ON A THEORY OF LINEAR DIFFERENTIAL EQUATIONS WITH AFTER-EFFECT

S. N. SHIMANOV

A system of linear differential equations with after-effect (or delay) is considered. The study of systems of this kind is of interest, in particular, for automatic control systems.

The investigation is based on the method, proposed by N. N. Krasovskii [1], of considering equations with time delay in the functional space of continuous functions \(C[-\tau, 0]\).

It is shown that in the space of continuous functions \(C[-\tau, 0]\), in which the solutions of system (1.1) are considered, a finite-dimensional basis can be chosen on which the motion of system (1.1) is described by a system of ordinary differential equations in canonical form and with constant coefficients. In the additional subspace, the norm of every solution decreases according to an exponential law with an arbitrarily large exponent, if a sufficient number of eigen and associated elements entering into the basis is chosen.

The investigation is based on certain facts valid for the original and adjoint systems, the general form of the first integral, and the general solution of system (1.1).

1. Statement of the problem. Equivalent system of “ordinary” differential equations. Consider a linear system of equations with after-effect of the form

\[ \frac{d x_s(t)}{dt} = \sum_{j=1}^{n} \int_{-\tau}^{0} x_j(t+\vartheta)\,d\eta_{sj}(\vartheta) \qquad (s=1,\ldots,n). \tag{1.1} \]

Here the integral on the right-hand side of equation (1.1) is understood in the Stieltjes sense [2]. In this form equations (1.1) were considered by A. D. Myshkis [2]. The functions \(\eta_{sj}(\vartheta)\) are functions of bounded variation defined on the interval \([-\tau,0]\). In particular, if the Stieltjes measure is chosen according to the condition \(d\eta_{sj}(\vartheta)=\zeta_{sj}(\vartheta)d\vartheta\) for \(\vartheta\ne0\), \(\vartheta\ne\tau\), \(d\eta_{sj}(0)=a_{sj}\) and \(d\eta_{sj}(-\tau)=b_{sj}\), then system (1.1) can be written in the form

\[ \frac{d x_s(t)}{dt} = \sum_{j=1}^{n} \left( a_{sj}x_j(t) + b_{sj}x_j(t-\tau) + \int_{-\tau}^{0} x_j(t+\vartheta)\zeta_{sj}(\vartheta)\,d\vartheta \right) \tag{1.2} \]

\[ (s=1,\ldots,n), \]

where \(a_{sj}\), \(b_{sj}\) are constants, and \(\zeta_{sj}(\vartheta)\) are continuous functions on the interval \([-\tau,0]\).

In what follows, for system (1.1) we shall use vector notation. Let

\[ x(t)=\{x_s(t),\ s=1,\ldots,n\},\qquad \|d\eta_{sj}(\vartheta)\|_1^n=d\eta(\vartheta). \]

Then system (1.1) will be written in the form

\[ \frac{dx(t)}{dt}=\int_{-\tau}^{0} x(t+\vartheta)\,d\eta(\vartheta). \tag{1.3} \]

The equivalent system of “ordinary” differential equations with an operator right-hand side is [1]:

\[ \frac{dx_t(\vartheta)}{dt}=Ax_t(\vartheta), \tag{1.4} \]

where \(x_t(\vartheta)=x(t+\vartheta)\), \(0\geq \vartheta \geq -\tau\), and the operator \(A\) is defined as follows:

\[ y(\vartheta)=Ax(\vartheta)= \begin{cases} \dfrac{dx_k(\vartheta)}{d\vartheta}, & -\tau<\vartheta<0\quad (k=1,\ldots,n),\\[1.2em] \displaystyle \sum_{j=1}^{n}\int_{-\tau}^{0} x_j(t+\vartheta)\,d\eta_{kj}(\vartheta), & \vartheta=0. \end{cases} \tag{1.5} \]

Consider some solution of systems (1.1), (1.4):
\(x(t+\vartheta)=x(x_0(\vartheta),t_0,t+\vartheta)\), where \(x_0(\vartheta)\) is the initial vector-function \((x(x_0(\vartheta),t_0,t_0+\vartheta)=x_0(\vartheta))\), \(t_0\) is the initial time, and \(x_0(\vartheta)\in C[-\tau,0]\).

For fixed \(t\geq t_0>0\), one may regard the element of the solution \(x(t+\vartheta)\), \(-\tau\leq \vartheta\leq 0\), as the image of the initial element \(x_0(\vartheta)\in C[-\tau,0]\) under the mapping

\[ x_t(\vartheta)=x(t+\vartheta)=T(t,t_0)x_0(\vartheta). \tag{1.6} \]

Let us state several basic properties of the operator \(T(t,t_0)\). The operator \(T(t,t_0)\) is linear, completely continuous, and possesses the semigroup property with respect to \(t\):

\[ T(t_1+t_2,t_0)=T(t_1+t_2,t_1)\cdot T(t_1,t_0) \]

\[ (t_1>t_0,\quad t_2>0). \tag{1.7} \]

For every solution of system (1.1) the estimate holds

\[ \|x(x_0(\vartheta),t_0,t+\vartheta)\|_{-\tau 0} \leq \|x_0(\vartheta)\|_{-\tau 0}\exp\{L(t-t_0)\} \tag{1.8} \]

for \(t\geq t_0\), where

\[ \|x(\vartheta)\|_{-\tau 0} = \sup\{|x_1(\vartheta)|,\ldots,|x_n(\vartheta)|\} \quad\text{for }-\tau\leq \vartheta\leq 0, \]

\[ L=\sup \left\{ \frac{ \left|\displaystyle\sum_{j=1}^{n}\int_{-\tau}^{0} x_j(\vartheta)\,d\eta_{sj}(\vartheta)\right| }{ \|x(\vartheta)\|_{-\tau 0} } \quad (s=1,\ldots,n) \right\}. \tag{1.9} \]

It follows from (1.8) that the norm of the operator \(T(t,t_0)\) has the estimate

\[ \|T(t,t_0)\|\leq \exp\{L(t-t_0)\},\qquad t>t_0, \tag{1.10} \]

where

\[ \|T(t,t_0)\|=\sup \frac{\|T(t,t_0)x(\vartheta)\|_{-\tau0}}{\|x(\vartheta)\|_{-\tau0}} . \]

The limit of \(T(t,t_0)\) as \(t\to t_0\) in the strong topology [1] is the identity operator \(J\) in the space \(C[-\tau,0]\), \((Jx(\vartheta)=x(\vartheta))\), i.e.,

\[ \lim\|T(t,t_0)x(\vartheta)-x(\vartheta)\|_{-\tau0}=0, \tag{1.11} \]

\[ t\to t_0\quad (t>t_0). \]

The operator \(A\) (1.5) is defined by the relation

\[ Ax(\vartheta)=\lim_{\xi\to 0}\left[\frac{1}{\xi}\bigl(T(t+\xi,t_0)x(\vartheta)-T(t,t_0)x(\vartheta)\bigr)\right]. \]

Thus, the operator \(A\) is the infinitesimal generating operator for the operator \(T(t,t_0)\).

