Arbitrary Coefficient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays

Abstract

We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function by linear static output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for the solvability of the arbitrary coefficient assignment problem by the static output feedback controller. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We provide an example illustrating our results.

1. Introduction

A large number of works have been devoted to the problem of stability of time-delay systems and problems of stabilization for control systems with delays (see reviews [1,2,3,4]). A number of methods have been developed to solve this problem. One of the methods, known as the Lyapunov–Krasovskii functional approach [5], allows one to obtain sufficient conditions for asymptotic and exponential stabilization of delayed systems [6,7,8]. This method historically traces back to the second Lyapunov method. Another approach to studying problems of stability and stabilization of time-delay systems is an eigenvalue-based approach [9]. This approach traces back to the first Lyapunov method. Here, the goal is to find conditions providing the desired assignment of the spectrum of the system, that is, the set of zeros of the characteristic function of the system.

The classical problem of spectrum assignment (for systems without delays) is usually studied as the problem of coefficient assignment for characteristic function (in another terminology as the problem of modal control) and is as follows: consider a linear time-invariant control system

$$\dot{x}=Ax+Bu,$$

(1)

$x\in {\mathbb{K}}^n$, $u\in {\mathbb{K}}^m$ (here $\mathbb{K}=\mathbb{C}$ or $\mathbb{K}=\mathbb{R}$). Let the controller have the form of linear static state feedback

$$u=Qx.$$

(2)

The closed-loop system (1), (2) takes the form

$$\dot{x}=(A+BQ)x.$$

(3)

Definition 1.

System (1) is called arbitrary spectrum assignable by means of static state feedback (2) if, for any ${\gamma }_i\in \mathbb{K}$, $i=\overline{1,n}$, there exists a gain matrix Q such that the characteristic polynomial of the matrix $A+BQ$ of the system (3) coincides with the polynomial

$${\lambda }^n+{\gamma }_1{\lambda }^{n-1}+\dots +{\gamma }_n.$$

(4)

The problem of arbitrary spectrum assignment was solved in [10] (for $\mathbb{K}=\mathbb{C}$) and in [11] (for $\mathbb{K}=\mathbb{R}$)—namely, it has been proven that complete controllability of (1) is a necessary and sufficient condition for arbitrary spectrum assignability (i.e., for modal controllability) of system (3). This property is a sufficient condition for exponential stabilization of the system (with any given decay of rate).

For systems with delays, the spectrum is in general infinite. The spectrum depends on coefficients of the characteristic function. The problem of arbitrary spectrum assignment of linear time-delay systems (in contrast to systems without delays) is not equivalent to the problem of arbitrary coefficient assignment (ACA) for the characteristic function of the closed-loop system. A number of early works dealt with coefficient assignment, spectrum assignment, and stabilizability problems for linear time-delay systems by means of static state feedback with delays: sufficient conditions for ACA were obtained in [12] for systems with multiple lumped delays; decoupling and canonical forms were used for coefficient assignment in [13,14]; stabilizability and spectrum assignment for linear autonomous systems with general time delays were studied in [15]; problems of stabilizability independent of delay were developed for the class of delay differential systems of the retarded type with commensurate time delays in [16]; spectrum placement problem was studied in [17] by using a ring of delay operators.

In a study [18], an approach was developed for assigning an arbitrary finite spectrum for linear systems with delays. Later, the finite spectrum assignment problem for time-delay systems by linear state feedback was studied in [19] for systems with one lumped delay in the states with the scalar controller, in [20] for systems with multiple commensurate lumped delays in the states with the scalar controller, in [21] for systems with multiple commensurate lumped delays in the states and control with the scalar controller, in [22] for systems with multiple commensurate lumped delays in the states and control with the multidimensional controller, in [23] for systems with multiple commensurate lumped and distributed delays in the states with the scalar controller, and in [24] for systems of neutral type with multiple commensurate lumped delays in the states with the scalar controller. In [25], the ACA problem was studied for single-input single-output (SISO) systems with commensurate lumped delays in the states by the dynamic output feedback controller. In [26], the problem of stabilization of linear systems with both input and state delays by observer–predictors was studied. In the paper [27], the assignment of the poles of a second-order vibrating system through state feedback with one lumped delay was studied by means of the linear matrix inequality (LMI) approach. A partial pole assignment approach was presented in [28] for second-order systems with time delay.

