2-\varepsilon}, \]
where \(\varepsilon\) is an arbitrarily small preassigned number, we have
\[ \left|\sum_{i=1}^{N}\frac{p_i}{(\lambda_i-\mu_{\rho})^2}\right| > K_2\rho^{2-\varepsilon} - \sum_{i=1}^{N}{}' \left|\frac{p_i}{(\lambda_i-\mu_{\rho})^2}\right|. \tag{2.17} \]
The prime on the summation sign \(\sum\) indicates that the term with number \(i=\rho\) is absent. But since, for sufficiently large \(\rho\),
\[ |\lambda_i-\mu_{\rho}|\geq 1 \qquad (i\neq \rho), \]
it follows that
\[ \sum_{i=1}^{N}{}' \left|\frac{p_i}{(\lambda_i-\mu_{\rho})^2}\right| \leq \sum_{i=1}^{N}|p_i| \leq K_3 \qquad (K_3=\mathrm{const}<\infty), \]
where \(K_3\) does not depend on the choice of the magnitude \(N\), in view of the fact that the series \(\sum_k |p_k|\), according to (1.28), converges and its sum is bounded for any number of terms. Therefore, for \(k\neq \rho\),
\[ \left|\frac{H_k(\mu_{\rho})}{D'(\mu_{\rho})}\right| \leq \frac{1}{K_2\rho^{2-\varepsilon}-K_3}. \tag{2.18} \]
If \(k=\rho\), then the quantity on the left in inequality (2.18) remains bounded by virtue of (1.24).
Next, in accordance with (2.5),
\[ |y_k(t)| \leq \sum_{\rho=1}^{N} \left|\frac{H_k(\mu_{\rho})}{D'(\mu_{\rho})}\right| |z_{\rho}(t)|. \tag{2.19} \]
But since (2.15)
\[ |z_i(t)|\leq |z_i(0)|\leq C_3\{\,\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\,\}, \]
it follows that
\[ |y_k(t)| \leq C_3 \sum_{\rho=1}^{N} \left|\frac{H_k(\mu_{\rho})}{D'(\mu_{\rho})}\right|. \tag{2.20} \]
The sum on the right in inequality (2.20), on the basis of (2.18), is bounded for any \(N\); therefore
\[ |y_k(t)| \leq A_1\{\,\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\,\} \tag{2.21} \]
\[ (A_1=\mathrm{const}<\infty) \]
and the magnitude \(A_1\) does not depend on \(N\). Using (2.2) and (2.21), we obtain
\[ |u| \leq \sum_{i=1}^{N}|p_i|\,|y_i| \leq A_1\sum_{i=1}^{N}|p_i|. \]
Since the sum of the series \(\sum_{k=1}^{N}|p_k|\) is bounded for every arbitrarily large \(N\), we finally obtain
\[ |u|\leq A\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\}\qquad (A=\mathrm{const}<\infty), \]
where the constant \(A\) does not depend on the choice of \(N\).
