UDC 517.949.22

ON THE QUESTION OF OBSERVABILITY OF SYSTEMS WITH RETARDATION

N. N. Krasovskii, A. B. Kurzhanskii

The question is considered of an operation that singles out the unstable coordinates of a second-order linear system with retardation from an available observation of a linear combination of phase coordinates. The work belongs to the class of problems on controllability and observability of dynamical systems [1, 2].

1. Let the system be given

\[ \frac{dx_1}{dt}=a_{11}x_1(t)+a_{12}x_2(t)+b_{11}x_1(t-h)+b_{12}x_2(t-h), \]

\[ \frac{dx_2}{dt}=a_{21}x_1(t)+a_{22}x_2(t)+b_{21}x_1(t-h)+b_{22}x_2(t-h) \tag{1.1} \]

with constant coefficients and constant retardation \(h>0\). Solutions \(\{x_1(t),x_2(t)\}\) of such a system for \(t\ge t_0=0\) are determined by initial vector-functions \(\varphi(t)=\{\varphi_1(t),\varphi_2(t)\}\), specified on the initial time interval \([-h,0]\). We shall regard the functions \(\varphi(t)\) as elements of the space \(C^{(1)}[-h,0]\) of continuously differentiable functions specified on the interval \([-h,0]\).

Consider the characteristic equation of system (1.1)

\[ \Delta(\lambda)= \left| \begin{array}{cc} a_{11}+b_{11}e^{-\lambda h}-\lambda, & a_{12}+b_{12}e^{-\lambda h}\\ a_{21}+b_{21}e^{-\lambda h}, & a_{22}+b_{22}e^{-\lambda h}-\lambda \end{array} \right|=0. \tag{1.2} \]

It is known that an equation of the form (1.2) has only a finite number \(k\) of roots \(\lambda_i\) for which

\[ \operatorname{Re}\lambda_i>0 \qquad (i=1,\ldots,k). \tag{1.3} \]

To simplify the exposition, suppose that the roots \(\lambda_i\) satisfying condition (1.3) are simple. Then the solution of system (1.1) can be represented in the form [3]

\[ x_s(t)=\sum_{i=1}^{k} d_s^{(i)} y_i(t)+z_s(t)\qquad (s=1,2). \tag{1.4} \]

Here \(y_i(t)=y_i(0)e^{\lambda_i t}\). The quantities \(d_s^{(i)}\) are determined by the equalities

\[ d_1^{(i)}=-(a_{12}+b_{12}e^{-\lambda_i h})\, \frac{1}{\Delta_{2s}(\lambda_i)\,d\Delta(\lambda_i)/d\lambda_i}, \tag{1.5} \]

\[ d_2^{(i)}=(a_{11}+b_{11}e^{-\lambda_i h}-\lambda_i)\, \frac{1}{\Delta_{2s}(\lambda_i)\,d\Delta(\lambda_i)/d\lambda_i}, \tag{1.5} \]

where \(\Delta_{2s}(\lambda_i)\) \((s=1,2)\) is that element of the first row of the matrix (1.2) which is nonzero. The numbers \(y_i(0)\) are determined as follows:

\[ \begin{aligned} y_i(0)=f_{\lambda_i}[\varphi(\vartheta)] &= \\ &=\sum_{s=1}^{2}\Delta_{2s}(\lambda_i) \left[ \varphi_s(0)+ \int_{-h}^{0} e^{-\lambda_i(\vartheta+h)} \sum_{l=1}^{2} b_{sl}\varphi_l(\vartheta)\,d\vartheta \right]. \end{aligned} \tag{1.6} \]

Here, as below, it is assumed that \(-h\le \vartheta \le 0\). The functionals \(f_{\lambda_i}[\varphi(\vartheta)]\) \((i=1,\ldots,k)\) satisfy the conditions

\[ f_{\lambda_i}\{d^{(i)}\exp(-\lambda_i\vartheta)\}=\delta_{ij} \quad (i,j=1,\ldots,k), \]

\[ f_{\lambda_j}[z(t+\vartheta)]\equiv 0 \quad \text{for every } t\ge 0,\ -h\le \vartheta\le 0, \quad \text{if and only if } f_{\lambda_j}[z(\vartheta)]\equiv 0, \]

where \(\delta_{ij}\) is the Kronecker symbol.