Some recent important works on stochastic time-delay systems have been reported in [29,30,31]. In [29], the event-triggered control problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control was studied. In [30], the stability problem for a class of stochastic delay nonlinear systems driven by G-Brownian motion was studied. The global stabilization of stochastic nonlinear systems with time-varying delay, unknown powers, and SISS stochastic inverse dynamics was studied in [31].

The problem of spectrum assignment by static output feedback (for systems without delays) is as follows: consider a linear time-invariant control system

$$\dot{x}=Fx+Gu,y=Hx,$$

(5)

$x\in {\mathbb{K}}^n$, $u\in {\mathbb{K}}^m$, $y\in {\mathbb{K}}^k$. Let the controller have the form of linear static output feedback where

$$u=Ky.$$

(6)

The closed-loop system (5), (6) takes the form

$$\dot{x}=(F+GKH)x.$$

(7)

Definition 2.

System (5) is called arbitrary spectrum assignable by means of static output feedback (6) if, for any ${\gamma }_i\in \mathbb{K}$, $i=\overline{1,n}$, there exists a gain matrix K such that the characteristic polynomial of the matrix $F+GKH$ of the system (7) coincides with the polynomial (4).

The static output feedback problem of eigenvalue assignment (in particular, of stabilization) is one of the most important open questions in control theory; see reviews [32,33]. This problem has been studied for over 40 years by various authors. The most significant results have been obtained in [34] for $\mathbb{K}=\mathbb{C}$, in [35,36,37] for $\mathbb{K}=\mathbb{R}$.

The problems of stabilization and spectrum assignment by static output feedback for time-delay systems are more difficult to study. A study [38] considered the problem of stabilization of system (5) by static output feedback with a delay $u\left(t\right)=Ky(t-\tau )$. For SISO systems with delays, necessary conditions for the existence of static output feedback stabilizing controllers were derived in [39]. Another study [40] considered the output feedback stabilization problem for a class of linear SISO systems with I/O network delays. The problem of stabilization of linear time-varying systems with input delays via delayed static output feedback is studied in [41].

Consider a control system defined by a linear differential equation of nth order where the input is a linear combination of m variables and its derivatives of order $\le n-p$, and the output is a k-dimensional vector of linear combinations of the state x and its derivatives of order $\le p-1$$(1\le p\le n)$:

$$\begin{array}{c}{x}^{\left(n\right)}+a_1{x}^{(n-1)}+\dots +a_{n}x=\\ \begin{array}{cc}\hfill & =b_{p1}{u}_1^{(n-p)}+b_{p+1,1}{u}_1^{(n-p-1)}+\dots +b_{n1}{u}_1+\dots \hfill \\ \hfill & +b_{pm}{u}_m^{(n-p)}+b_{p+1,m}{u}_m^{(n-p-1)}+\dots +b_{nm}{u}_m,\hfill \end{array}\end{array}$$

(8)

$$\begin{array}{cc}\hfill y_1& =c_{11}x+c_{21}{x}^{\prime }+\dots +c_{p1}{x}^{(p-1)},\dots ,\hfill \\ \hfill y_k& =c_{1k}x+c_{2k}{x}^{\prime }+\dots +c_{pk}{x}^{(p-1)}.\hfill \end{array}$$

(9)