3. Application. Consider the problem of A. M. Letov [5] for the system of differential equations truncated with respect to (1.3)
\[ \frac{dy_i}{dt}=\lambda_i y_i+u_N \qquad (i=1,2,\ldots,N). \tag{3.1} \]
The control \(u_N\) of system (3.1) is sought from the condition of minimizing the integral
\[ I(u_N)=\frac12\int_{0}^{\infty}\left[\left(\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}\right)^2+u_N^2\right]\,dt. \tag{3.2} \]
Let us form the function
\[ H=\frac12\left(\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}\right)^2+\frac12u_N^2+\sum_{i=1}^{N}z_i(\dot{y}_i-\lambda_i y_i-u_N), \]
then, in accordance with the method of solving the problem proposed in [5], we obtain the following system of equations:
\[ \dot{y}_i=\lambda_i y_i+\sum_{i=1}^{N}z_i, \]
\[ (i,j=1,2,\ldots,N) \tag{3.3} \]
\[ \dot{z}_j=\frac{1}{\Delta'(\lambda_j)} \left(\sum_{k=1}^{N}\frac{y_k(t)}{\Delta'(\lambda_k)}\right)-\lambda_j z_j. \]
The characteristic determinant of system (3.3) has the form
\[ \Delta= \left| \begin{array}{cccccccc} \lambda_1-\mu & 0 & \cdots & 0 & \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)}\\ 0 & \lambda_2-\mu & \cdots & 0 & \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)}\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \lambda_N-\mu & \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)}\\ \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)} & -(\lambda_1+\mu) & 0 & \cdots & 0\\ \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)} & 0 & -(\lambda_2+\mu) & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\ \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)} & 0 & 0 & \cdots & -(\lambda_N-\mu) \end{array} \right|. \]
or
\[ \Delta = \left| \begin{array}{cccccccc} (\lambda_1-\mu) & 0 & \cdots & 0 & \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)} \\ -(\lambda_1-\mu) & (\lambda_2-\mu) & \cdots & 0 & 0 & 0 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ -(\lambda_1-\mu) & 0 & \cdots & (\lambda_N-\mu) & 0 & 0 & \cdots & 0 \\ \dfrac{1}{\Delta'(\lambda_1)} & \dfrac{1}{\Delta'(\lambda_2)} & \cdots & \dfrac{1}{\Delta'(\lambda_N)} & -(\lambda_1+\mu) & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 & +(\lambda_1+\mu) & -(\lambda_2+\mu) & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & 0 & +(\lambda_1+\mu) & 0 & \cdots & -(\lambda_N+\mu) \end{array} \right|. \tag{3.4} \]
Expanding the determinant (3.4) by the elements of the first row and canceling the factor
\[ \prod_{i=1}^{N}(\lambda_i^2-\mu^2), \]
we obtain
\[ \Delta_1(\mu)=1+ \sum_{i=1}^{N} \frac{1}{\Delta'(\lambda_i)(\lambda_i-\mu)} \sum_{j=1}^{N} \frac{1}{\Delta'(\lambda_j)(\lambda_j+\mu)}. \tag{3.5} \]
It follows from (3.5) that the roots of the characteristic determinant (1.8) have the form
\[ \mu_k=\lambda_k+\theta_1(\operatorname{Re}\lambda_{k+1}-\operatorname{Re}\lambda_k) +\theta_2(\operatorname{Im}\lambda_{k+2}-\operatorname{Im}\lambda_k)i, \tag{3.6} \]
where
\[ |\theta_i|\leq 1 \quad (i=1,2). \]
If the value \(\mu\) is sufficiently large and differs from the nearest value \(\lambda_k\) in modulus by a finite amount, i.e.,
\[ |\lambda_k-\mu|=d \quad (d=\mathrm{const}), \tag{3.7} \]
then it can be shown that the series on the right in (3.5) converge in modulus, for any \(N\), to an arbitrarily small quantity \(q\). Therefore, for sufficiently large values of \(k\) in (3.6), the quantities
\[ |\theta_i|\leq \frac{1}{2} \quad (i=1,2). \]
The optimal control of the problem (3.1), (3.2) has the form
\[ u_N=\sum_{i=1}^{N}p_i y_i. \tag{3.8} \]
In the present paper it is shown that the optimal control \(u_N\), for any arbitrarily large \(N\), is bounded by some finite number \(A\) independent of the choice of the number \(N\), i.e.,
\[ |u_N|\leq A\{ \|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\}, \]
where
\[ \|\varphi(\vartheta)\|^{(h)}=\sup [\varphi(\vartheta)][-\tau\leq \vartheta\leq 0]. \]
References
- N. N. Krasovskii, PMM, vol. XXVI, no. 1, 1962.
- S. N. Shimanov, PMM, vol. XXIV, no. 1, 1960.
- S. N. Shimanov, PMM, vol. XXVII, no. 3, 1963.
- E. Pinney, Ordinary Differential-Difference Equations. IL, 1961.
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- A. I. Lur’e, Some Nonlinear Problems in the Theory of Automatic Control. 1951.
Received by the editors
June 5, 1965
Ural State
University