On Stabilization of Controlled Systems with Delay

Yu. S. Osipov

The problem of stabilizing steady motions of a controlled system described by nonlinear differential equations with time delay is considered. On the basis of the theory of stability of motion [1, 2, 5—10], the theory of controllability and stabilizability (see [4] and [11], where the corresponding bibliography is given), a stabilizability criterion [14] is proved for a linear controlled system with delay, the problem of stabilizing a nonlinear system with delay in the first approximation is solved, and critical cases of stabilization are singled out.

§ 1. Statement of the problem. Consider a nonlinear controlled system whose perturbed motion is described by the equations

\[ \frac{d x(t)}{dt}=A x(t)+A_\tau x(t-\tau)+Bu+X\bigl(x(t),x(t-\tau),u\bigr), \tag{1.1} \]

where \(x\) is an \(n\)-vector of phase coordinates; \(u\) is an \(m\)-vector of the control action; \(\tau=\mathrm{const}>0\) is the magnitude of the delay; \(A=\{a_{ki}\}\), \(A_\tau=\{a^\tau_{ki}\}\), \(B=\{b_{ki}\}\) are constant matrices of dimensions, respectively, \(n\times n\), \(n\times n\), \(n\times m\); \(X(x,(t),x(t-\tau),u)\) is a sufficiently smooth \(n\)-vector function denoting the collection of terms of order higher than the first of smallness with respect to the quantities \(x(t)\), \(x(t-\tau)\), \(u\). It is assumed that \(X(0,0,0)\equiv 0\).

Suppose that in the absence of the control action (\(u\equiv 0\) in (1.1)) the unperturbed motion \(x=0\) of system (1.1) is unstable in the sense of Lyapunov ([2], pp. 155—156).

The problem of stabilization of the controlled system (1.1) can be formulated as follows [4, 11]: let, at each instant \(t\) of the process, the control action \(u\) be formed on the basis of information about the current state \(x(t)\) of the system; it is required to construct a control law \(u=u[x]\) under which the unperturbed motion \(x=0\) of system (1.1) would be asymptotically stable ([2], pp. 155—156).

As an element of the trajectory corresponding to the instant of time \(t\), we shall take segments of the trajectory \(x_t(\vartheta)=x(t+\vartheta)=\{x_{kt}(\vartheta);\ k=1,\ldots,n;\ -\tau\leq \vartheta\leq 0\}\) and consider the motion in the space \(C_{[-\tau,0]}\) of continuous functions with the metric

\[ \|x(\vartheta)\|_\tau = \sup \left[ \left( \sum_{k=1}^{n} |x_k(\vartheta)|^2 \right)^{1/2} \quad \text{for } -\tau\leq \vartheta\leq 0 \right]. \]

We shall also denote

\[ \|x(t)\|= \left( \sum_{k=1}^{n} |x_k(t)|^2 \right)^{1/2}. \]

Everywhere in what follows we shall regard the argument \(\vartheta\) as varying within the limits \(-\tau\leq \vartheta\leq 0\).

In the case of systems with delay, it is expedient to choose the quantity \(u\) in (1.1) in the form of some functional \(u[x_t(\vartheta)]\), defined on the curves \(x_t(\vartheta)\) [3].

We shall seek a functional \(u[x(\vartheta)]\) solving the stabilization problem in the class of functionals \(u[x(\vartheta)]\) satisfying the Lipschitz conditions

\[ \left|u_i[x^{(1)}(\vartheta)]-u_i[x^{(2)}(\vartheta)]\right| \leq l\left\|x^{(1)}(\vartheta)-x^{(2)}(\vartheta)\right\|_\tau \quad (l=\mathrm{const}). \tag{1.2} \]

In the present paper a stabilizability criterion is proved for the linear system

\[ \frac{dx(t)}{dt}=Ax(t)+A_\tau x(t-\tau)+Bu \tag{1.3} \]

and the problem of stabilization of the nonlinear controlled system (1.1) is solved by means of the linear approximation (1.3); critical cases of stabilization are singled out, when its possibility is determined by the terms of the nonlinear addition \(X(x(t),x(t-\tau),u)\).

§ 2. Preliminary remarks. Stabilizability criterion for linear controlled systems with delay. We transform the equations of motion similarly to how this was done in [11].

In the space of functions \(\{x_t(\vartheta)\}\), an equation with aftereffect corresponds to an equivalent system of differential-operator equations [2], which in the case under consideration can be represented in the form

\[ \frac{dx_t(\vartheta)}{dt} = Px_t(\vartheta)+\sum_{i=1}^{m}R_i[B]u_i + F[X(x_t(0),x_t(-\tau),u)], \tag{2.1} \]

\[ Px(\vartheta)= \begin{cases} \dfrac{dx(\vartheta)}{d\vartheta}, & -\tau\leq \vartheta<0,\\[6pt] Ax(0)+A_\tau x(-\tau), & \vartheta=0, \end{cases} \tag{2.2} \]

\[ R_i[B]= \begin{cases} 0, & -\tau\leq \vartheta<0,\\ b_{(i)}, & \vartheta=0, \end{cases} \tag{2.3} \]

\(b_{(i)}\) is the \(i\)-th column vector of the matrix \(B\),

\[ F[X]= \begin{cases} 0, & -\tau\leq \vartheta<0,\\ X(x(0),x(-\tau),u), & \vartheta=0. \end{cases} \tag{2.4} \]

To the system of equations

\[ \frac{dx(t)}{dt}=Ax(t)+A_\tau x(t-\tau) \tag{2.5} \]

there may be associated the adjoint system of equations [7, 10]

\[ \frac{dx^*(t)}{dt}=-A'x^*(t)-A_\tau'x^*(t+\tau), \tag{2.6} \]

where the prime sign, here and throughout what follows, denotes transposition.

In the space of functions \(\{x_t(-\vartheta)\}\), the equations (2.6) correspond to the system of differential-operator equations

\[ \frac{dx_t^*(-\vartheta)}{dt}=-P^*x_t^*(-\vartheta), \tag{2.7} \]

On Stabilization of Controlled Systems with Delay

where

\[ -P^{*}x^{*}(-\vartheta)= \begin{cases} \dfrac{dx^{*}(-\vartheta)}{d(-\vartheta)}, & -\tau \leqslant \vartheta < 0,\\[1.2ex] -A'x^{*}(0)-A'_{\tau}x^{*}(\tau), & \vartheta=0. \end{cases} \tag{2.8} \]

Let us single out from (2.1) the finite-dimensional subsystem corresponding to the proper unstable degrees of freedom, applying the method of canonical representation of motions of linear systems with aftereffect, developed in [8–10].