Here, $x\in \mathbb{K}$ is a state variable, $u_{\alpha }\in \mathbb{K}$ are control variables, $y_{\beta }\in \mathbb{K}$ are output variables, $a_i$, $b_{l\alpha }$, $c_{\nu \beta }\in \mathbb{K}$, $i=\overline{1,n}$, $l=\overline{p,n}$, $\nu =\overline{1,p}$, $\alpha =\overline{1,m}$, $\beta =\overline{1,k}$. Construct the vectors $u=\mathrm{col}(u_1,\dots ,u_m)\in {\mathbb{K}}^m$, $y=\mathrm{col}(y_1,\dots ,y_k)\in {\mathbb{K}}^k$. Let the control in system (8), (9) have the form of linear static output feedback:

$$\begin{array}{c}u=Qy.\end{array}$$

(10)

System (8), (9), (10) can be rewritten in the form (5), (6). We say that system (8), (9) is arbitrary coefficient assignable by linear static output feedback (10) if for any ${\gamma }_i\in \mathbb{K}$, $i=\overline{1,n}$, there exists a linear static output feedback control (10) such that the characteristic polynomial of the closed-loop system (8), (9), (10) has the form (4). The conditions imposed on the orders of derivatives in (8) and (9) are natural because one needs the orders of the derivatives on the right-hand side of the closed-loop system to be less than n.

For the scalar system (8), (9), (10), the arbitrary coefficient assignment problem by static output feedback has been solved in [42]. Construct the matrices $B=\left\{b_{l\alpha }\right\}$, $l=\overline{1,n}$, $\alpha =\overline{1,m}$, and $C=\left\{c_{\nu \beta }\right\}$, $\nu =\overline{1,n}$, $\beta =\overline{1,k}$, where $b_{l\alpha }:=0$ for $l<p$ and $c_{\nu \beta }:=0$ for $\nu >p$. Let $J:=\left\{{\vartheta }_{ij}\right\}\in M_n\left(\mathbb{R}\right)$ where ${\vartheta }_{ij}=1$ for $j=i+1$ and ${\vartheta }_{ij}=0$ for $j\ne i+1$. Let T denote the transposition of a matrix. The following theorem holds [42]:

Theorem 1.

System (8), (9) is arbitrary coefficient assignable by linear static output feedback (10) if and only if the matrices

$$C^{T}B,C^{T}JB,\dots ,C^T{J}^{n-1}B$$

are linearly independent.

Extension of Theorem 1 to the case when the state, input, and output are multidimensional (i.e., $x\in {\mathbb{K}}^s$, $u_{\alpha }\in {\mathbb{K}}^s$, $y_{\beta }\in {\mathbb{K}}^s$, $s\ge 1$) was obtained in [43].

In the present paper, we extend Theorem 1 on arbitrary coefficient assignment by static output feedback to systems with non-commensurate lumped and distributed delays in the state variable.

Notation. Denote $\mathbb{K}=\mathbb{C}$ or $\mathbb{K}=\mathbb{R}$; ${\mathbb{K}}^n=\{x=\mathrm{col}(x_1,\dots ,x_n):x_i\in \mathbb{K}\}$ is the linear space of column vectors over $\mathbb{K}$; $M_{m,n}\left(\mathbb{K}\right)$ is the space of $m\times n$-matrices over $\mathbb{K}$; $M_n\left(\mathbb{K}\right):=M_{n,n}\left(\mathbb{K}\right)$; $I\in M_n\left(\mathbb{K}\right)$ is the identity matrix; $\overline{a}$ is the complex conjugation of a; T is the transposition of a vector or a matrix; * is the Hermitian conjugation, i.e., $A^{*}={\overline{A}}^T$; $\mathrm{Sp}H$ is the trace of a matrix $H\in M_n\left(\mathbb{K}\right)$; for a matrix $H\in M_n\left(\mathbb{K}\right)$, we use the denotation $H^0:=I$; denote $J:=\left\{{ϵ}_{ij}\right\}\in M_n\left(\mathbb{R}\right)$ where ${ϵ}_{ij}=1$ for $j=i+1$ and ${ϵ}_{ij}=0$ for $j\ne i+1$; denote by $\mathrm{vec}:M_{p,q}\left(\mathbb{K}\right)\to {\mathbb{K}}^{pq}$ the mapping, which “unrolls” a matrix $Z=\left\{z_{ij}\right\}$, $i=\overline{1,p}$, $j=\overline{1,q}$, by rows into the column vector $\mathrm{vec}Z=\mathrm{col}(z_{11},\dots ,z_{1q},\dots ,z_{p1},\dots ,z_{pq})\in {\mathbb{K}}^{pq}$.