Our task consists in carrying over certain known facts of the general theory of ordinary differential equations to systems of differential equations with delay.

2. The adjoint system and first integrals of equations (1.1)

For systems of ordinary differential equations with constant coefficients there always exists a so-called adjoint system. Every particular solution makes it possible to write a first integral of the original system of equations, since the scalar product of a vector particular solution of the original system and a vector particular solution of the adjoint system does not depend on time. An analogous fact also holds for linear systems with after-effect [3–5].

Consider the following system of linear “ordinary” differential equations with an operator right-hand side:

\[ \frac{dy_t(\vartheta)}{dt}=-A^*y_t(\vartheta), \tag{2.1} \]

where \(y_t(\vartheta)=y(t+\vartheta)\).

The operator \(A^*\) is defined by the relation

\[ y(\vartheta)=-A^*x(\vartheta)= \begin{cases} \dfrac{dx_k(\vartheta)}{d\vartheta}, & 0<\vartheta\leq \tau\quad (k=1,\ldots,n),\\[1.2ex] -\displaystyle\int_{\tau}^{0}\sum_{j=1}^{n} x_j(\vartheta)\,d\eta_{jk}(-\vartheta), & \vartheta=0, \end{cases} \tag{2.2} \]

where the \(\eta_{jk}(\vartheta)\) are the same as in (1.1), (1.5). We shall call system (2.1) the adjoint to system (1.4). Accordingly, the system

\[ \frac{dy_s}{dt} = -\sum_{j=1}^{n}\int_{\tau}^{0} y_j(t+\vartheta)\,d\eta_{js}(-\vartheta) \qquad (s=1,\ldots,n) \tag{2.3} \]

will be called the adjoint to system (1.1).

The system (2.3) is a system with advance in time. All its solutions are determined in the negative direction of change of time. However, there exists a countable set of particular solutions defined on the entire numerical axis \(t\). The system of “ordinary” equations (2.1), (2.2) is equivalent to the system (2.3). If \(y(y_0(\vartheta), t_0, t)\) is a solution of the system (2.3), then \(y(y_0(\vartheta), t_0, t+\vartheta)\), \(0\leq \vartheta\leq \tau\), will be a solution of the system (2.1), and conversely.

Let us form the “scalar” product of two vectors \(x(\vartheta)\) \((-\tau\leq \vartheta\leq 0)\) and \(y(\vartheta)\) \((0\leq \vartheta\leq \tau)\):

\[ (x(\vartheta),y(\vartheta))=\sum_{j=1}^{n}x_j(0)y_j(0) -\sum_{l,j=1}^{n}\int_{-\tau}^{0}\left[\int_{0}^{\vartheta}x_l(\xi)y_j(-\vartheta+\xi)\,d\xi\right]d\eta_{jl}(\vartheta). \]

By direct calculation we find that

\[ (Ax(\vartheta),y(\vartheta))=(x(\vartheta),A^*y(\vartheta)). \tag{2.5} \]

It follows from (2.5) that if \(x_t(\vartheta)\) is a solution of system (1.4), and \(y_t(\vartheta)\) is a solution of system (2.1), continued in the direction of increasing \(t\), then

\[ \frac{d(x_t(\vartheta),y_t(\vartheta))}{dt}=0. \tag{2.6} \]

Hence

\[ (x_t(\vartheta),y_t(\vartheta))=\mathrm{const} \tag{2.7} \]

is a first integral of system (1.4) and system (1.1) when \(x_t(\vartheta)=x(t+\vartheta)\) and \(y_t(\vartheta)=y(t+\vartheta)\).

In particular, the scalar product of the vectors \(x(\vartheta)\) \((-\tau\leq \vartheta\leq 0)\) and \(y(\vartheta)\) \((0\leq \vartheta\leq \tau)\) for system (1.2) will be

\[ (x(\vartheta),y(\vartheta))=\sum_{j=1}^{n}x_j(0)y_j(0) +\sum_{j=1}^{n}\sum_{l=1}^{n}\int_{0}^{\tau}x_l(\xi-\tau)y_j(\xi)b_{jl}\,d\xi- \]

\[ -\sum_{j=1}^{n}\sum_{l=1}^{n}\int_{-\tau}^{0} \left[\int_{0}^{\vartheta}x_l(\xi)y_j(-\vartheta+\xi)\zeta_{jl}(\vartheta)\right]d\xi\,d\vartheta. \]

Consider the characteristic equation of systems (1.1) and (1.4)

\[ \Delta(\lambda)\equiv \det\left|-\delta_{ij}\lambda+\int_{-\tau}^{0}e^{\lambda\vartheta}\,d\eta_{ij}(\vartheta)\right|=0. \tag{2.8} \]

Let us also write the characteristic equation of system (1.3), or (2.1),

\[ \Delta^*(\lambda)\equiv \det\left|-\delta_{ij}\mu+\int_{\tau}^{0}e^{\mu\vartheta}\,d\eta_{ji}(-\vartheta)\right|=0. \tag{2.9} \]

It is easy to observe that if \(\lambda_\sigma\) is a root of equation (2.8), then \(\mu_\sigma=-\lambda_\sigma\) is a root of equation (2.9). The roots \(\lambda_\sigma\) and \(\mu_\sigma\) have the same multiplicity, and the matrices of the form (2.8) and (2.9) corresponding to the values \(\lambda=\lambda_\sigma\) and \(\mu=\mu_\sigma\) have the same rank. The complex numbers \(\{\lambda_\sigma\}\) constitute the spectrum of the operator \(A\), and the numbers \(\{\mu_\sigma\}=\{-\lambda_\sigma\}\) constitute the spectrum of the operator \(-A^*\).