The spectrum \(\{\lambda_i\}\) of the operator \(P\) (2.2), determined by the roots of the characteristic equation

\[ \Delta(\lambda)=\det[A-\lambda E+A_{\tau}e^{-\lambda\tau}]=0 \tag{2.9} \]

of system (2.5), contains only a finite number of eigenvalues \(\lambda\) with \(\operatorname{Re}\lambda \geqslant 0:\{\lambda_{\sigma};\ \sigma=1,\ldots,N\}\).

Let the eigenvalue \(\lambda_{\sigma}\) of multiplicity \(m_{\sigma}\) correspond to \(l_{\sigma}\) Jordan chains of root elements of the operator \(P\) (2.2):
\(d_{\sigma,i_{\sigma}}^{(k_{\sigma})}(\vartheta)\)
\((i_{\sigma}=0,\ldots,m_{\sigma}^{(k_{\sigma})};\ k_{\sigma}=1,\ldots,l_{\sigma};\ \sigma=1,\ldots,r;\ l_{\sigma}+\sum_{k_{\sigma}=1}^{l_{\sigma}} m_{\sigma}^{(k_{\sigma})}=m_{\sigma};\ m_1+\cdots+m_r=N)\), satisfying the equations

\[ (P-\lambda_{\sigma}J)d_{\sigma,0}^{(k_{\sigma})}(\vartheta)=0, \]

\[ (P-\lambda_{\sigma}J)d_{\sigma,j_{\sigma}+1}^{(k_{\sigma})}(\vartheta) =d_{\sigma,j_{\sigma}}^{(k_{\sigma})}(\vartheta) \quad (j_{\sigma}=0,\ldots,m_{\sigma}^{(k_{\sigma})}-1;\tag{2.10} \]

\[ k_{\sigma}=1,\ldots,l_{\sigma};\ \sigma=1,\ldots,r). \]

Then the spectrum \(\{\lambda_i^{*}\}\) of the operator \(-P^{*}\) (2.8) contains exactly \(N\) eigenvalues \(\lambda^{*}\) with \(\operatorname{Re}\lambda^{*}\leqslant 0\):
\(\{\lambda_i^{*}=-\lambda_i;\ i=1,\ldots,N\}\); to the eigenvalue \(\lambda_{\sigma}^{*}\) of multiplicity \(m_{\sigma}\) there correspond \(l_{\sigma}\) Jordan chains of root elements of the operator \(-P^{*}\) (2.8):
\(d_{\sigma,i_{\sigma}}^{*(k_{\sigma})}(-\vartheta)\)
\((i_{\sigma}=0,\ldots,m_{\sigma}^{(k_{\sigma})};\ k_{\sigma}=1,\ldots,l_{\sigma};\ \sigma=1,\ldots,r;\ l_{\sigma}+\sum_{k_{\sigma}=1}^{l_{\sigma}}m_{\sigma}^{(k_{\sigma})}=m_{\sigma};\ m_1+\cdots+m_r=N)\), satisfying the equations

\[ (-P^{*}+\lambda_{\sigma}J)d_{\sigma,0}^{*(k_{\sigma})}(-\vartheta)=0, \]

\[ (-P^{*}+\lambda_{\sigma}J)d_{\sigma,j_{\sigma}+1}^{*(k_{\sigma})}(-\vartheta) =d_{\sigma,j_{\sigma}}^{*(k_{\sigma})}(-\vartheta) \quad (j_{\sigma}=0,\ldots,m_{\sigma}^{(k_{\sigma})}-1;\tag{2.11} \]

\[ k_{\sigma}=1,\ldots,l_{\sigma});\ \sigma=1,\ldots,r). \]

We denote by the symbol \((\varphi,\psi)\) the quantity

\[ (\varphi,\psi)=\varphi'(0)\psi(0)+\int_{-\tau}^{0}\varphi'(\vartheta)A'_{\tau}\psi(\vartheta+\tau)\,d\vartheta . \tag{2.12} \]

As shown in [7,10], the systems
\(\{d_{\sigma,i_{\sigma}}^{(k_{\sigma})}(\vartheta)\}\) and
\(\{d_{\sigma,i_{\sigma}}^{*(k_{\sigma})}(-\vartheta)\}\)
of linearly independent root elements of the operators \(P\) (2.2) and \(-P^{*}\) (2.8),

corresponding to the selected part of the spectrum can be constructed so that the equalities hold:
\[ \bigl(d_{\sigma,i_\sigma}^{(k_\sigma)},\,d_{j,i_j}^{*(k_j)}\bigr)=0 \quad\text{for }\sigma\ne j \quad (i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma, \]
\[ \sigma=1,\ldots,r),\quad (i_j=0,\ldots,m_j^{(k_j)};\ k_j=1,\ldots,l_j;\ j=1,\ldots,r), \]
\[ \bigl(d_{\sigma,i_\sigma}^{(k_\sigma)},\,d_{\sigma,j_\sigma}^{*(k_\sigma)}\bigr) = \begin{cases} 1 & \text{if } i_\sigma+j_\sigma=m_\sigma^{(k_\sigma)},\\ 0 & \text{if } i_\sigma+j_\sigma\ne m_\sigma^{(k_\sigma)}, \end{cases} \tag{2.13} \]
\[ (i_\sigma,j_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r), \]
\[ \bigl(d_{\sigma,i_\sigma}^{(k_\sigma)},\,d_{\sigma,j_\sigma}^{*(n_\sigma)}\bigr)=0 \quad\text{for }k_\sigma\ne n_\sigma \]
\[ (i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma,n_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r). \]

Let us consider in the space \(C[-\tau,0]\) the \(N\) linear functionals [7, 10]
\[ f_{\sigma,i_\sigma}^{(k_\sigma)}[\varphi(\vartheta)] = \bigl(\varphi,d_{\sigma,i_\sigma}^{*(k_\sigma)}\bigr) \tag{2.14} \]
\[ (i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r) \]
and we shall represent the elements \(\psi(\vartheta)\in C[-\tau,0]\) in the form
\[ \psi(\vartheta) = z(\vartheta) + \sum_{\sigma=1}^{r} \sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta)\, y_{\sigma,i_\sigma}^{(k_\sigma)}, \tag{2.15} \]
where the constants \(y_{\sigma,i_\sigma}^{(k_\sigma)}\) are equal to
\[ y_{\sigma,i_\sigma}^{(k_\sigma)} = f_{\sigma,m_\sigma^{(k_\sigma)}-i_\sigma}^{(k_\sigma)}[\psi,(\vartheta)] \tag{2.16} \]
\[ (i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r), \]
and \(z(\vartheta)\) is an element of the functional space \(L_f\), defined by the equalities
\[ L_f:\ f_{\sigma,i_\sigma}^{(k_\sigma)}[z(\vartheta)]=0 \tag{2.17} \]
\[ (i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r). \]

Then, for the element of the trajectory \(x_t(\vartheta)\) of the system, we have the representation
\[ x_t(\vartheta) = z_t(\vartheta) + \sum_{\sigma=1}^{r} \sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta)\, y_{\sigma,i_\sigma}^{(k_\sigma)}(t), \tag{2.18} \]
where
\[ y_{\sigma,i_\sigma}^{(k_\sigma)}(t) = f_{\sigma,m_\sigma^{(k_\sigma)}-i_\sigma}^{(k_\sigma)}[x_t(\vartheta)] \]
and \(z_t(\vartheta)\in L_f\), if and only if \(z_0(\vartheta)\in L_f\).