2. Main Results

Consider a control system defined by a linear time-invariant differential equation of nth order with multiple non-commensurate lumped and distributed delays in the state variable $x\in \mathbb{K}$; the input is a linear combination of m variables and its derivatives of order $\le n-p$; the output is a k-dimensional vector of linear combinations of the state x and its derivatives of order $\le p-1$,

$$\begin{array}{cc}\hfill & x^{\left(n\right)}\left(t\right)+a_{10}{x}^{(n-1)}\left(t\right)+a_{11}{x}^{(n-1)}(t-h_1)+\dots +a_{1s}{x}^{(n-1)}(t-h_s)+\dots \hfill \\ \hfill & +a_{n0}x\left(t\right)+a_{n1}x(t-h_1)+\dots +a_{ns}x(t-h_s)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }\left(\tau \right)x^{(n-i)}(t+\tau )d\tau \hfill \\ \hfill & =b_{p1}{u}_1^{(n-p)}\left(t\right)+b_{p+1,1}{u}_1^{(n-p-1)}\left(t\right)+\dots +b_{n1}{u}_1\left(t\right)\hfill \\ \hfill & +\dots +b_{pm}{u}_m^{(n-p)}\left(t\right)+\dots +b_{nm}{u}_m\left(t\right),t>0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill y_1\left(t\right)& ={\overline{c}}_{11}x\left(t\right)+{\overline{c}}_{21}{x}^{\prime }\left(t\right)+\dots +{\overline{c}}_{p1}{x}^{(p-1)}\left(t\right),\dots ,\hfill \\ \hfill y_k\left(t\right)& ={\overline{c}}_{1k}x\left(t\right)+{\overline{c}}_{2k}{x}^{\prime }\left(t\right)+\dots +{\overline{c}}_{pk}{x}^{(p-1)}\left(t\right),\hfill \end{array}$$

(12)

with initial conditions $x^{(n-i)}\left(\tau \right)={\varphi }_i\left(\tau \right)$, $\tau \in [-h_s,0]$; here $0=h_0<h_1<\dots <h_s$ are constant delays, ${\varphi }_i:[-h_s,0]\to \mathbb{K}$ are continuous functions; $a_{ij}$, $b_{l\alpha }$, $c_{\nu \beta }\in \mathbb{K}$, $i=\overline{1,n}$, $j=\overline{0,s}$, $l=\overline{p,n}$, $\alpha =\overline{1,m}$, $\nu =\overline{1,p}$, $\beta =\overline{1,k}$; $g_{i\eta }:[-h_{\eta },-h_{\eta -1}]\to \mathbb{K}$ are integrable functions ($i=\overline{1,n}$, $\eta =\overline{1,s}$); $u=\mathrm{col}(u_1,\dots ,u_m)\in {\mathbb{K}}^m$ is a control vector and $y=\mathrm{col}(y_1,\dots ,y_k)\in {\mathbb{K}}^k$ is an output vector; $p\in \left\{\overline{1,n}\right\}$; the complex conjugation to $c_{\nu \beta }$ is used for convenience of notation (for consistency with previous works).

Let the controller in system (11), (12) have the form of linear static output feedback with lumped and distributed delays:

$$u\left(t\right)=\sum _{\rho =0}^{\theta }{Q}_{\rho }y(t-{\sigma }_{\rho })+\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}{R}_{\varkappa }\left(\tau \right)y(t+\tau )d\tau ,$$

(13)