Consider the equation

\[ (A-\lambda_\sigma J)x(\vartheta)=0,\qquad -\tau\leq \vartheta\leq 0. \tag{2.10} \]

It can be shown that the solution of equation (2.10) has the form

\[ x_\sigma = a_\sigma e^{\frac{\lambda}{\sigma}\vartheta}, \quad -\tau \leqslant \vartheta \leqslant 0, \tag{2.11} \]

where \(a_\sigma\) are constants. Then \(x_\sigma(t+\vartheta)\), \(-\tau \leqslant \vartheta \leqslant 0\), satisfies equation (1.4) and equation (1.1). Similarly, for the adjoint operator \(-A^*\) (2.2) we find that the equation

\[ (-A^*-\mu_\sigma J)y(\vartheta)=0,\quad 0\leqslant \vartheta \leqslant \tau, \tag{2.12} \]

has a solution of the form

\[ y_\sigma(\vartheta)=b_\sigma e^{\lambda_\sigma \vartheta}, \quad 0\leqslant \vartheta \leqslant \tau, \tag{2.13} \]

and \(y_\sigma(t+\vartheta)\) will satisfy equations (2.1) and (2.3) for \(-\infty < t < +\infty\).

Next consider two solutions of equations (2.10) and (2.12): \(x_\sigma(\vartheta)\) and \(y_k(\vartheta)\), corresponding to different \(\lambda_\sigma\) and \(\mu_k=-\lambda_k \ne -\lambda_\sigma\). Then

\[ (x_\sigma(\vartheta),\, y_k(\vartheta))=0. \tag{2.14} \]

Indeed, we have

\[ (A-\lambda_\sigma J)x_\sigma(\vartheta)=0,\quad (-A^*-\mu_k J)y_k(\vartheta)=0. \]

Multiplying the first identity scalarly on the right by \(y_k(\vartheta)\), and multiplying the second identity scalarly on the left by \(x_\sigma(\vartheta)\), we obtain

\[ (Ax_\sigma(\vartheta),\, y_k(\vartheta))-\lambda_\sigma (x_\sigma(\vartheta),\, y_k(\vartheta))=0, \]

\[ (x_\sigma(\vartheta),\, -A^*y_k(\vartheta))-\mu_k(x_\sigma(\vartheta),\, y_k(\vartheta))=0. \]

Adding the first and second identities and taking account of relation (2.5), we find

\[ (-\lambda_\sigma-\lambda_k)(x_\sigma(\vartheta),\, y_k(\vartheta))=0. \]

And since \(\lambda_\sigma-\lambda_k\ne 0\), from this we obtain (2.14).

Consider the equation

\[ (A-\lambda_\sigma J)x(\vartheta)=z(\vartheta),\quad -\tau \leqslant \vartheta \leqslant 0, \tag{2.15} \]

where \(z(\vartheta)\in C[-\tau,0]\).

In order for equation (2.15) to be solvable with respect to \(x(\vartheta)\), it is necessary and sufficient that the conditions

\[ (z(\vartheta),\, y_\sigma^{(j)}(\vartheta))=0 \quad (j=1,\ldots,m). \tag{2.16} \]

be fulfilled.

Here \(y^{(j)}\) is a basis of the subspace of the space \(C[0,+\tau]\) determined by the condition

\[ (-A^*+\lambda_\sigma J)y(\vartheta)=0,\quad 0\leqslant \vartheta \leqslant \tau. \tag{2.17} \]

Indeed, solving equation (2.15), we find that for \(-\tau \leq \vartheta < 0\) \(x(\vartheta)\) has the form

\[ x(\vartheta)=\{x_i(\vartheta),\ i=1,\ldots,n\}, \]

\[ x_i(\vartheta)=x_i^* e^{\lambda_\sigma \vartheta} +\int_0^\vartheta e^{\lambda_\sigma(\vartheta-\xi)} z_i(\xi)\,d\xi, \]

\[ -\tau \leqslant \vartheta \leqslant 0 \quad (i=1,\ldots,n), \]

where \(x_i^*\) are arbitrary constants. For \(\vartheta=0\), from (2.15) we obtain a system of nonhomogeneous algebraic equations with respect to \(x_i^*\):

\[ \sum_{j=1}^{n} x_j^* \int_{-\tau}^{0} e^{\lambda_\sigma \vartheta}\, d\eta_{kj}(\vartheta) - x_k^* \lambda_\sigma = z_k(0) - \]

\[ {}-\sum_{j=1}^{n}\int_{-\tau}^{0} \left[\int_{0}^{\vartheta} e^{\lambda_\sigma(\vartheta-\xi)} z_j(\xi)\,d\xi\right] d\eta_{kj}(\vartheta) \tag{2.18} \]

\[ (k=1,\ldots,n). \]

Let, further, a complete system \(b_1^{(j)},\ldots,b_n^{(j)}\) \((j=1,\ldots,m)\) of \(m\) linearly independent solutions of the system of homogeneous algebraic equations

\[ \sum_{j=1}^{n} b_j \int_{-\tau}^{0} e^{\lambda_\sigma \vartheta}\, d\eta_{jk}(\vartheta) - b_k \lambda_\sigma = 0 \qquad (k=1,\ldots,n). \tag{2.19} \]

Then the system (2.18), and hence also the system (2.15), is solvable provided the \(m\) conditions

\[ \sum_{k=1}^{n} b_k^{(j)} \left\{ z_k(0)- \sum_{l=1}^{n}\int_{-\tau}^{0} \left[\int_{0}^{\vartheta} e^{\lambda_\sigma(\vartheta-\xi)} z_l(\xi)\,d\xi\right] d\eta_{kl}(\vartheta) \right\} \tag{2.20} \]

\[ (j=1,\ldots,m). \]

Let us next show that \(y_\sigma^{(j)}=\{b_k^{(j)}e^{-\lambda_\sigma\vartheta},\ k=1,\ldots,n\}\) form a basis of the subspace of the space \(C[0,\tau]\) defined by equation (2.17). Solving equation (2.17), for \(0\leq \vartheta \leq \tau\) we obtain

\[ y_l(\vartheta)=y_l^* e^{-\lambda_\sigma\vartheta}, \qquad 0\leq \vartheta \leq \tau \quad (l=1,\ldots,n), \]

where \(y_l^*\) are arbitrary constants. For \(\vartheta=0\) we obtain a system of linear homogeneous equations with respect to

\[ -\sum_{l=1}^{n} y_l^* \int_{\tau}^{0} e^{-\lambda_\sigma\vartheta}\, d\eta_{lk}(-\vartheta) +\lambda_\sigma y_k^* = 0 \tag{2.21} \]

\[ (k=1,\ldots,n). \]

Comparing equations (2.21) with equations (2.19), we find that a complete system of independent particular solutions will be

\[ \{b_l^{(j)},\ l=1,\ldots,n\}. \]

Therefore \(\{y_l^{(j)}(\vartheta)\}=\{b_l^{(j)}e^{-\lambda_\sigma\vartheta},\ l=1,\ldots,n\}\) constitute a complete system of independent vectors of the subspace defined by equation (2.17). Thus, the solvability condition (2.16) for equation (2.15) is proved.