In equality (1.4) the functions \(y_i(t)\) denote the proper unstable motions of the original system (1.1) and satisfy the equations

\[ dy_i(t)/dt=\lambda_i y_i(t). \tag{1.7} \]

The function \(z(t)\) is a solution of system (1.1) on the set of initial functions \(z(\vartheta)\) lying in the linear subspace \(\Lambda_k\) of the space \(C^{(1)}[-h,0]\), determined by the relations

\[ f_{\lambda_i}[z(\vartheta)]=0\quad (i=1,\ldots,k). \tag{1.8} \]

Denote the Euclidean norm of the vector \(z(t)\) by the symbol \(\|z(t)\|\). The norm of the continuous function \(z(t+\vartheta)\), given on the interval \([t-h,t]\), taken in the metric of the space \(C[-h,0]\), will be denoted by

\[ \|z(t+\vartheta)\|_{C}^{(h)} =\max_{\vartheta}\{|z_1(t+\vartheta)|,\ |z_2(t+\vartheta)|\}. \]

As shown in [3], the solution \(z(t)\) satisfies the condition

\[ \|z(t)\|<K\exp(-\beta t)\,\|z(\vartheta)\|_{C}^{(h)}. \]

Here \(K,\beta\) are constants independent of \(\varphi(\vartheta)\), \(\beta=q\gamma\), \(0<q<1\), where \(\gamma>0\) is such that \(\operatorname{Re}\lambda_i<-\gamma\) \((i=k+1,\ldots)\).

1. Suppose that on the interval \([0,T]\) of finite length the quantity

\[ \xi(t)=b_1x_1(t)+b_2x_2(t) \]

is known. Then on this interval we have the function

\[ \xi(t)= b_1\sum_{i=1}^{k} d_1^{(i)}y_i(t) + b_2\sum_{i=1}^{k} d_2^{(i)}y_i(t) + b_1z_1(t)+b_2z_2(t). \tag{2.1} \]

Let us formulate the observation problem.

Problem A. Suppose that on the interval \([0,T]\) the function \(\xi(t)\), generated by an arbitrary initial curve \(\varphi(\vartheta)\) from \(C^{(1)}[-h,0]\), is known. It is required to find an operation \(\psi\) acting on the function \(\xi(t)\) \((0 \leq t \leq T)\) in such a way that

\[ \psi[\xi(t)] = y_1(T). \tag{2.2} \]

Owing to the linearity of system (1.1), we seek the operation as a linear one. Namely, we shall seek an operation solving Problem A in the form of a linear functional defined on continuous functions \(\xi(t)\) and satisfying condition (2.2) for arbitrary initial curves \(\varphi(\vartheta)\) from \(C^{(1)}[-h,0]\) and arbitrary numbers \(y_1(T)\). In an analogous way one can seek an operation singling out any of the unstable coordinates

\[ y_i(T)\quad (i=1,\ldots,k) \]

or a linear combination

\[ \sum_{i=1}^{k} p_i y_i(T) \]

of these coordinates. The extraction of the unstable coordinates proper turns out to be important, for example, in constructing a control action stabilizing system (1.1) (see [4]).

We shall call a system of the form (1.1) observable with respect to a coordinate if it is possible to specify an interval \([0,T]\) of finite length such that there exists an operation solving Problem A for this coordinate. In the present paper conditions are given for observability of system (1.1) with respect to the coordinates \(y_i(t)\).

1. Let us formulate these conditions.

Theorem 3.1. System (1.1) is observable with respect to the coordinate \(y_i(t)\), \((i=1,\ldots,k)\), if and only if

\[ b_1 d_1^{(i)} + b_2 d_2^{(i)} \ne 0. \tag{3.1} \]

Here we note that the numbers \(d_1^{(i)}, d_2^{(i)}\) are the components of the vectors \(d^{(i)}\)—solutions of the systems of equations

\[ (A + B \exp(-\lambda_i h)-\lambda_i E)d^{(i)} = 0 \quad (i=1,\ldots,k). \]

Conditions (2.3), thus, mean that the vector \(b\) is not orthogonal to any of the vectors \(d^{(i)}\) corresponding to roots \(\lambda_i\) with positive real part of the characteristic equation of the observed system.