$y\left(t\right)=0$, $t<-h_s$. Here $\theta \ge 0$ is an integer, $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$ are constant delays, $Q_{\rho }=\left\{q_{\alpha \beta }^{\rho }\right\}\in M_{m,k}\left(\mathbb{K}\right)$ are constant matrices ($\rho =\overline{0,\theta }$), $R_{\varkappa }\left(\tau \right)=\left\{r_{\alpha \beta }^{\varkappa }\left(\tau \right)\right\}\in M_{m,k}\left(\mathbb{K}\right)$, $r_{\alpha \beta }^{\varkappa }:[-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}]\to \mathbb{K}$ are integrable functions ($\varkappa =\overline{1,\theta }$), $\alpha =\overline{1,m}$, $\beta =\overline{1,k}$. By (12), we have

$$y_{\beta }\left(t\right)=\sum _{\nu =1}^p{\overline{c}}_{\nu \beta }{x}^{(\nu -1)}\left(t\right),\beta =\overline{1,k}.$$

Hence, for any $\alpha =\overline{1,m}$,

$$\begin{array}{cc}\hfill & {\displaystyle u_{\alpha }(t)=\sum _{\beta =1}^k[\sum _{\rho =0}^{\theta }{q}_{\alpha \beta }^{\rho }(\sum _{\nu =1}^p{\overline{c}}_{\nu \beta }{x}^{(\nu -1)}(t-{\sigma }_{\rho }))}\hfill \\ \hfill & +\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}{r}_{\alpha \beta }^{\varkappa }(\tau )(\sum _{\nu =1}^p{\overline{c}}_{\nu \beta }{x}^{(\nu -1)}(t+\tau ))d\tau ].\hfill \end{array}$$

The closed-loop system (11), (12), (13) takes the form

$$\begin{array}{cc}\hfill & x^{(n)}(t)+\sum _{i=1}^n\sum _{j=0}^s{a}_{ij}{x}^{(n-i)}(t-h_j)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }(\tau )x^{(n-i)}(t+\tau )d\tau \hfill \\ \hfill & -\sum _{\alpha =1}^m\sum _{l=p}^n{b}_{l\alpha }(\sum _{\beta =1}^k[\sum _{\rho =0}^{\theta }{q}_{\alpha \beta }^{\rho }(\sum _{\nu =1}^p{\overline{c}}_{\nu \beta }{x}^{(\nu -1)}(t-{\sigma }_{\rho }))\hfill \\ \hfill & +\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}{r}_{\alpha \beta }^{\varkappa }(\tau )(\sum _{\nu =1}^p{\overline{c}}_{\nu \beta }{x}^{(\nu -1)}(t+\tau ))d\tau ]{)}^{(n-l)}=0.\hfill \end{array}$$

(14)

Denote by $\phi (\lambda )$ the characteristic function of the closed-loop system (14). Then

$$\begin{array}{c}\phi (\lambda )={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{j=0}^s{a}_{ij}{e}^{-\lambda h_j}+\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }(\tau )e^{\lambda \tau }d\tau \right)\\ -\sum _{\alpha =1}^m\sum _{l=p}^n{b}_{l\alpha }\left(\sum _{\nu =1}^p[\sum _{\beta =1}^k(\sum _{\rho =0}^{\theta }{q}_{\alpha \beta }^{\rho }{\overline{c}}_{\nu \beta }{e}^{-\lambda {\sigma }_{\rho }})+\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}{r}_{\alpha \beta }^{\varkappa }(\tau ){\overline{c}}_{\nu \beta }{e}^{\lambda \tau }d\tau ]{\lambda }^{n-l+\nu -1}\right).\end{array}$$

(15)

The set $\mathrm{\Lambda }=\{\lambda \in \mathbb{C}:\phi (\lambda )=0\}$ is the spectrum of system (14). If $\mathrm{\Lambda }$ is contained in the open left half-plane, then system (14) is exponentially stable. The spectrum of system (14) is uniquely determined by the coefficients of system (14). We study the problem of assigning an arbitrary coefficients to the characteristic function (15) of the closed-loop system.

Definition 3.