Consider a root \(\lambda_\sigma\) of equation (2.8). If \(\lambda_\sigma\) is a simple root, then to it there corresponds a unique vector satisfying equation (2.10). If \(\lambda_\sigma\) is a multiple root, then we shall write it in the sequence \(\{\lambda_\sigma\}\) as many times as its multiplicity. To the multiple root \(\lambda_\sigma\) there will correspond

there correspond either eigenvectors, or eigenvectors and associated vectors decomposing into Jordan chains of the form: to the roots \(\lambda_\sigma=\lambda_{\sigma+1}=\cdots=\lambda_{\sigma+m}\) there correspond vectors satisfying the equations

\[ (A-\lambda_\sigma J)x_\sigma(\vartheta)=0, \]

\[ (A-\lambda_\sigma J)x_{\sigma+1}(\vartheta)=x_\sigma(\vartheta), \tag{2.22} \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ (A-\lambda_\sigma J)x_{\sigma+m}(\vartheta)=x_{\sigma+m-1}(\vartheta). \]

The number of elements \(m\) in a chain is not greater than the multiplicity of the root \(\lambda_\sigma\); the number of chains is likewise not greater than the multiplicity of the root \(\lambda_\sigma\). But the total number of eigen and associated elements corresponding to the multiple root \(\lambda_\sigma\), in all chains, is equal to the multiplicity of the root \(\lambda_\sigma\).

The same can be said also concerning the eigenvectors and associated vectors of the adjoint system (2.12). Namely, putting \(\mu_\sigma=-\lambda_\sigma\), we find that to the root \(\mu_\sigma\) there correspond as many Jordan chains as to the root \(\lambda_\sigma\), and each chain contains as many vectors as there are in the corresponding chain (2.22). The latter follows from the fact that the matrix \(\Delta(\lambda_\sigma)\) of the form (2.8), which determines the structure of the eigenvectors and associated vectors of equation (2.10), differs from the matrix \(\Delta^*(\mu_\sigma)=\Delta^*(-\lambda_\sigma)\) by transposition of the elements relative to the main diagonal and by a change of sign in all elements.

To the roots \(-\lambda_\sigma=-\lambda_{\sigma+1}=\cdots=-\lambda_{\sigma+m}\) there will correspond a Jordan chain of vectors \(y_\sigma(\vartheta),\ldots,y_{\sigma+m}(\vartheta)\), \(0\leqslant\vartheta\leqslant\tau\), satisfying the equations

\[ (-A^*+\lambda_\sigma J)y_\sigma(\vartheta)=0, \]

\[ (-A^*+\lambda_\sigma J)y_{\sigma+1}(\vartheta)=y_\sigma(\vartheta), \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \tag{2.23} \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ (-A^*+\lambda_\sigma J)y_{\sigma+m}(\vartheta)=y_{\sigma+m-1}(\vartheta). \]

If \(\lambda_\sigma\) is a simple root, then the vector \(x_\sigma(\vartheta)\) satisfies the conditions

\[ (x_\sigma(\vartheta),y_j(\vartheta))= \begin{cases} 1, & \sigma=j,\\ 0, & \sigma\ne j \end{cases} \tag{2.24} \]

\[ (j,\sigma=1,2,\ldots), \]

where \(\{y_j(\vartheta)\}\) is the collection of eigenvectors and associated vectors of the operator \(-A^*\). For the case of the eigenvector \(y_j(\vartheta)\), formula (2.24) follows from (2.14). We shall show that formula (2.24) also holds for associated elements constituting a chain of the form (2.23) when \(\sigma=j\) \((\sigma,\sigma+1,\ldots,\sigma+m)\).

Multiplying equation (2.10) by \(y_{j+k}(\vartheta)\) on the right scalarly, we obtain

\[ (Ax_\sigma(\vartheta),y_{j+k}(\vartheta))=\lambda_\sigma(x_\sigma(\vartheta),y_{j+k}(\vartheta)). \tag{2.25} \]

Multiplying (2.23) for \(j=\sigma\) scalarly on the left by \(x_\sigma(\vartheta)\), we obtain

\[ (x_\sigma(\vartheta), -A^*y_j(\vartheta))=-\lambda_j(x_\sigma(\vartheta), y_j(\vartheta)), \]

\[ (x_\sigma(\vartheta), -A^*y_{j+1}(\vartheta)) = -\lambda_j(x_\sigma(\vartheta), y_{j+1}(\vartheta)) + (x_\sigma(\vartheta), y_j(\vartheta)), \tag{2.26} \]

\[ \cdots \]

\[ \cdots \]

\[ (x_\sigma(\vartheta), -A^*y_{j+m}(\vartheta)) = -\lambda_j(x_\sigma(\vartheta), y_{j+m}(\vartheta)) + (x_\sigma(\vartheta), y_{j+m-1}(\vartheta)). \]

Adding the identities (2.25) to the identities (2.26) corresponding to them in number, and taking (2.5) into account, we obtain

\[ (\lambda_\sigma-\lambda_j)(x_\sigma(\vartheta), y_j(\vartheta))=0, \]

\[ (\lambda_\sigma-\lambda_j)(x_\sigma(\vartheta), y_{j+1}(\vartheta)) + (x_\sigma(\vartheta), y_j(\vartheta))=0, \tag{2.27} \]

\[ \cdots \]

\[ \cdots \]

\[ (\lambda_\sigma-\lambda_j)(x_\sigma(\vartheta), y_{j+m}(\vartheta)) + (x_\sigma(\vartheta), y_{j+m-1}(\vartheta))=0. \]

For \(\lambda_\sigma\ne\lambda_j\), from the first identity we obtain \((x_\sigma(\vartheta), y_j(\vartheta))=0\), and from the following identities
\((x_\sigma(\vartheta), y_{j+1}(\vartheta))=0,\ldots,(x_\sigma(\vartheta), y_{j+m}(\vartheta))=0\). For \(j=\sigma\) the product \((x_\sigma(\vartheta), y_j(\vartheta))\) is different from zero, since in this case the system of equations (2.15) with \(z(\vartheta)=x_\sigma(\vartheta)\) must not be solvable. Multiplying \(x_\sigma(\vartheta)\) by a constant factor, it is easy to ensure that \((x_\sigma(\vartheta), y_\sigma(\vartheta))=1\). Suppose now that \(\lambda_\sigma\) is a multiple root and that one or several Jordan chains correspond to it. In an analogous way one can show that for \(\lambda_\sigma\ne\lambda_j\) the conditions hold:

\[ (x_{\sigma+k}(\vartheta),\ y_{j+l}(\vartheta))=0 \]

\[ (k=1,\ldots,m_\sigma;\quad l=1,\ldots,m_j^*), \tag{2.28} \]

where \(m_\sigma\) is the “length” of the chain \(x_\sigma(\vartheta),\ldots,x_{\sigma+m}(\vartheta)\), and \(m_j^*\) is the “length” of the chain \(y_j(\vartheta),\ldots,y_{j+m}(\vartheta)\).