In the variables \(z_t(\vartheta)\) and \(y(t)\) the system of equations of the first approximation (1.3) has the form:
\[ \frac{dy(t)}{dt}=Gy(t)+Cu, \tag{2.19} \]

\[ \frac{d z_t(\vartheta)}{dt}=Pz_t(\vartheta)+\sum_{i=1}^{m} Z_i[B,\vartheta]u_i, \tag{2.20} \]

and the nonlinear equations (1.1) are thereby transformed into the form:

\[ \frac{dy(t)}{dt}=Gy(t)+Cu+Y[y(t),z_t(0),z_t(-\tau),u], \tag{2.21} \]

\[ \frac{dz_t(\vartheta)}{dt}=Pz_t(\vartheta)+\sum_{i=1}^{m} Z_i[B,\vartheta]u_i +Z[y(t),z_t(0),z_t(-\tau),u,\vartheta]. \tag{2.22} \]

Here \(y(t)=\{y_{\sigma,i_\sigma}^{(k_\sigma)}(t);\ i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\sigma=1,\ldots,r\}\) is an \(N\)-vector; \(G\) is a constant Jordan matrix of dimension \(N\times N\), consisting of \(l_1+\cdots+l_r\) blocks \(G_\sigma^{(k_\sigma)}\) of dimensions \(m_\sigma^{(k_\sigma)}+1\times m_\sigma^{(k_\sigma)}+1\), of the form

\[ G= \begin{bmatrix} G_1^{(1)} & & & & 0\\ & \ddots & & & \\ & & G_1^{(l_1)} & & \\ & & & \ddots & \\ & & & & G_r^{(1)}\\ 0 & & & & \ddots\\ & & & & G_r^{(l_r)} \end{bmatrix}, \qquad G_\sigma^{(k_\sigma)}= \begin{bmatrix} \lambda_\sigma & 1 & 0 & \cdots & 0\\ 0 & \lambda_\sigma & 1 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdots & \cdot\\ \cdot & \cdot & \cdot & \cdots & \cdot\\ \cdot & \cdot & \cdot & \cdots & \cdot\\ 0 & 0 & 0 & \cdots & \lambda_\sigma \end{bmatrix} \tag{2.23} \]

\[ (\sigma=1,\ldots,r;\ k_\sigma=1,\ldots,l_\sigma). \]

\(C\) is a constant matrix of dimension \(N\times m\), consisting of \(l_1+\cdots+l_r\) blocks \(C_\sigma^{(k_\sigma)}\) of dimensions \(m_\sigma^{(k_\sigma)}+1\times m\), of the form

\[ C= \begin{bmatrix} C_1^{(1)}\\ \cdot\\ \cdot\\ \cdot\\ C_1^{(l_1)}\\ \cdot\\ \cdot\\ \cdot\\ C_r^{(1)}\\ \cdot\\ \cdot\\ \cdot\\ C_r^{(l_r)} \end{bmatrix}, \qquad C_\sigma^{(k_\sigma)}= \begin{bmatrix} f_{\sigma,m_\sigma^{(k_\sigma)}}^{(k_\sigma)}[R_i[B]]\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ f_{\sigma,0}^{(k_\sigma)}[R_i[B]] \end{bmatrix}_{i=1}^{m} \tag{2.24} \]

\[ (\sigma=1,\ldots,r;\ k_\sigma=1,\ldots,l_\sigma); \]

the quantity

\[ Y[y(t),z_t(0),z_t(-\tau),u]= \{Y_{\sigma,i_\sigma}^{(k_\sigma)}[y(t),z_t(0),z_t(-\tau),u]\}= \]

\[ = f_{\sigma,m_\sigma^{(k_\sigma)}-i_\sigma}^{(k_\sigma)} \left[ F\left(X\left(z_t(0)\right) +\sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{j_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,j_\sigma}^{(k_\sigma)}(0)y_{\sigma,j_\sigma}^{(k_\sigma)}(t),\right. \]

\[ z_t(-\tau)+\sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(-\tau)y_{\sigma,i_\sigma}^{(k_\sigma)}(t,u)\Bigg]; \]

\[ i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\quad k_\sigma=1,\ldots,l_\sigma;\quad \sigma=1,\ldots,r \tag{2.25} \]

is an \(N\)-vector functional; the quantity

\[ \begin{aligned} Z[y(t),z_t(0),z_t(-\tau),u,\vartheta] &=F\Bigg[ X\Big(z_t(0)+\sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(0) y_{\sigma,i_\sigma}^{(k_\sigma)}(t),\\ &\qquad\qquad z_t(-\tau)+\sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(-\tau) y_{\sigma,i_\sigma}^{(k_\sigma)}(t),u\Big)\Bigg]\\ &\quad-\sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta) Y_{\sigma,i_\sigma}^{(k_\sigma)}[y(t),z_t(0),z_t(-\tau),u] \end{aligned} \tag{2.26} \]

is a real \(n\)-vector operator; the quantity

\[ Z_i[B,\vartheta]=R_i[B]- \sum_{\sigma=1}^{r}\sum_{k_\sigma=1}^{l_\sigma} \sum_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}} d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta) f_{\sigma,m_\sigma^{(k_\sigma)}-i_\sigma}^{(k_\sigma)}[R_i[B]] \tag{2.27} \]

is a real \(n\)-vector operator.

The problem of stabilization of motions described by equations (1.1) is equivalent to the problem of stabilization of motions described by equations (2.21), (2.22). Therefore, in what follows we consider equations (2.21), (2.22).