Let us prove the necessity of the conditions of the theorem. Consider expression (2.1). The existence of a linear operation \(\psi\) singling out the coordinate \(y_i(T)\) then means that the conditions

\[ \psi[(b_1 d_1^{(j)} + b_2 d_2^{(j)})\exp(\lambda_j(t-T))] = \delta_{ij} \quad (j=1,\ldots,k), \tag{3.2} \]

\[ \psi(b_1 z_1(t) + b_2 z_2(t)) = 0 \]

are satisfied for all functions \(z(t)\) generated by initial curves belonging to the subspace \(\Lambda_k\) (1.8). The latter conditions are compiled with account taken of relations (1.7). Suppose \(b_1 d_1^{(i)} + b_2 d_2^{(i)}=0\). Then, in view of the linearity of the operation \(\psi\), we have

\[ \psi[(b_1 d_1^{(i)} + b_2 d_2^{(i)})\exp(\lambda_i(t-T))] = 0, \]

which contradicts condition (3.2). The necessity of the conditions of the theorem is proved.

Let us pass to the question of the sufficiency of the conditions of Theorem 3.1. Consider problem A for the first coordinate. To this end, write an expression analogous to (2.1), but with only the coordinate \(y_1(t)\) singled out:

\[ \xi(t)=b_1x_1(t)+b_2x_2(t)=(b_1d_1^{(1)}+b_2d_2^{(1)})y_1(t)+b_1z_1(t)+b_2z_2(t). \]

Here the quantity \(z(t)\) is the solution of system (1.1) generated by the initial curve \(z(\vartheta)\), lying in the subspace \(\Lambda_1\), defined by the relation

\[ f_{\lambda_1}[z(\vartheta)]=0,\quad z(\vartheta)\in C^{(1)}[-h,0]. \]

We initially assume the root \(\lambda_1\) to be real and assume that \(\operatorname{Re}\lambda_2<\operatorname{Re}\lambda_1\). The linear functional \(\psi\) must satisfy the conditions

\[ \begin{aligned} \psi[(b_1d_1^{(1)}+b_2d_2^{(1)})\exp(\lambda_1(t-T))]&=1,\\ \psi[b_1z_1(t)+b_2z_2(t)]&=0 \end{aligned} \tag{3.3} \]

for all \(z(\vartheta)\) from \(\Lambda_1\). In the further considerations, however, we shall seek such a functional \(\psi^0\) for which the first of equalities (3.3) has the form

\[ \psi^0[(b_1d_1^{(1)}+b_2d_2^{(1)})\exp\lambda_1t]=1, \]

while the second remains unchanged. In the case when the number \(T\) has been found, the functional \(\psi\) is obtained from the functional \(\psi^0\) by stretching by a factor of \(\exp\lambda_1T\), \(\psi=\exp(\lambda_1T)\psi^0\). The functional \(\psi^0\) then singles out the quantity \(y_1(0)\).

Let us note at once that functions of the form \(b_1z_1(t)+b_2z_2(t)\) form a linear set. Denote it by the symbol \(\Omega\). Thus, in fact, we have the problem of constructing a linear functional which takes the value 1 on the element \((b_1d_1^{(1)}+b_2d_2^{(1)})e^{\lambda_1t}\) and vanishes on \(\Omega\). Consequently, it is necessary next to show that condition (3.1) is sufficient for the solvability of the indicated problem (3.3). Problem (3.3) will be solvable if the functional \(\psi^0\), defined on the functions \((b_1d_1^{(1)}+b_2d_2^{(1)})\exp\lambda_1t,\ b_1z_1(t)+b_2z_2(t)\in\Omega\), considered as elements of some functional space \(B\), can be extended to the whole space. For this, in turn, it is necessary and sufficient [5] that

\[ \rho[(b_1d_1^{(1)}+b_2d_2^{(1)})e^{\lambda_1t},\Omega]_B=d>0. \tag{3.4} \]

Here the symbol \(\rho[\ ]\) denotes the distance between the elements \((b_1d_1^{(1)}+b_2d_2^{(1)})e^{\lambda_1t}\) and the set \(\Omega\) in the metric of the space \(B\). When condition (3.4) is satisfied, the norm \(\|\psi\|_{B^*}\) of the functional \(\psi^0\) in the conjugate metric is equal to

\[ \|\psi^0\|_{B^*}=\frac{1}{d}. \tag{3.5} \]

1. We shall prove relation (3.4) for the space \(C[0,T]\) of continuous functions considered on the interval \(0\leq t\leq T\).