Consider the case \(\lambda_\sigma=\lambda_j\), and suppose that the root \(\lambda_\sigma\) corresponds to the chain of vectors \(x_\sigma(\vartheta),\ldots,x_{\sigma+m}(\vartheta)\). Take one of the chains of vectors \(y_j(\vartheta),\ldots,y_{j+m_j}(\vartheta)\) corresponding to the root \(\lambda_i=\lambda_\sigma\). Let altogether \(l\) Jordan chains correspond to the root \(\lambda_j\). For convenience we denote all these chains as follows: \(y_j^{(k)},\ldots,y_{j+m_j}^{(k)}\) \((k=1,\ldots,l)\). Then from the solvability condition of equations (2.22) there follow the conditions

\[ (x_{\sigma+\nu}(\vartheta),\ y_j^{(k)}(\vartheta))=0 \tag{2.29} \]

\[ (\nu=1,\ldots,m_{\sigma-1},\quad k=1,\ldots,l). \]

From the nonsolvability of the equation of the form

\[ (A-\lambda_\sigma I)x(\vartheta)=x_{\sigma+m}(\vartheta) \]

it follows that not all the products

\[ (x_{\sigma+m}(\vartheta),\ y_j^{(k)}(\vartheta))\quad (k=1,\ldots,l) \tag{2.30} \]

are equal to zero.

First of all, let us note that only those products from (2.30) can be different from zero which correspond to chains with an equal number of elements. Let \((x_{\sigma+m}(\vartheta), y_j^{(k)}(\vartheta))=1\) and suppose that the number of elements \(m_1\) in the chain \(y_j^{(1)}, \ldots, y_{j+m_1}^{(1)}\) is greater than the number \(m\).

Then

\[ (x_{m+\sigma}(\vartheta), y_j^{(1)}(\vartheta)) = (x_{\sigma+m-1}(\vartheta), y_{j+1}^{(1)}(\vartheta)) =\ldots= (x_{\sigma+1}(\vartheta), y_{j+m-1}^{(1)}(\vartheta)) = \]

\[ = ((-A+\lambda_\sigma J)x_\sigma(\vartheta),\, y_{j+m}^{(1)}(\vartheta))=0 \quad (m \leq m_1). \tag{2.31} \]

Thus, \(m_1 \leq m\). But then the conditions

\[ (x_\sigma(\vartheta), y_{j+m_1}^{(1)}(\vartheta))=0,\quad (x_\sigma(\vartheta), y_{j+m_1-1}^{(1)}(\vartheta))=0,\ldots \]

\[ \ldots,\ (x_\sigma(\vartheta), y_{j+m_1-2}(\vartheta))=0,\ldots \]

\[ \ldots,\ (x_\sigma(\vartheta), y_j(\vartheta))=0 \]

are satisfied. The latter conditions contradict the inequality

\[ \sum_{k=1}^{l} (x_{m+\sigma}(\vartheta), y_j^{(k)}(\vartheta))^2 \neq 0. \]

Thus, if the scalar product \((x_{\sigma+m}(\vartheta), y_j(\vartheta))\) is equal to one (different from zero), then in the chains \(x_\sigma(\vartheta), \ldots, x_{\sigma+m}(\vartheta)\), \(y_j(\vartheta), \ldots, y_{j+m_1}(\vartheta)\) there is the same number of elements.

Let us further note that if there are two or more chains \(y_j(\vartheta)\), \(y_{j+1}(\vartheta), \ldots\), also containing \(m\) elements, then we may regard the product \((x_{\sigma+m}(\vartheta), y_j(\vartheta))\) as equal to one for only one of these chains, namely for that chain which corresponds to the chain \(x_\sigma(\vartheta), \ldots, x_{\sigma+m}(\vartheta)\). For the remaining chains this product may be considered equal to zero. Indeed, suppose there are two chains \(y_j^{(1)}(\vartheta), \ldots, y_{j+m}^{(1)}(\vartheta)\) and \(y_j^{(2)}(\vartheta), \ldots, y_{j+m}^{(2)}(\vartheta)\), for which \((x_{\sigma+m}, y_j^{(1)})=1\) and \((x_{\sigma+m}, y_j^{(2)})=1\). Then, instead of the chain \(y_j^{(2)}, \ldots, y_{j+m}^{(2)}\), introduce the chain \(\overline{y}_j=y_j^{(1)}-y_j^{(2)}, \ldots, \overline{y}_{j+m}=y_{j+m}^{(1)}-y_{j+m}^{(2)}\). The new chain, obviously, satisfies the condition \((x_{\sigma+m}(\vartheta), \overline{y}_j(\vartheta))=0\) and equations of the form (2.23).

Thus, all eigen- and associated elements of the operator \(A\), and the eigenvectors and associated elements of the operator \(-A^*\), can be so chosen and arranged in the sequence of elements \(x_\sigma(\vartheta), \ldots, x_{\sigma+m}(\vartheta)\), \(y_\sigma(\vartheta), \ldots, y_{\sigma+m}(\vartheta)\) satisfying the following conditions. If \(x_\sigma(\vartheta)\) is an eigen-element without associated elements, then

\[ (x_\sigma(\vartheta), y_j(\vartheta)) = \begin{cases} 0, & \sigma \neq j,\\ 1, & \sigma = j . \end{cases} \tag{2.32} \]

If \(x_\sigma(\vartheta), \ldots, x_{\sigma+m}(\vartheta)\) form a Jordan chain, then the corresponding adjoint Jordan chain will be \(y_\sigma(\vartheta), \ldots, y_{\sigma+m}(\vartheta)\), and the relations

\[ (x_{\sigma+k}(\vartheta), y_{\sigma+m-j}(\vartheta)) = \begin{cases} 1, & j=k,\\ 0, & j\neq k \end{cases} \tag{2.33} \]

hold,

\[ (j=0,\ldots,m;\ k=0,\ldots,m), \]

\[ (x_{\sigma+k}(\vartheta), y_j(\vartheta))=0 \quad \begin{matrix} j<\sigma\\ j>\sigma+m. \end{matrix} \]

For the operators \(A\) and \(-A^{*}\) one can always construct two countable sequences of root elements \(\{x_{\sigma}(\vartheta)\}\) and \(\{y_{\sigma}(\vartheta)\}\) (proper and associated elements) satisfying conditions (2.32) and (2.33).