Let us transform the equations of motion (2.21), (2.22) to real form. Suppose, without loss of generality, that the eigenvalues \(\lambda_\sigma\) \((\sigma=1,\ldots,r)\) are ordered so that the first \(s\) of them are real, and the remaining \(q=r-s\) are complex, with \(\lambda_{p+1}=\overline{\lambda_p}\) \((p=s+1,s+3,\ldots,r-1)\) (\(q\) is an even number, since the matrices \(A,A_\tau\) in (1.1) are real). Then the system of equations (2.21), (2.22) can be represented in the form:

\[ \frac{dv(t)}{dt}=Qv(t)+Du+V[v(t),z_t(0),z_t(-\tau),u], \tag{2.28} \]

\[ \frac{dz_t(\vartheta)}{dt}=Pz_t(\vartheta)+\sum_{i=1}^{m} Z_i[B,\vartheta]u_i +Z[v(t),z_t(0),z_t(-\tau),u,\vartheta], \tag{2.29} \]

where
\[ v=\{v_j;\ j=1,\ldots,N\}= \{y_{\sigma,i_\sigma}^{(k_\sigma)};\ i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,s;\ \operatorname{Re}y_{p,i_p}^{(k_p)};\ i_p=0,\ldots,m_p^{(k_p)};\ \operatorname{Im}y_{p,i_p}^{(k_p)};\ i_p=0,\ldots,m_p^{(k_p)};\ k_p=1,\ldots,l_p;\ p=s+1,s+3,\ldots,r-1\} \]
is an \(N\)-vector; \(Q\) is a constant quasidiagonal \(N\times N\) matrix of the form

\[ Q= \begin{bmatrix} \cdots & & & & \\ & \overline{G_\sigma^{(k_\sigma)}} & & & 0\\ & & \cdots & & \\ 0 & & & \begin{vmatrix} \operatorname{Re} G_p^{(k_p)} & -\operatorname{Im} G_p^{(k_p)}\\ \operatorname{Im} G_p^{(k_p)} & \operatorname{Re} G_p^{(k_p)} \end{vmatrix} & \\ & & \cdots & & \end{bmatrix} \tag{2.30} \]

\[ (k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,s;\ k_p=1,\ldots,l_p;\ p=s+1,\ s+3,\ldots,r-1); \]

\(D\) is a constant \(N\times m\)-matrix of the form

\[ D= \begin{bmatrix} \cdot\\ \cdot\\ \cdot\\ C_\sigma^{(k_\sigma)}\\ \cdot\\ \cdot\\ \cdot\\ \operatorname{Re} C_p^{(k_p)}\\ \operatorname{Im} C_p^{(k_p)}\\ \cdot\\ \cdot\\ \cdot \end{bmatrix} \qquad \begin{gathered} (k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,s;\ k_p=1,\ldots,l_p;\\ p=s+1,\ s+3,\ldots,r-1); \end{gathered} \tag{2.31} \]

\[ \begin{aligned} V[v(t),z_t(0),z_t(-\tau),u] &=\{V_j[v(t),z_t(0),z_t(-\tau),u];\ j=1,\ldots,N\}\\ &=\{Y_{\sigma,i_\sigma}^{(k_\sigma)}[v(t),z_t(0),z_t(-\tau),u];\ i_\sigma=0,\ldots,m_\sigma^{(k_\sigma)};\ k_\sigma=1,\ldots,l_\sigma;\\ &\quad \sigma=1,\ldots,s;\ \operatorname{Re}Y_{p,i_p}^{(k_p)}[v(t),z_t(0),z_t(-\tau),u];\ i_p=0,\ldots,m_p^{(k_p)};\\ &\quad \operatorname{Im}Y_{p,i_p}^{(k_p)};\ i_p=0,\ldots,m_p^{(k_p)};\ k_p=1,\ldots,l_p;\ p=s+1,\ s+3,\ldots,r-1\} \end{aligned} \]

is an \(N\)-vector-functional.

In [11] the problem of stabilizing linear periodic systems with delay was considered and, in particular, a stabilizability criterion was established for linear systems (1.3) with constant coefficients, when the selected part of the spectrum of the operator \(P\) (2.2) has a simple structure and the control action \(u\) is a scalar (\(m=1\)). In the general case, the stabilizability criterion for system (1.3) can be formulated as follows:

Theorem 2.1. For the linear controllable system with delay (1.3) to be stabilizable, it is necessary and sufficient that the finite-dimensional subsystem

\[ \frac{dv}{dt}=Qv+Du, \tag{2.32} \]

corresponding to the unstable part of the spectrum of the operator \(P\) \((2.2)\)—\(\{\lambda_k;\ \operatorname{Re}\lambda_k \geqslant 0;\ k=1,\ldots,N\}\)—be completely controllable, i.e., that the rank of the matrix

\[ W=[D\ QD\ \ldots\ Q^{N-1}D] \tag{2.33} \]

be equal to \(N\).

In this case the stabilizing control action can be constructed on the curves \(x_t(\vartheta)\) in the form of a linear functional

\[ u=\Gamma v= \]

\[ =\Gamma\left( \begin{bmatrix} \cdots\\ \cdots\\ \left[x_t'(0)d_{\sigma,i_\sigma}^{*(k_\sigma)}(0)\right]_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}}\\ \cdots\\ \left[\operatorname{Re}x_t'(0)d_{p,i_p}^{*(k_p)}(0)\right]_{i_p=0}^{m_p^{(k_p)}}\\ \left[\operatorname{Im}x_t'(0)d_{p,i_p}^{*(k_p)}(0)\right]_{i_p=0}^{m_p^{(k_p)}}\\ \cdots\\ \cdots \end{bmatrix} + \int_{-\tau}^{0} \begin{bmatrix} \left[x_t'(\vartheta)A_\tau' d_{\sigma,i_\sigma}^{*(k_\sigma)}(\vartheta)\right]_{i_\sigma=0}^{m_\sigma^{(k_\sigma)}}\\ \cdots\\ \cdots\\ \left[\operatorname{Re}x_t'(\vartheta)A_\tau' d_{p,i_p}^{*(k_p)}(\vartheta)\right]_{i_p=0}^{m_p^{(k_p)}}\\ \left[\operatorname{Im}x_t'(\vartheta)A_\tau' d_{p,i_p}^{*(k_p)}(\vartheta)\right]_{i_p=0}^{m_p^{(k_p)}}\\ \cdots \end{bmatrix} \,d\vartheta \right) \tag{2.34} \]

\[ (k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,s;\ k_p=1,\ldots,l_p;\ p=s+1,s+3,\ldots,r-1), \]

where \(\Gamma\) is a constant \(m\times N\) matrix.

The proof of Theorem 2.1 in its main features repeats the proof of Theorem 3.1 in [11], and therefore is not given here.