Let us note that equation (1.1) can always be orthogonally transformed so that the vector \(b\) is directed along the axis \(Ox_1\). Let this be so. Then relation (3.4) will mean that

\[ \inf\rho[b_1d_1^{(1)}\exp\lambda_1t,\ b_1z_1(t)]_{C[0,T]}=d>0\quad (0\leq t\leq T). \tag{4.1} \]

Here the infimum is taken over all initial curves \(z(\vartheta)\) lying in \(\Lambda_1\).

Suppose that (4.1) is not satisfied on \([0,3h]\). Then for any \(\varepsilon>0\) there exists an initial curve \(z^{(\varepsilon)}(\vartheta)\in\Lambda_1\) such that

\[ \left||d_1^{(1)}|\exp \lambda_1 t-|z^{(\varepsilon)}(t)|\right|<\varepsilon \tag{4.2} \]

for \(0\le t\le 3h\).

Here we consider two cases. Suppose first that there exists a finite number \(N>0\) such that inequality (4.2) is satisfied for any \(\varepsilon>0\) only by means of those functions \(z^{(\varepsilon)}(t)\in\Lambda_1\), considered on the interval \(h\le t\le 2h\), which satisfy the condition \(\|z^{(\varepsilon)}(2h+\vartheta)\|_{C}^{(h)}<N\). Then, noting that

\[ \|z(t)\|<KN\exp \operatorname{Re}\lambda_2(t-2h),\qquad (\operatorname{Re}\lambda_2<\lambda_1,\quad t\ge 2h), \]

we obtain that, for \(t\ge T\), the inequality

\[ |b_1z_1(t)|\le \frac{1}{2}\,|b_1d_1^{(1)}|\exp \lambda_1 t \]

is satisfied if the number \(T\) is determined as follows:

\[ T=\frac{1}{\lambda_1-\operatorname{Re}\lambda_2} \left|\ln\frac{2KN}{|d_1^{(1)}|}\right|. \]

Thus, in the present case condition (4.1) is valid for this choice of \(T\).

Consider the second case. Suppose the number \(N\) indicated above does not exist. Then, taking a sequence of positive numbers \(\varepsilon_i\to0\) as \(i\to\infty\), we obtain that for each \(\varepsilon_i\) there exists a function \(z_1^{(i)}(t)\) from the interval \([h,3h]\), generated by an initial curve \(z^{(i)}(\vartheta)\in\Lambda_1\), for which condition (4.2) is satisfied, with \(\varepsilon_i\) in place of \(\varepsilon\) and the function \(z_1^{(i)}(t)\) in place of \(z^{(\varepsilon)}(t)\). Moreover, we also obtain that

\[ \|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)} =\max |z_2^{(i)}(2h+\vartheta)|\to\infty \quad (-h\le \vartheta<0)\quad \text{as } i\to\infty . \]

Make in system (1.1) the change of variables

\[ u_1^{(i)}=\frac{z_1^{(i)}(t)} {\|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}}, \qquad u_2^{(i)}=\frac{z_2^{(i)}(t)} {\|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}}, \]

taking into account that \(z^{(i)}(t)\) is the solution of this system generated by the function \(z^{(i)}(\vartheta)\in C_h^{(1)}[-h,0]\). Then we obtain the equations

\[ \frac{du_1^{(i)}}{dt} =\gamma_1^{(i)}(t)+a_{12}u_2^{(i)}(t)+b_{12}u_2^{(i)}(t-h), \]

\[ \frac{du_2^{(i)}}{dt} =\gamma_2^{(i)}(t)+a_{22}u_2^{(i)}(t)+b_{22}u_2^{(i)}(t-h). \tag{4.3} \]