We note further that if \(x_{\sigma}(\vartheta)\) \((y_{\sigma}(\vartheta))\) is a root element (proper or associated) of the operator \(A\) \((-A^{*})\), corresponding to the root \(\lambda_{\sigma}\) \((-\lambda_{\sigma})\), then \(x_{\sigma}(t+\vartheta)\), \(-\tau\leq \vartheta\leq 0\) \((y_{\sigma}(t+\vartheta)\) for \(0\leq \vartheta\leq \tau)\), will be a particular solution of the system of equations (1.5) \(((2.1))\), and for \(\vartheta=0\), a particular solution of the system (1.1) \(((1.3))\).

This can be verified by directly constructing particular solutions of the systems (1.1) and (1.3) by Euler’s method [3], and also by constructing root elements of the operator \(A(-A^{*})\) with subsequent comparison of the systems of algebraic equations to which both the first and the second problems lead.

1. Splitting of the operator \(A\). Extraction of a canonical system of ordinary differential equations.

The characteristic equation (2.8) has a finite number \(N=N(\alpha)\) of roots \(\lambda_{\sigma}\) situated to the right of the line \(\operatorname{Re}\lambda=\alpha\) in the complex \(\lambda\)-plane (\(\alpha\) is a positive or negative number). Let \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) be the roots of equation (2.8) (the eigenvalues of the operator \(A\)), arranged in decreasing order of their real parts
\(\operatorname{Re}\lambda_{1}\geq \operatorname{Re}\lambda_{2}\geq \operatorname{Re}\lambda_{3}\geq \cdots\). In the indicated sequence the root \(\lambda_{\sigma}\) occurs as many times as its multiplicity. Construct a system of eigenvectors and associated vectors of the operator \(A\):

\[ x_{1}(\vartheta),\ldots,\quad x_{\sigma}(\vartheta),\ldots, \tag{3.1} \]

corresponding to the eigenvalues \(\lambda_{1},\ldots,\lambda_{\sigma},\ldots\). We shall also construct a system of eigenvectors and associated vectors of the operator \(-A^{*}\):

\[ y_{1}(\vartheta),\ldots,\quad y_{\sigma}(\vartheta),\ldots, \tag{3.2} \]

corresponding to the eigenvalues \(-\lambda_{1},\ldots,-\lambda_{\sigma},\ldots\) of the operator \(-A^{*}\). Suppose that conditions (2.32), (2.33) are satisfied. Then, if \(x_{\sigma}(\vartheta)\) is an eigenvector of the operator \(A\) without associated vectors, then

\[ (x_{\sigma}(\vartheta)y_{j}(\vartheta))= \begin{cases} 1, & \sigma=j,\\ 0, & \sigma\ne j. \end{cases} \tag{3.3} \]

If \(x_{\sigma}(\vartheta),\ldots,x_{\sigma+m}(\vartheta)\) is a Jordan chain consisting of the eigenelement \(x_{\sigma}(\vartheta)\) and the associated elements \(x_{\sigma+1}(\vartheta),\ldots,x_{\sigma+m}(\vartheta)\), then the corresponding chain of elements of the adjoint operator \(-A^{*}\) will be \(y_{\sigma}(\vartheta),\ldots,y_{\sigma+m}(\vartheta)\), and the conditions

\[ (x_{\sigma+k}(\vartheta),\, y_{\sigma+m-j}(\vartheta))= \begin{cases} 1, & j=k,\\ 0, & j\ne k \end{cases} \]

\[ (0\leq j\leq m,\; 0\leq k\leq m), \]

\[ (x_{\sigma+k}(\vartheta),\, y_{j+\sigma}(\vartheta))=0\quad (j<0,\; j>m), \tag{3.4} \]

\[ \lambda_{\sigma}=\lambda_{\sigma+1}=\cdots=\lambda_{\sigma+m}. \]

Consider the set \(N_{\alpha}\) of elements \(x_{i}(\vartheta)\) \((i=1,\ldots,N_{\alpha})\) (3.1), which-

which correspond to the roots \(\lambda_i\) \((i=1,\ldots,N_\alpha)\), \(\operatorname{Re}\lambda_i>\alpha\), where \(\alpha\) is a negative or positive number. Let us write the \(N_\alpha\) functionals

\[ f_j[x(\vartheta)]=(x(\vartheta),\,y_j(\vartheta)) \quad (j=1,\ldots,N_\alpha). \]

The conditions

\[ f_j[x(\vartheta)]=0 \quad (j=1,\ldots,N_\alpha) \]

define a subspace \(L_\alpha\) in the linear normed space \(C[-\tau,0]\).

Next take an arbitrary element \(x(\vartheta)\in C[-\tau,0]\) and represent it in the form

\[ x(\vartheta)=\sum_{j=1}^{N_\alpha} y_j x_j(\vartheta)+z(\vartheta),\qquad y(\vartheta)=x(\vartheta)-z(\vartheta). \tag{3.5} \]

We choose the coefficients \(y_j\) so that \(z(\vartheta)\) belongs to the subspace \(L_\alpha\). For this, the conditions

\[ f_j[z(\vartheta)]\equiv f_j[x(\vartheta)]-\sum_{l=1}^{N_\alpha} y_l (x_l(\vartheta),\,y_j(\vartheta))=0 \tag{3.6} \]

\[ (j=1,\ldots,N_\alpha) \]

must be satisfied.