Remark 2.2. The conditions of Theorem 2.1 can be checked, more conveniently, not for the matrix (2.28), but for the matrix

\[ W^*=[C\ GC\ \ldots\ G^{N-1}C], \tag{2.35} \]

corresponding to the equations of motion of the system in complex form (2.19), (2.20).

Remark 2.3. For a concrete computation of the matrix \(\Gamma\) in (2.34) one can use the solution of the problem of optimal stabilization [12], minimizing, for example, the functional

\[ J_u=\int_{0}^{\infty} \left[ \sum_{i=1}^{N}v_i^2(t)+\sum_{k=1}^{m}u_k^2(t) \right]dt \]

by the control (2.34) on the motions of the subsystem (2.32), by the method described in [13]. This problem has a solution if and only if the subsystem (2.32) is completely controllable [4].

Verification of the conditions of Theorem 2.1 is quite difficult. We therefore present here a more effective criterion for stabilizability of the system (1.3). This criterion is given without proof in the work of N. N. Krasovskii [15].

Let us consider \(r\) constant matrices \(\Delta^{(\sigma)}\) of dimensions \(l_\sigma \times m\) \((\sigma=1,\ldots,r)\) of the form

\[ \Delta^{(\sigma)}=\{\Delta_{k_\sigma i}\}=\{b'_{(i)}d_{\sigma,0}^{*(k_\sigma)}(0)\} \tag{2.36} \]

\[ (k_\sigma=1,\ldots,l_\sigma;\ i=1,\ldots,m;\ \sigma=1,\ldots,r). \]

Here \(d_{\sigma,0}^{*(k_\sigma)}(0)\) is the \(k_\sigma\)-th eigenvector of the adjoint system (2.7), corresponding to the eigenvalue \(\lambda_\sigma^*=-\lambda_\sigma\) \((\operatorname{Re}\lambda_\sigma^*\leq 0)\), determined according to (2.11), \((\sigma=1,\ldots,r;\ k_\sigma=1,\ldots,l_\sigma)\); \(b_{(i)}\) is the \(i\)-th column vector of the matrix \(B\) from (1.3). The following assertion holds:

Theorem 2.2. The linear controllable system with delay (1.3) is stabilizable if and only if the rank of each matrix \(\Delta^{(\sigma)}\) (2.36) is equal to \(l_\sigma\):

\[ r[\Delta^{(\sigma)}]=l_\sigma \qquad (\sigma=1,\ldots,r). \tag{2.37} \]

In this case the stabilizing control action can be constructed in the form (2.34).

Proof. Necessity. In accordance with Remark 2.2, the verification of the stabilization conditions may be carried out using the matrix \(W^*\) (2.35). Let the sign \(\sim\) between matrices mean that they have equal ranks. Performing block multiplication in (2.35), we have

\[ W\sim W^*= \left[ \left[ C_\sigma^{(k_\sigma)}\, G_\sigma^{(k_\sigma)} C_\sigma^{(k_\sigma)} \ldots \left(G_\sigma^{(k_\sigma)}\right)^{N-1} C_\sigma^{(k_\sigma)} \right] \right]_{\sigma=1}^{r} = W_\sigma \big|_{\sigma=1}^{r}. \tag{2.38} \]

From the form of the matrix (2.38), the necessity of the conditions of the theorem follows directly. Indeed, if the conditions (2.37) were not satisfied for at least one eigenvalue \(\lambda_j^*\), then the rank of the submatrix \(W_j\) would be less than

\[ \sum_{k_j=1}^{l_j} \left(m_j^{(k_j)}+1\right), \]

and, consequently, the rank of the matrix \(W\) (2.33) would be less than \(N\), which, according to Theorem 2.1, is impossible.

Sufficiency. Suppose now that the conditions of Theorem 2.2 are satisfied. Each Jordan block \(G_n^{(i)}\) can be represented in the form

\[ G_n^{(i)}=\lambda_n E+H_n^{(i)} \qquad (i=1,\ldots,l_n;\ n=1,\ldots,r), \tag{2.39} \]

where \(E\) and \(H_n^{(i)}\) are, respectively, the identity and superdiagonal ([15], p. 20) matrices.

In (2.38), from the second block column subtract the first multiplied by \(\lambda_1\); from the third, subtract the first multiplied by \(\lambda_1^2\cdot 1\) and the transformed second multiplied by \(\lambda_1\cdot 2\), and so on, subtracting from each \(k\)-th \((k=2,\ldots,N)\) block column the first multiplied by \(\lambda_1^{k-1}\), and each transformed \(i\)-th \((i=2,\ldots,k-1)\) multiplied by \(\lambda_1^{k-i} C_{k-i}^{\,k-1}\) \((C_n^m\) is the number of combinations of \(n\) taken \(m\) at a time). Using here the expansion (2.34) and taking into account that the matrix \(H_n^{(i)}\) is nilpotent with nilpotency index \(m_n^{(i)}+1\), we obtain

\[ W\sim W^*\sim \left[ \begin{array}{c} \left[C_1^{(k_1)}H_1^{(k_1)}C_1^{(k_1)}\ldots \left(H_1^{(k_1)}\right)^{m_1^{(k_1)}} C_1^{(k_1)}\,0\ldots 0\right]\sim W_1 \\[4pt] \left[ C_\sigma^{(k_\sigma)}G_\sigma^{(k_\sigma)}C_\sigma^{(k_\sigma)} \ldots \left(G_\sigma^{(k_\sigma)}\right)^{N-1}C_\sigma^{(k_\sigma)} \right]_{k_\sigma=1,\ \{\lambda_\sigma-\lambda_1\}}^{\sim W_\sigma} \end{array} \right]_{\sigma=2}^{r}. \tag{2.40} \]

(Here and everywhere below, the index \(\{\varepsilon\}\) on matrices means that the quantity \(\varepsilon\) stands on the main diagonals of the Jordan blocks \(G_\sigma^{(k_\sigma)}\) entering into them.)

We shall call the elementary-operations procedure carried out above a \(\lambda_1\)-operation.