System (11), (12) is said to be arbitrary coefficient assignable by static output feedback (13) if, for any integer $\ell \ge 0$, for any given $0={\omega }_0<{\omega }_1<\dots <{\omega }_{\ell }$, for any numbers ${\gamma }_{i\mu }\in \mathbb{K}$, $i=\overline{1,n}$, $\mu =\overline{0,\ell }$, and for any integrable functions ${\delta }_{i\xi }:[-{\omega }_{\xi },-{\omega }_{\xi -1}]\to \mathbb{K}$, $i=\overline{1,n}$, $\xi =\overline{1,\ell }$, there exist an integer $\theta \ge 0$, numbers $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$, constant matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, and integrable matrix functions $R_{\varkappa }:[-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}]\to M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$, such that the characteristic function (15) of the closed-loop system (14) satisfies the equality

$$\phi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{\mu =0}^{\ell }{\gamma }_{i\mu }{e}^{-\lambda {\omega }_{\mu }}+\sum _{\xi =1}^{\ell }{\int }_{-{\omega }_{\xi }}^{-{\omega }_{\xi -1}}{\delta }_{i\xi }\left(\tau \right)e^{\lambda \tau }d\tau \right).$$

Remark 1.

The problem of arbitrary coefficient assignment was studied and solved in [44] for system (11), (12), (13) with only lumped delays ($g_{i\eta }\left(\tau \right)\equiv 0$, $i=\overline{1,n}$, $\eta =\overline{1,s}$; $R_{\varkappa }\left(\tau \right)\equiv 0\in M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$); in [45] for systems with only one lumped and one distributed delays ($s=1$, $\theta =1$, $h_1={\sigma }_1=h>0$); in [46] for systems with multiple commensurate delays ($h_j=jh$, $j=\overline{0,s}$; ${\sigma }_{\rho }=\rho h$, $\rho =\overline{0,\theta }$; $h>0$). Here, we consider a more general case.

From system (11), (12), construct the matrices $B=\left\{b_{l\alpha }\right\}$, $l=\overline{1,n}$, $\alpha =\overline{1,m}$, and $C=\left\{c_{\nu \beta }\right\}$, $\nu =\overline{1,n}$, $\beta =\overline{1,k}$, where $b_{l\alpha }:=0$ for $l<p$ and $c_{\nu \beta }:=0$ for $\nu >p$.

Theorem 2.

System (11), (12) is arbitrary coefficient assignable by the static output feedback controller (13) if and only if the matrices

$$\begin{array}{c}{C}^{*}B,C^{*}JB,,\dots ,C^{*}{J}^{n-1}B\end{array}$$

(16)

are linearly independent.

Proof. 

Let a function

$$\psi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{\mu =0}^{\ell }{\gamma }_{i\mu }{e}^{-\lambda {\omega }_{\mu }}+\sum _{\xi =1}^{\ell }{\int }_{-{\omega }_{\xi }}^{-{\omega }_{\xi -1}}{\delta }_{i\xi }\left(\tau \right)e^{\lambda \tau }d\tau \right)$$

(17)

be given, where $\ell \ge 0$ is an arbitrary integer, $0={\omega }_0<{\omega }_1<\dots <{\omega }_{\ell }$ are arbitrary given delays, ${\gamma }_{i\mu }\in \mathbb{K}$ are arbitrary numbers, and ${\delta }_{i\xi }:[-{\omega }_{\xi },-{\omega }_{\xi -1}]\to \mathbb{K}$ are arbitrary integrable functions. One needs to construct a number $\theta \ge 0$, numbers $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$, matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, and integrable functions $R_{\varkappa }:[-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}]\to M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$, such that the characteristic function (15) of the closed-loop system (14) satisfies the equality

$$\phi \left(\lambda \right)=\psi \left(\lambda \right).$$

(18)

Let us write the characteristic function (15) of the closed-loop system (14) in the form

$$\phi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{j=0}^s{a}_{ij}{e}^{-\lambda h_j}+\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }\left(\tau \right)e^{\lambda \tau }d\tau \right)-\Delta ,$$