Hence, taking into account the conditions (3.3) and (3.4), we find: if \(x_j(\vartheta)\) is an eigen-element without adjoint elements of the operator \(A\), then

\[ y_j=f_j[x(\vartheta)]=(x(\vartheta),\,y_j(\vartheta)); \tag{3.7} \]

if \(x_j(\vartheta)\) is an eigen-element and there corresponds to it a chain of adjoint elements \(x_{j+1}(\vartheta),\ldots,x_{j+m}(\vartheta)\), then

\[ y_{j+k}=f_{j+m-k}[x(\vartheta)]=(x(\vartheta),\,y_{j+m-k}(\vartheta)). \tag{3.8} \]

We note that, in the presence of complex roots \(\lambda_\sigma\), complex terms \(y_jx_j(\vartheta)\) will enter into formula (3.5). However, since to each complex \(y_jx_j(\vartheta)\) in the sum (3.6) there will correspond the conjugate term, the sum of all elements \(y_jx_j(\vartheta)\) in (3.6) is real. Therefore \(z(\vartheta)\) is always a real element of \(C[-\tau,0]\).

Let us show that formulas (3.5), (3.7) uniquely determine the decomposition of an arbitrary element \(x(\vartheta)\in C[-\tau,0]\) into two elements \(z(\vartheta)\) and \(y(\vartheta)=x(\vartheta)-z(\vartheta)\), belonging respectively to the subspace \(L_\alpha\) and to \(C[-\tau,0]-L_\alpha\). The latter subspace is finite-dimensional and has as a basis the vector \(\{x_j(\vartheta),\,j=1,\ldots,N_\alpha\}\).

Indeed, suppose that the representation of the element \(x(\vartheta)\) by formulas (3.5) and (3.7) is not unique. Then one can indicate distinct \(y'_1,\ldots,y'_{N_\alpha},z'\) and \(y''_1,\ldots,y''_{N_\alpha},z''\) such that

\[ x(\vartheta)=\sum_{j=1}^{N_\alpha} y'_j x_j(\vartheta)+z'(\vartheta) =\sum_{j=1}^{N_\alpha} y''_j x_j(\vartheta)+z''(\vartheta). \]

Taking into account that \(f_k[z'(\vartheta)]=f_k[z''(\vartheta)]=0\), we have

\[ \sum_{j=1}^{N_\alpha} (y'_j-y''_j) f_k[x_j(\vartheta)]=0, \]

and since the determinant \(|f_k[x_j(\vartheta)]|\) is equal to one, it follows that \(y'_j=y''_j\) \((j=1,\ldots,N_\alpha)\). But then \(z'(\vartheta)=z''(\vartheta)\).

Thus, the representation of the element \(x(\vartheta)\) in the form (3.5) and (3.7) is unique.

From (3.5) and (3.7) we have

\[ z(\vartheta)=P_z[x(\vartheta)] =x(\vartheta)-\sum_{j=1}^{N_\alpha} y_j[x(\vartheta)]\,x_j(\vartheta), \tag{3.9} \]

\[ y(\vartheta)=P_y[x(\vartheta)] =\sum_{j=1}^{N_\alpha} y_j[x(\vartheta)]\,x_j(\vartheta). \tag{3.10} \]

Formulas (3.9) and (3.10) define the projection operators \(P_z\) and \(P_y\). Obviously,

\[ P_z+P_y=J, \]

where \(J\) is the identity operator, \(Jx(\vartheta)=x(\vartheta)\). It is not difficult to verify the equalities

\[ P_z^2=P_z,\qquad P_y^2=P_y,\qquad P_yP_z=0,\qquad P_zP_y=0. \tag{3.11} \]

Here the operator “\(0\)” denotes \(0x(\vartheta)\equiv0\), \(-\tau\leq\vartheta\leq0\). Thus, \(P_z\) (3.9) is the projection operator in the “direction” \(y(\vartheta)\), defined in \(C[-\tau,0]\), with range \(\Pi_\alpha\{z(\vartheta)\}\). And the operator \(P_y\) (3.10) is the projection operator in the “direction” \(z(\vartheta)\), defined in the subspace \(C[-\tau,0]-\Pi_\alpha\) of dimension \(N_\alpha\) with basis \(\{x_j(\vartheta),\ j=1,\ldots,N_\alpha\}\).

Using the operators \(P_y\) and \(P_z\), one can split the operator \(A\):

\[ A=JA=P_zA+P_yA. \tag{3.12} \]

Introduce the notation \(P_zA=A_z,\ P_yA=A_y\):

\[ A_y[x(\vartheta)] =P_yAx(\vartheta) =\sum_{j=1}^{N_\alpha}(Ax(\vartheta),y_j(\vartheta))x_j(\vartheta) =\sum_{j=1}^{N_\alpha}(x(\vartheta),A^*y_j(\vartheta))x_j(\vartheta)= \]

\[ =\sum_{j=1}^{N_\alpha}\left\{ \begin{array}{l} (x(\vartheta),y_j(\vartheta))\lambda_j x_j(\vartheta),\\[2mm] (x(\vartheta),y_{j+m}(\vartheta))\lambda_j x_j(\vartheta) +(x(\vartheta),y_{j+m-1}(\vartheta))x_j(\vartheta)+\\ \qquad \cdots\\ +(x(\vartheta),y_j(\vartheta))\lambda_j x_{j+m}(\vartheta). \end{array} \right. \tag{3.13} \]

Here the comma before the summation sign means that the terms written in the first line enter under the summation sign if the root \(\lambda_j\) corresponds to one eigen-element without associated elements. If, however, the element \(x_j(\vartheta)\) corresponds to a Jordan group of elements \(x_j(\vartheta),\ldots,x_{j+m}(\vartheta)\), then in the sum in the positions from the \(j\)-th to the \(j+m\)-th elements there stands the second line, determined through the eigen-element \(x_j(\vartheta)\) and the associated elements \(x_{j+1}(\vartheta),\ldots,x_{j+m}(\vartheta)\).

It is easy to verify the validity of the equalities

\[ P_y A = A P_y,\qquad P_z A = A P_z . \tag{3.14} \]

Let us now consider equation (1.4). Put \(x(\vartheta)=y(\vartheta)+z(\vartheta)\). (3.5) Then

\[ \frac{d x_t(\vartheta)}{dt} = \frac{d y_t(\vartheta)}{dt} + \frac{d z_t(\vartheta)}{dt} = A\bigl(y_t(\vartheta)+z_t(\vartheta)\bigr)= \]

\[ = (A_y+A_z)\bigl(y_t(\vartheta)+z_t(\vartheta)\bigr) = A_y y_t(\vartheta)+A_z z_t(\vartheta). \]

Obviously, if \(z_t(\vartheta)\in \mathcal L_\alpha\), then \(\dfrac{d z_t(\vartheta)}{dt}\in \mathcal L_\alpha\). Thus, equation (1.4) splits into two equations

\[ \frac{d y_t(\vartheta)}{dt}=A_y y_t(\vartheta), \tag{3.15} \]

\[ \frac{d z_t(\vartheta)}{dt}=A_z z_t(\vartheta). \tag{3.16} \]

Here \(A_y\) is a finite-dimensional (\(N_\alpha\)-dimensional) operator, which can be represented, according to (3.13), in the following way.