Denote by \(n_\sigma\) the greatest nilpotency index of the matrix \(H_\sigma^{(k_\sigma)}\) \((k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r)\), \(\bigl(n_\sigma=\max_{k_\sigma} m_\sigma^{(k_\sigma)}+1\bigr)\). Obviously,

\[ n_1+\cdots+n_r \leq N. \]

From (2.40) select the submatrix \(M_1\) formed by the block columns beginning with the \((n_1+1)\)-st and by the block rows beginning with the \((l_1+1)\)-st. Apply to \(M_1\) the \((\lambda_2-\lambda_1)\)-operation. From the transformed matrix \(M_1\), select the submatrix \(M_2\) formed by the block columns beginning with the \((n_2+1)\)-st and by the block rows beginning with the \((l_2+1)\)-st, and apply to \(M_2\) the \((\lambda_3-\lambda_2)\)-operation, and so on. We note that each elementary \((\lambda_k-\lambda_{k-1})\)-operation on \(M_k\) is elementary both for the submatrices \(W_\sigma\) \((\sigma=1,\ldots,r)\) from (2.38), and for the matrix \(W\) (2.33) itself. Thus we have

\[ \left[ \begin{array}{ccc} [[T_1]] & [0_1] & \sim W_1\\ [[F_\sigma]] & [T_\sigma] & [0_\sigma]\sim W_\sigma \end{array} \right]_{\sigma=2}^{r}. \tag{2.41} \]

Here \([T_\sigma]\) is a \(\left(\sum_{k_\sigma=1}^{l_\sigma}(m_\sigma^{(k_\sigma)}+1)\times mn_\sigma\right)\)-matrix equal to

\[ [T_\sigma]= \left[ T_\sigma^{(k_\sigma)} H_\sigma^{(k_\sigma)} T_\sigma^{(k_\sigma)} \ldots H_\sigma^{(k_\sigma)\,m_\sigma^{(k_\sigma)}} T_\sigma^{(k_\sigma)} \right]_{k_\sigma=1}^{l_\sigma} \tag{2.42} \]

\[ (\sigma=1,\ldots,r); \]

\[ T_1^{(k_1)}=C_1^{(k_1)}\quad (k_1=1,\ldots,l_1);\qquad T_i^{(k_i)}= \prod_{p=1}^{i-1} \left[G_i^{(k_i)\,n_i-p}\right]_{\{\lambda_i-\lambda_{i-p}\}} C_i^{(k_i)} \tag{2.43} \]

\[ (i=2,\ldots,r;\ k_i=1,\ldots,l_i); \]

\([0_\sigma]\) is the zero

\[ \left(\sum_{k_\sigma=1}^{l_\sigma}(m_\sigma^{(k_\sigma)}+1)\times m\left(N-\sum_{j=1}^{\sigma} n_j\right)\right) \]

matrix \((\sigma=1,\ldots,r)\);

\([F_\sigma]\) is a

\[ \left(\sum_{k_\sigma=1}^{l_\sigma}(m_\sigma^{(k_\sigma)}+1)\times m\sum_{j=1}^{\sigma-1} n_j\right) \]

matrix of the form

\[ [F_\sigma]= \left[ F_{\sigma 1}^{(k_\sigma)} F_{\sigma 2}^{(k_\sigma)} \ldots F_{\sigma\,\sigma-1}^{(k_\sigma)} \right]_{k_\sigma=1}^{l_\sigma} \quad (\sigma=2,\ldots,r); \tag{2.44} \]

\[ \left[F_{\sigma i}^{(k_\sigma)}\right]= \left[ C_{\sigma i}^{(k_\sigma)} \right] \left[ G_\sigma^{(k_\sigma)} \right]_{\{\lambda_\sigma-\lambda_i\}} C_{\sigma i}^{(k_\sigma)} \ldots \left[ G_\sigma^{(k_\sigma)\,n_i-1} \right]_{\{\lambda_\sigma-\lambda_i\}} C_{\sigma i}^{(k_\sigma)} \biggr]_{k_\sigma=1}^{l_\sigma}; \]

\[ C_{\sigma,1}^{(k_\sigma)}=C_\sigma^{(k_\sigma)};\qquad C_{\sigma i}^{(k_\sigma)}= \prod_{p=1}^{i-1} \left[ G_\sigma^{(k_\sigma)\,n_i-p} \right]_{\{\lambda_\sigma-\lambda_{i-p}\}} C_\sigma^{(k_\sigma)} \biggr]_{k_\sigma=1}^{l_\sigma} \]

\[ (k_\sigma=1,\ldots,l_\sigma;\ i=2,\ldots,\sigma-1;\ \sigma=2,\ldots,r). \]

From (2.42) and the inequalities \(\lambda_k-\lambda_{k-1}\neq 0\) \((k=2,\ldots,r)\), it follows that

\[ [T_\sigma]\sim \left[ C_\sigma^{(k_\sigma)} H_\sigma^{(k_\sigma)} C_\sigma^{(k_\sigma)} \ldots H_\sigma^{(k_\sigma)\,m_\sigma^{(k_\sigma)}} C_\sigma^{(k_\sigma)} \right]_{k_\sigma=1}^{l_\sigma}. \tag{2.45} \]

By the condition of the theorem, the equalities (2.37), (2.36) hold. But then from (2.45), the equalities (2.24), \(f_{\sigma,0}^{(k_\sigma)}[R_i[B]]=b_{(i)}'d_{\sigma,0}^{*(k_\sigma)}(0)\) \((k_\sigma=1,\ldots,l_\sigma;\ \sigma=1,\ldots,r)\), it follows that

\[ r[T_\sigma]=\sum_{k_\sigma=1}^{l_\sigma}\left(m_\sigma^{(k_\sigma)}+1\right)\quad(\sigma=1,\ldots,r). \tag{2.46} \]

When the relations (2.46) are satisfied, the structure of the matrix (2.41) implies the equality

\[ r[W]=N, \tag{2.47} \]

which, on the basis of Theorem 2.1, completes the proof.

Remark 2.4. Let us note several equalities that are used below and follow from (2.41)—(2.45):

\[ r[W_\sigma]=r[F_\sigma]=r[T_\sigma]= r\left[ C_\sigma^{(k_\sigma)}\ldots H_\sigma^{(k_\sigma)m_\sigma^{(k_\sigma)}}C_\sigma^{(k_\sigma)} \right]_{k_\sigma=1}^{l_\sigma} \tag{2.48} \]

\[ (\sigma=1,\ldots,r), \]

\[ r[W]=\sum_{\sigma=1}^{r} r\left[ C_\sigma^{(k_\sigma)}H_\sigma^{(k_\sigma)}C_\sigma^{(k_\sigma)}\ldots H_\sigma^{(k_\sigma)m_\sigma^{(k_\sigma)}} \right]_{k_\sigma=1}^{l_\sigma}. \tag{2.49} \]

Denote

\[ \Gamma_\sigma= \left[ C_\sigma^{(k_\sigma)}H_\sigma^{(k_\sigma)}C_\sigma^{(k_\sigma)}\ldots H_\sigma^{(k_\sigma)m_\sigma^{(k_\sigma)}}C_\sigma^{(k_\sigma)} \right]_{k_\sigma=1}^{l_\sigma}. \tag{2.50} \]

§ 3. Stabilization of nonlinear control systems with delay in the first approximation.