Here \(\gamma_j^{(i)}(t)=a_{j1}u_1^{(i)}(t)+b_{j1}u_1^{(i)}(t-h)\) \((j=1,2)\). If we now take into account inequality (4.2), then the quantities \(\gamma_j^{(i)}(t)\) can be estimated on the interval \(h\le t\le 3h\) by means of the relation

\[ |\gamma_j^{(i)}(t)| < (|a_{j1}|+|b_{j1}|)(1+d_1^{(i)}\exp 3\lambda_1 h)\, \frac{1}{\|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}} = \]
\[ = Q[\|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}]^{-1}. \quad (Q=\mathrm{const}). \tag{4.4} \]

Introduce the notation

\[ \varphi_1^{(i)}(t)=a_{12}u_2^{(i)}(t)+b_{12}u_2^{(i)}(t-h). \]

Integrating the first of equations (4.3), we obtain, taking into account inequalities (4.4),

\[ \int_h^t \varphi_1^{(i)}(\tau)\,d\tau < (2d_1\exp 3\lambda_1 h+3h)\, \frac{1}{\|z_2(2h+\vartheta)\|_{C}^{(h)}} = \]
\[ = Q_1[\|z_2(2h+\vartheta)\|_{C}^{(h)}]^{-1} \quad (h\leq t\leq 3h). \tag{4.5} \]

Let \(b_{12}\ne 0\). Then for the second equation of system (4.3) we obtain

\[ \frac{du_2^{(i)}}{dt}=\varphi_2^{(i)}(t)+\eta u_2^{(i)}(t). \tag{4.6} \]

Here the notation

\[ \eta=a_{22}-\frac{a_{12}}{b_{12}}\,b_{22},\qquad \varphi_2^{(i)}(t)=\gamma_2^{(i)}(t)+\frac{b_{22}}{b_{12}}\,\varphi_1^{(i)}(t) \]

has been introduced.

From (4.4), (4.5) it further follows that

\[ \left|\int_h^t \varphi_2^{(i)}(\tau)\,d\tau\right| < Q_2[\|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}]^{-1}. \]
\[ (Q_2>0,\ h\leq t\leq 3h). \tag{4.7} \]

Let \(\xi_i\) be a point of the interval \([h,2h]\) satisfying the condition

\[ |u_2^{(i)}(\xi_i)|=\max |u_2^{(i)}(t)|=1. \]

In other words,

\[ |u_2^{(i)}(\xi_i)|=\|u_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}. \]

Then the solution of equation (4.6) can be written in the following form:

\[ u_2^{(i)}(t)=u_2^{(i)}(\xi_i)e^{\eta(t-\xi_i)}+ \]
\[ +\int_{\xi_i}^{t}\varphi_2^{(i)}(\tau)\,d\tau +\int_{\xi_i}^{t}\eta e^{\eta(t-\tau)} \int_{\xi_i}^{\tau}\varphi_2^{(i)}(\vartheta)\,d\vartheta\,d\tau. \tag{4.8} \]

The sequence of points \(\xi_i\) is situated on the interval \([h,2h]\). Therefore, passing if necessary to a subsequence of these numbers, we obtain that \(\xi_i\to \xi_0\) as \(i\to\infty\). The indicated subsequence can always be chosen so that \(u_2^{(i)}(\xi_i)\to u_2^{0}(\xi_0)\) as \(i\to\infty\). Further, passing to the limit as \(i\to\infty\) in the integral equation (4.8), we arrive at the relation

\[ u_2^0(t)=\lim_{i\to\infty}u_2^{(i)}(t)=u_2^0(\xi_0)e^{\eta(t-\xi_0)} . \tag{4.9} \]

Here the convergence of the functions \(u_2^{(i)}(t)\) to the function \(u_2^0(t)\) will be uniform in \(t\) for \(\xi_0\leq t\leq 3h\). The latter property follows from the estimates (4.7). Noting that, as \(i\to\infty\), \(u^{(i)}(t)\to 0\) uniformly in \(t\) on \([0,3h]\), we find that on the interval \(2h\leq t\leq 3h\) system (4.3) has a particular solution of the form

\[ u^0(t)= \begin{pmatrix} 0\\ u_2^0(\xi_0) \end{pmatrix} e^{\eta(t-\xi_0)} . \tag{4.10} \]