Let

\[ y_t(\vartheta)=\sum_{j=1}^{N_\alpha} y_j(t)x_j(\vartheta). \tag{3.17} \]

Then

\[ A_y y_t(\vartheta)= \sum_{j=1}^{N_\alpha} \left\{ \begin{array}{l} \lambda_j y_j(t)x_j(\vartheta),\\[2mm] \bigl(\lambda_j y_j(t)-y_{j+1}(t)\bigr)x_j(\vartheta)+\\[1mm] \quad+\bigl(\lambda_j y_{j+1}(t)-y_{j+2}(t)\bigr)x_{j+1}(\vartheta)+\cdots\\ \quad\cdots+\lambda_j y_{j+m}(t)x_{j+m}(\vartheta). \end{array} \right. \tag{3.18} \]

It follows from (3.18) that \(A_y\) is representable in the form of an \(N_\alpha\)-dimensional matrix \(B\) with Jordan blocks of the form \(\|\lambda_i\|\) or of the form

\[ \left\| \begin{array}{ccccc} \lambda_j-1 & 0 & \ldots & 0\\ 0 & \lambda_j-1 & \ldots & 0\\ 0 & 0 & \lambda_j & \ldots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \ldots & \lambda_j \end{array} \right\| \tag{3.19} \]

and arranged along the main diagonal. The spectrum of the operator \(A_y\) consists of \(N_\alpha\) eigenvalues \(\lambda_1,\ldots,\lambda_{N_\alpha}\) of the operator \(A\). The spectrum of the operator \(A_z\) consists of the remaining eigenvalues of the operator \(A\): \(\lambda_{N+1},\lambda_{N+2},\ldots\)

The variables \(y(\vartheta)\), \(z(\vartheta)\) are defined in (3.5), and the operator \(A_y\) is defined in (3.13). From equation (3.15) there follow \(N_\alpha\) ordinary differential equations

\[ \frac{dy_j}{dt}=\lambda_j y_j, \]

\[ \left\{ \begin{aligned} \frac{dy_j}{dt}&=\lambda_j y_j-y_{j+1}\\ \frac{dy_{j+1}}{dt}&=\lambda_j y_{j+1}-y_{j+2}\\ &\cdots\\ \frac{dy_{j+m}}{dt}&=\lambda_j y_{j+m}. \end{aligned} \right. \tag{3.20} \]

Here \(y_j\) are the components of the vector

\[ y_t(\vartheta)=\sum_{j=1}^{N_\alpha} y_j x_j(\vartheta). \]

Or, in vector notation,

\[ \frac{dy}{dt}=By,\qquad y=\begin{pmatrix} y_1\\ \vdots\\ y_{N_\alpha} \end{pmatrix}, \tag{3.21} \]

where the matrix \(B\) is composed of matrices of the form (3.19).

Theorem 3.1. Choose \(\alpha\le -\gamma\), where \(\gamma\) is a positive number. Then \(N_\alpha\) roots \(\lambda_\sigma\) will lie to the right of the line \(\operatorname{Re}\lambda=-\gamma\). The following assertion holds.

If \(\alpha\le -\gamma\) in (3.5), (3.7), then the spectrum of the operator \(A_z\) satisfies the condition \(\operatorname{Re}\lambda_\sigma<-\gamma\) \((\gamma>0)\), and every solution of the system (3.16) \(z_t=z(t,z_0(\vartheta),t_0,\vartheta)\) with an arbitrary initial function \(z_0(\vartheta)\) belonging to \(L_\alpha:\ f_j[z_0(\vartheta)]=0\) \((j=1,\ldots,N_\alpha)\), for all \(t>t_0\), satisfies the inequality

\[ \|z_t(\vartheta)\|=\|z(t,z_0(\vartheta),t_0,\vartheta)\|_{t=0} \le C e^{-\beta(t-t_0)}\|z_0(\vartheta)\|_{t=0}, \tag{3.22} \]

where \(C,\beta\) are positive numbers independent of the choice of \(z_0(\vartheta)\), \(\beta=q\gamma,\ 0<q<1\).

This theorem is a direct generalization of theorem (29.1) of N. N. Krasovskii from [1, p. 165]. The proof is carried out in the same way, only taking into account that on \(L_\alpha\) the conditions \(f_j[z_0(\vartheta)]=0\) \((j=1,\ldots,N_\alpha)\) are fulfilled.

Thus, we arrive at the following main result.

Theorem 3.2. In the space of continuous functions \(C[-\tau,0]\), \(x(\vartheta)=\{x_i(\vartheta),\ -\tau\le\vartheta\le0,\ i=1,\ldots,n\}\), in which the motions of system (1.1) are considered, a finite-dimensional basis \(\{x_j(\vartheta),\ j=1,\ldots,N_\alpha\}\) can be chosen, on which the motion of system (1.1) is described by a system of \(N_\alpha\) ordinary differential equations of canonical form (3.20) or (3.21) with constant coefficients. In the complementary subspace \(L_\alpha\), \(f_j[x(\vartheta)]=0\) \((j=1,\ldots,N_\alpha)\), the motion of system (1.1) is described by an ordinary differential equation of the form (3.16), and the norm of every solution \(z_t(\vartheta)\) decreases according to an exponential law with an arbitrarily large exponent, if the number \(\alpha\) is chosen negative and sufficiently large in magnitude.

We note that questions of the canonical decomposition of solutions of systems with delay were considered, in another aspect, by J. Hale.

References

1. Krasovskii N. N. Some problems in the theory of stability of motion. Gostekhizdat, Moscow–Leningrad, 1959.

2. Myshkis A. D. Linear differential equations with retarded argument. Gostekhizdat, Moscow–Leningrad, 1951.

3. Elsgolts L. E. Qualitative methods in mathematical analysis. Gostekhizdat, Moscow, 1955.

4. Shimanov S. N. PMM, 23, issue 5, 1959, pp. 836–844.

5. Shimanov S. N. PMM, 23, issue 1, 1960, pp. 55–63.

6. Shimanov S. N. DAN SSSR, 133, No. 1, 1960, pp. 36–39.

7. Hale K. J. Linear functional-differential equations with constant coefficients. Technical report, 63-6, 1963, RJAS.

Received by the editors
September 5, 1964

Ural State University
named after S. M. Kirov