It is assumed here that the vector-function \(X(x(t),x(t-\tau),u)\) from (1.1) satisfies the Lipschitz conditions

\[ \left|X_i(x^{(1)},y^{(1)},u^{(1)})- X_i(x^{(2)},y^{(2)},u^{(2)})\right|\leq \]

\[ \leq l\left(\|x^{(1)}-x^{(2)}\|+\|y^{(1)}-y^{(2)}\|+\|u^{(1)}-u^{(2)}\|\right), \tag{3.1} \]

\[ l=l_1\left(\|x^{(1)}\|+\|x^{(2)}\|+\|y^{(1)}\|+\|y^{(2)}\|+\|u^{(1)}\|+\|u^{(2)}\|\right)^\alpha \]

\[ (l_1=\mathrm{const}>0,\quad \alpha=\mathrm{const}>0). \]

Suppose that the finite-dimensional subsystem (2.32) is completely controllable, i.e., the rank of each matrix \(\Delta^{(\sigma)}\) (2.36) is equal to \(l_\sigma\). Then, on the basis of Theorem 2.2, the system of the first approximation (1.3) can be stabilized by the control action (2.34). But then, for the nonlinear system (1.1), all conditions of Theorem 33.2 from the book [2] (pp. 194—195) are satisfied and, consequently, the control action (2.34) stabilizes the system (1.1).

Remark 3.1. To prove the assertion just stated, one may also use a functional of the type (3.8) from [11], which, as can be shown, satisfies along the motions of systems (2.32), (2.20) all the conditions of Theorem 30.1 from the book [2]. For this it is enough simply to repeat the arguments of Theorem 33.2 from [2].

Now suppose that, for the finite-dimensional subsystem (2.32), the conditions of complete controllability are not satisfied and, consequently, the system of the first approximation, according to Theorem 2.2, cannot be stabilized.

This will take place if and only if the rank of at least one of the matrices \(\Delta^{(\sigma)}\) (2.36) is not equal to \(l_\sigma\):

\[ r\left[\Delta^{(i_n)}\right] < l_{i_n} \tag{3.2} \]

\[ (n=1,\ldots,p;\ 1 \leq p \leq r). \]

The finite-dimensional subsystem (2.32), by means of a nonsingular linear transformation, can be represented in the form [4]:

\[ \frac{dv^{[1]}}{dt}=Q_1v^{[1]}+Q_2v^{[2]}+D_1u, \tag{3.3} \]

\[ \frac{dv^{[2]}}{dt}=Q_3v^{[2]}, \tag{3.4} \]

\[ v= \begin{bmatrix} v^{[1]}\\ v^{[2]} \end{bmatrix}, \]

where the subsystem (3.3) is completely controllable; and \(Q_3\) is a

\[ \left(\sum_{n=1}^{p}\left(\sum_{k_{i_n}}^{l_{i_n}}\left(m_{i_n}^{k_{i_n}}+1\right)-r[\Gamma_{i_n}]\right)\right) \times \left(\sum_{n=1}^{p}\left(\sum_{k_{i_n}}^{l_{i_n}}\left(m_{i_n}^{k_{i_n}}+1\right)-r[\Gamma_\sigma]\right)\right) \]

matrix whose spectrum consists of the numbers \(\lambda_{i_n}\) \((n=1,\ldots,p;\ 1 \leq p \leq r)\) (see Remark 2.4).

Consider two possible cases.

Suppose that at least one of the inequalities (3.2) is satisfied for an eigenvalue \(\lambda_{i_j}^{*}\) with \(\operatorname{Re}\lambda_{i_j}^{*}\neq 0\) \((\operatorname{Re}\lambda_{i_j}^{*}<0)\). Then the spectrum of the matrix \(Q_3\) contains a number \(\lambda_{i_j}\) with \(\operatorname{Re}\lambda_{i_j}>0\), and, consequently, the first-approximation system (1.3), for any choice of the control action \(u[x_t(\vartheta)]\), will be unstable and will have a trajectory tending away from the point \(x=0\) as \(t\to\infty\) like an exponential. But then [5] the nonlinear system (1.1) will also be unstable.

Now suppose that conditions (3.2) are satisfied only for eigenvalues \(\lambda_{i_n}^{*}\) with \(\operatorname{Re}\lambda_{i_n}^{*}=0\) \((n=1,\ldots,p;\ 1\leq p\leq r)\). Then the spectrum of \(Q_3\) consists of numbers \(\lambda_{i_n}\) with \(\operatorname{Re}\lambda_{i_n}=0\) \((n=1,\ldots,p;\ 1\leq p\leq r)\). For the subsystem (3.3) the conditions of complete controllability are satisfied and, consequently, it can be stabilized [4] by a control action of the form (3.34). For each such fixed control action, the characteristic equation of the finite-dimensional subsystem (2.32), and therefore also of the first-approximation system (1.3), will have all roots \(\lambda_\sigma\) with \(\operatorname{Re}\lambda_\sigma\leq 0\), and among them at least one root \(\lambda_{i_n}\) \((n=1,\ldots,p;\ 1\leq p)\) with \(\operatorname{Re}\lambda_{i_n}=0\). Thus, in this case the possibility of stabilizing the nonlinear system (1.1) will be determined [6] by the terms of the nonlinear addition \(X(x(t), x(t-\tau), u)\).

The following assertion has been proved:

Theorem 3.1. Let the spectrum of the operator \(-P^{*}\) (2.8) contain \(r\) distinct eigenvalues \(\lambda_\sigma^{*}\) with \(\operatorname{Re}\lambda_\sigma^{*}\leq 0\). Let each eigenvalue \(\lambda_\sigma^{*}\) correspond to \(l_\sigma\) linearly independent eigenvectors \(d_{\sigma,0}^{*(k_\sigma)}(\vartheta)\) of the operator \(-P^{*}\) (2.8).

(1). If the rank of each matrix \(\Delta^{(\sigma)}\) (2.31) is equal to \(l_\sigma\):

\[ r\left[\Delta^{(\sigma)}\right]=l_\sigma \quad (\sigma=1,\ldots,r), \]

then the unperturbed motion \(x=0\) of system (1.1) is stabilized by the control action (2.34), independently of the terms of the nonlinear addition \(X(x(t),x(t-\tau),u)\).