Equality (4.10) is impossible if \(\eta\) is not a root of the characteristic equation (1.2). If, however, \(\eta\) is a root of equation (1.2), then the reasoning described above is carried out for all initial functions \(z^{(i)}(\vartheta)\) lying in the subspace \(\Lambda^{(2)}\), whose elements satisfy the conditions

\[ f_{\lambda_1}[z(\vartheta)]=0,\qquad f_{\eta}[z(\vartheta)]=0,\qquad z(\vartheta)\in C[-h,0]. \tag{4.11} \]

Then the result obtained in (4.9) means that the function \(u_0(t)\) \((2h\leq t\leq 3h)\) must belong to \(\Lambda^{(2)}\), which is impossible.

Thus, the assumption that relation (4.2) can be realized by means of initial functions \(z^{(i)}(\vartheta)\) for which
\[ \|z_2^{(i)}(2h+\vartheta)\|_{C}^{(h)}\to\infty \quad\text{as } i\to\infty \]
is false. Comparing this conclusion with that obtained earlier for the first case, we find that there exists an interval \([0,T]\) for which condition (4.1) is fulfilled. System (1.1), consequently, turns out to be observable with respect to the coordinate \(y_1(t)\).

If the function \(\xi(t)\) is regarded as an element of the space \(C[0,T]\), then the functional \(\psi\) solving problem A should be sought in the form [5]

\[ \psi[\xi(t)]=\int_0^T \xi(t)\,dv(t), \]

where \(v(t)\) is a function of bounded variation on \([0,T]\). Conditions (3.3) then mean that the functional \(\psi\) separates the element
\[ (b_1d_1^{(1)}+b_2d_2^{(1)})e^{\lambda_1(t-T)} \]
from the linear manifold \(\Omega\). The norm of the functional \(\psi\) is determined by the equality

\[ \|\psi\|= \left\{ \inf \max \left| (b_1d_1^{(1)}+b_2d_2^{(1)})\exp(\lambda_1(t-T)) - (b_1z_1(t)+b_2z_2(t)) \right| \right\}^{-1}, \tag{4.12} \]

where the maximum is taken over all \(0\leq t\leq T\), and the infimum over all \(z(\vartheta)\in\Lambda_1\).

The question of the concrete construction of the operation \(\psi\) is a separate problem and lies beyond the scope of the present work. We note here only the circumstance that the operation \(\psi\), or, in other words, the function \(v(t)\), is in general determined for a known value of the norm
\[ \|\psi\|_{B^*}=\int_0^T |dv| \]
on the extremal element \(b_1z_1^0(t)+b_2z_2^0(t)\) of problem (4.12), analogously to how this is done, for example, when using the \(L\)-problem of moments. Condition (4.1) may, by analogy with [2], be interpreted in such a way that the least intensity \(\|\xi(t)\|\) of the signal

\(\xi(t)\), carrying the value \(y_1(T)=1\), equal to \(d\), is a positive quantity. The difficulty here consists in the fact that the element \(b_1 z_1^0(t)+b_2 z_2^0(t)\) may not be a solution of system (1.1), or may even not be an element of the space \(C[0,T]\).

Let now the root \(\lambda_1\) be complex. In this case we also have the root \(\lambda_2=\overline{\lambda}_1\), conjugate to \(\lambda_1\). We have \(\operatorname{Re}\lambda_1=\operatorname{Re}\lambda_2>0\). Let us now write condition (1.4) with the first two coordinates separated out. Then we obtain that, for \(t\geq 0\),

\[ \xi(t)=(b_1 d_1^{(1)}+b_2 d_2^{(1)})\exp \lambda_1 t\, y_1(0)+ \]
\[ +(b_1 \overline{d}_1^{(1)}+b_2 \overline{d}_2^{(1)})\exp \overline{\lambda}_1 t\, y_2(0)+z(t), \tag{4.13} \]

where the function \(z(t)\) is the solution of system (1.1) generated by the initial curve \(z(\vartheta)\), satisfying the conditions

\[ f_{\lambda_1}(z(\vartheta))=0,\qquad f_{\lambda_2}(z(\vartheta))=f_{\overline{\lambda}_1}(z(\vartheta)). \tag{4.14} \]

We note that the quantities \(d_1^{(1)}, d_2^{(1)}, y_1(T)\) are complex numbers, whereas the sum of the first two terms on the right-hand side of (4.13) is a real quantity. Writing equality (4.13) in terms of real quantities, we obtain

\[ \xi(t)=\exp \mu t\,((b_1\operatorname{Re}d_1^{(1)}+b_2\operatorname{Re}d_2^{(1)})\cos \nu t- \]
\[ -(b_1\operatorname{Im}d_1^{(1)}+b_2\operatorname{Im}d_2^{(1)})\sin \nu t)\,y_1(0)- \]
\[ -\exp \mu t\,((b_1\operatorname{Re}d_1^{(1)}+b_2\operatorname{Re}d_2^{(1)})\sin \nu t+ \]
\[ +(b_1\operatorname{Im}d_1^{(1)}+b_2\operatorname{Im}d_2^{(1)})\cos \nu t)+b_1z_1(t)+b_2z_2(t). \]

Introduce the notation:

\[ b_1\operatorname{Re}d_1^{(1)}+b_2\operatorname{Re}d_2^{(1)}=\alpha, \]

\[ b_1\operatorname{Im}d_1^{(1)}+b_2\operatorname{Im}d_2^{(1)}=\beta. \]

The condition of Theorem 3.1 then means that \(\alpha^2+\beta^2\ne0\). Further, we have

\[ x(t)=\exp \mu t[\cos(\nu t+\varphi)y_1(0)-\sin(\nu t+\varphi)y_2(0)]+ \]
\[ +b_1z_1(t)+b_2z_2(t), \]

where \(\varphi=\operatorname{arc\,tg}\dfrac{\beta}{\alpha}\). We shall show that, under the condition \(\alpha^2+\beta^2\ne0\), there exists an operation \(\psi_1^0\) which separates out the coordinate \(y_1(0)\). This operation will be defined by the equalities

\[ \psi_1^0[\exp \mu t\cos(\nu t+\varphi)]=1, \]

\[ \psi_1^0[\exp \mu t\sin(\nu t+\varphi)]=0, \]

\[ \psi_1^0[b_1z_1(t)+b_2z_2(t)]=0. \]

The operation \(\psi_1\) will exist if the condition is fulfilled

\[ \rho[\exp \mu t\cos(\nu t+\varphi),\ b_1z_1(t)+b_2z_2(t)+ \]

\[ {}+ k \exp \mu t \sin(\nu t+\varphi)] = l > 0, \]

where \(k\) is an arbitrary real number. Assuming that the system has been transformed so that the vector \(b\) is directed along the axis \(Ox_1\), we obtain the condition

\[ \rho[\exp \mu t \cos(\nu t+\varphi),\ b_1 z_1(t)+k\exp \mu t \sin(\nu t+\varphi)] = l > 0, \tag{4.15} \]

where

\[ \varphi=\operatorname{arc\,tg}\frac{b_1\operatorname{Im} d_1^{(1)}}{b_1\operatorname{Re} d_1^{(1)}} . \]

The proof of relation (4.15) uses the scheme given above. Namely, suppose that (4.15) is false on \([0,3h]\). Then, for \(0\le t\le 3h\) and for any \(\varepsilon\), the inequality

\[ \left|\exp \mu t|\cos(\nu t+\varphi)|-|b_1 z_1(t)+k\exp \mu t\sin(\nu t+\varphi)|\right|<\varepsilon \tag{4.16} \]

holds.

If there exists a number \(N>0\) such that inequality (4.16) is satisfied for arbitrary \(\varepsilon\) only by means of functions \(z^{(\varepsilon)}(t)\in \Lambda_2\) (4.14), considered on the interval \(h\le t\le 2h\), which satisfy the condition
\[ \|z^{(\varepsilon)}(2h+\vartheta)\|_{C}^{(h)}<N, \]
then we use the inequality

\[ \|z(t)\|<N\exp[\operatorname{Re}\lambda_3(t-2h)],\qquad \operatorname{Re}\lambda_3<\operatorname{Re}\lambda_1\quad (t\ge 2h). \]