(2). If the rank of at least one matrix \(\Delta^{(p_i)}\) from (2.31), corresponding to an eigenvalue \(\lambda_{p_i}^{*}\) with \(\operatorname{Re}\lambda_{p_i}^{*}<0\), is not equal to \(l_{p_i}\):

\[ r\left[\Delta^{(p_i)}\right]<l_{p_i}\quad (\operatorname{Re}\lambda_{p_i}^{*}<0;\ i=1,\ldots,q;\ 1\leq q\leq r), \]

then the unperturbed motion \(x=0\) of system (1.1) cannot be stabilized by any control action from the class (1.2), independently of the terms of the nonlinear addition \(X(x(t),x(t-\tau),u)\).

(3). If the rank of each matrix \(\Delta^{(n_j)}\) from (2.31), corresponding to an eigenvalue \(\lambda_{n_j}^{*}\) with \(\operatorname{Re}\lambda_{n_j}^{*}<0\), is equal to \(l_{n_j}\):

\[ r\left[\Delta^{(n_j)}\right]=l_{n_j} \]

\[ (\operatorname{Re}\lambda_{n_j}^{*}<0;\ j=1,\ldots,p;\ 0\leq p<r), \]

then the rank of at least one matrix \(\Delta^{(s_k)}\) from (2.31), corresponding to an eigenvalue \(\lambda_{s_k}^{*}\) with \(\operatorname{Re}\lambda_{s_k}^{*}=0\), is not equal to \(l_{s_k}\):

\[ r\left[\Delta^{(s_k)}\right]<l_{s_k}\quad (\operatorname{Re}\lambda_{s_k}^{*}=0;\ k=1,\ldots,n;\ 1\leq n\leq r), \]

and the possibility of stabilizing system (1.1) is determined by the terms of the nonlinear addition \(X(x(t),x(t-\tau),u)\).

By analogy with [4], where systems of ordinary differential equations are considered, we introduce the following definition (see Remark 2.4):

Definition 3.1. Cases in which the possibility of stabilization of the controlled system (1.1) is determined by the terms of the nonlinear addition \(X(x(t),x(t-\tau),u)\) will be called critical.

Let the spectrum of the operator \(P\) (2.2) contain \(l\) distinct eigenvalues \(\lambda_{p_i}\) with \(\operatorname{Re}\lambda_{p_i}>0\) \((i=1,\ldots,l)\), \(q\) distinct purely imaginary eigenvalues \(\lambda_{r_k}\ne0\) with \(\operatorname{Re}\lambda_{r_k}=0\) \((k=1,\ldots,q)\), and a zero eigenvalue \(\lambda_j=0\) of multiplicity \(m_j\).

We shall say that a critical case of \(n\) zero roots and \(s\) purely imaginary roots \(\lambda_{r_k}\) \((k=1,\ldots,s)\) \((0\leq n\leq m_j;\ 0\leq s\leq q;\ n+s>1)\) occurs if the following conditions are satisfied:

\[ (1)\quad \sum_{i=1}^{l} m_i-\sum_{i=1}^{l} r[\Gamma_{p_i}]=0 \tag{3.5} \]

i.e., the rank of each matrix \(\Delta^{(p_i)}\) (2.36) is equal to \(l_{p_i}\),

\[ (2)\quad m_j-r[\Gamma_j]=n, \tag{3.6} \]

\[ (3)\quad \sum_{k=1}^{q} m_k-\sum_{k=1}^{q} r[\Gamma_{r_k}]=s \quad \left(\text{provided } r\left[\Delta^{(r_k)}\right]<l_{r_k};\ k=1,\ldots,s\right). \tag{3.7} \]

Remark 3.1. In accordance with the terminology of [4], the critical case of \(n\) zero roots and \(s\) purely imaginary roots \(\lambda_{r_k}\) \((k=1,\ldots,s)\) occurs if the spectrum of the matrix \(Q_3\) from (3.4) consists of the zero eigenvalue \(\lambda_j\) of multiplicity \(n\) and \(s\) purely imaginary numbers \(\lambda_{r_k}\) \((k=1,\ldots,s)\), but contains no number \(\lambda_{p_i}\) with \(\operatorname{Re}\lambda_{p_i}>0\).

Remark 3.2. To verify the stabilizability conditions of the controlled system (1.1), and also to construct the stabilizing control action \(u[x_t(\vartheta)]\), it is necessary to know the root elements \(d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta)\) and \(d_{\sigma,i_\sigma}^{*(k_\sigma)}(-\vartheta)\) of the operators \(P\) (2.2) and \(-P^*\) (2.8).

The problem of determining the vectors \(d_{\sigma,i_\sigma}^{(k_\sigma)}(\vartheta)\), \(d_{\sigma,i_\sigma}^{*(k_\sigma)}(-\vartheta)\) by (2.10), (2.11) reduces to solving the equations

\[ [A-\lambda_\sigma E + A_\tau e^{-\lambda_\sigma \tau}]\,d_{\sigma,0}^{(k_\sigma)}(0)=0, \]

\[ [A-\lambda_\sigma E + A\tau e^{-\lambda_\sigma \tau}]\,d_{\sigma,i}^{(k_\sigma)}(0) = d_{\sigma,i-1}^{(k_\sigma)}(0) - A_\tau e^{-\lambda_\sigma \tau} \sum_{n=1}^{i}\frac{(-\tau)^n}{n!}\, d_{\sigma,i-n}^{(k_\sigma)}(0) \tag{3.8} \]

and

\[ [A' + \lambda_\sigma^* E + A_\tau' e^{\lambda_\sigma^* \tau}]\, d_{\sigma,0}^{*(k_\sigma)}(0)=0, \tag{3.9} \]

\[ [A' + \lambda_\sigma^* E + A_\tau' e^{\lambda_\sigma^* \tau}]\, d_{\sigma,i}^{*(k_\sigma)}(0) = d_{\sigma,i-1}^{*(k_\sigma)}(0) - A_\tau' e^{\lambda_\sigma^* \tau} \sum_{n=1}^{i}\frac{(-\tau)^n}{n!}\, d_{\sigma,i-n}^{*(k_\sigma)}(0) \]

\[ (\operatorname{Re}\lambda_\sigma \geq 0;\quad \lambda_\sigma^*=-\lambda_\sigma;\quad \sigma=1,\ldots,r;\quad k_\sigma=1,\ldots,l_\sigma;\quad i=1,\ldots,m_\sigma^{(k_\sigma)}). \]

For constructing solutions of the systems of equations (3.8), (3.9), one may use the arguments of N. G. Chetaev from the book [1] (pp. 57–64), whose validity, as can be shown, also holds for equations of the type (3.8), (3.9).

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Received by the editors
February 8, 1965

Ural State University
named after A. M. Gorky