The last inequality allows one to choose a sufficiently large number \(T\) such that on the interval \([0,T]\) there necessarily exists a point \(t^*\), where
\[ \cos(\nu t^*+\varphi)=1,\qquad \sin(\nu t^*+\varphi)=0 \]
and
\[ \exp \mu t^* \ge \frac{1}{2}|b_1 z_1(t^*)|. \]
But then on \([0,T]\) condition (4.16) is fulfilled. If the number \(N\) does not exist, then, following the scheme indicated above, we choose a sequence of positive numbers \(\varepsilon_i\to 0\) and corresponding solutions \(z^{(i)}(t)\) with increasing norms
\[ \|z^{(i)}(2h+\vartheta)\|_{C}^{(h)}\to \infty \]
and numbers \(k_i\), for which inequality (4.16) is valid with \(\varepsilon=\varepsilon_i\). Making in system (1.1) a change of variables of the form

\[ u_j^{(i)}= \frac{z_j^{(i)}(t)} {\|z_j^{(i)}(2h+\vartheta)\|_{C}^{(h)}+|k_i|} \qquad (j=1,2) \]

and investigating the solution of the new system as \(i\to\infty\), we arrive at a contradiction in the same way as in the case of a real root. It follows from what has been said that \(\psi_1^0\) exists.

In an analogous way one can prove the existence of the operation \(\psi_2^0\), which selects the coordinate \(y_2(0)\). Knowing \(\psi_1^0\) and \(\psi_2^0\), it is easy to pass to the operations \(\psi_1\) and \(\psi_2\), which select respectively the quantities \(y_1(T)\) and \(y_2(T)\). This is done according to the equalities

\[ \psi_1=\exp \mu T(\psi_1^0\cos \nu T-\psi_2^0\sin \nu T), \]

\[ \psi_2=\exp \mu T(\psi_1^0\sin \nu T+\psi_2^0\cos \nu T). \]

Remarks:

4.1. Above it was assumed that \(\operatorname{Re}\lambda_1>\operatorname{Re}\lambda_2\). If the indicated condition is not fulfilled and \(\operatorname{Re}\lambda_1\ge \operatorname{Re}\lambda_2\), then, since the roots \(\lambda_i\) are simple, we obtain that the imaginary parts \(\operatorname{Im}\lambda_i\) of these roots are distinct. The separation of the subspaces corresponding to different roots is then carried out by separating the corresponding harmonics, similarly to how this was done above in selecting the operation \(\psi_1\).

4.2. In proving the sufficiency of the conditions of Theorem 3.1 it was assumed that \(b_{12} \ne 0\). If, however, \(b_{12}=0\) or \(b_{12}=a_{12}=0\), then the arguments are carried out analogously to the case considered, and the conclusions of the theorem remain valid.

4.3. In the case when it is necessary to single out the \(i\)-th unstable coordinate \(y_i(t)\), we proceed as follows. Assuming that conditions (3.1) have been fulfilled for \(m\) coordinates \((m \leq i-1)\), we find the corresponding motions \(y_j(t)\) \((j=1,\ldots,m)\) and, subtracting

\[ \sum_{j=1}^{m}\bigl(b_1 d_1^{(j)}+b_2 d_2^{(j)}\bigr)y_j(t) \]

from the quantity \(\xi(t)\), we arrive at the problem of singling out the coordinate \(y_i(t)\) from the quantity

\[ \xi(t)-\sum_{j=1}^{m}\bigl(b_1 d_1^{(j)}+b_2 d_2^{(j)}\bigr)y_j(t). \]

This problem is again solved by the procedure considered in the proof of the theorem.

References

1. Kalman R. E. On the general theory of control systems. Proceedings of the 1st IFAC Congress, 2, Moscow, 1962.

2. Krasovskii N. N. On the theory of controllability and observability of linear dynamical systems. PMM, 28, issue 1, 1964.

3. Shimanov S. N. Differential equations, vol. I, No. 1, 102—116, 1965.

4. Krasovskii N. N., Osipov Yu. S. Izv. Acad. Sci. USSR, Technical Cybernetics, No. 6, 1963.

5. Kantorovich L. V., Akilov G. P. Functional Analysis in Normed Spaces. Fizmatgiz, 1959.

Received by the editors
31 July 1965

Ural State University