(19)

where $\Delta ={\Delta }_1+{\Delta }_2$, and

$${\Delta }_1=\sum _{\rho =0}^{\theta }\sum _{\alpha =1}^m\sum _{\beta =1}^k\sum _{l=p}^n\sum _{\nu =1}^p{b}_{l\alpha }{\overline{c}}_{\nu \beta }{q}_{\alpha \beta }^{\rho }{\lambda }^{n-l+\nu -1}{e}^{-\lambda {\sigma }_{\rho }},$$

$${\Delta }_2=\sum _{\varkappa =1}^{\theta }\sum _{\alpha =1}^m\sum _{\beta =1}^k\sum _{l=p}^n\sum _{\nu =1}^p{b}_{l\alpha }{\overline{c}}_{\nu \beta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}{r}_{\alpha \beta }^{\varkappa }\left(\tau \right){\lambda }^{n-l+\nu -1}{e}^{\lambda \tau }d\tau .$$

By using the proof of [44] (Theorem 2) (see the reasoning from Formula (11) to Formula (15) in [44]), we obtain that

$${\Delta }_1=\sum _{\rho =0}^{\theta }\sum _{i=1}^n\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right){\lambda }^{n-i}{e}^{-\lambda {\sigma }_{\rho }}.$$

By the same reasoning, we obtain that

$${\Delta }_2=\sum _{\varkappa =1}^{\theta }\sum _{i=1}^n{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}\mathrm{Sp}\left(C^{*}{J}^{i-1}B{R}_{\varkappa }\left(\tau \right)\right){\lambda }^{n-i}{e}^{\lambda \tau }d\tau .$$

Substituting ${\Delta }_1$ and ${\Delta }_2$ in (19), we obtain

$$\begin{array}{cc}\hfill & \phi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{j=0}^s{a}_{ij}{e}^{-\lambda h_j}+\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }\left(\tau \right)e^{\lambda \tau }d\tau \right)\hfill \\ \hfill & -\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{\rho =0}^{\theta }\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right)e^{-\lambda {\sigma }_{\rho }}+\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}\mathrm{Sp}\left(C^{*}{J}^{i-1}B{R}_{\varkappa }\left(\tau \right)\right)e^{\lambda \tau }d\tau \right).\hfill \end{array}$$

(20)

Taking into account (20), (18), and (17), we obtain that system (11), (12) is arbitrary coefficient assignable by (13) iff there exist $\theta \ge 0$, numbers $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$, matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, and integrable matrix functions $R_{\varkappa }:[-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}]\to M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$, such that for all $i=\overline{1,n}$ the following equalities hold:

$$\begin{array}{cc}\hfill & \sum _{\mu =0}^{\ell }{\gamma }_{i\mu }{e}^{-\lambda {\omega }_{\mu }}=\sum _{j=0}^s{a}_{ij}{e}^{-\lambda h_j}-\sum _{\rho =0}^{\theta }\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right)e^{-\lambda {\sigma }_{\rho }},\hfill \\ \hfill & \sum _{\xi =1}^{\ell }{\int }_{-{\omega }_{\xi }}^{-{\omega }_{\xi -1}}{\delta }_{i\xi }\left(\tau \right)e^{\lambda \tau }d\tau =\sum _{\eta =1}^s{\int }_{-h_{\eta }}^{-h_{\eta -1}}{g}_{i\eta }\left(\tau \right)e^{\lambda \tau }d\tau \hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & -\sum _{\varkappa =1}^{\theta }{\int }_{-{\sigma }_{\varkappa }}^{-{\sigma }_{\varkappa -1}}\mathrm{Sp}\left(C^{*}{J}^{i-1}B{R}_{\varkappa }\left(\tau \right)\right)e^{\lambda \tau }d\tau .\hfill \end{array}$$

(22)

Denote

$$T_1:=\{{\omega }_1,\dots ,{\omega }_{\ell }\},T_2:=\{h_1,\dots ,h_s\},$$

$$S_1:=\{\mu \in \left\{\overline{1,\ell }\right\}:{\omega }_{\mu }\in T_1\backslash T_2\},S_2:=\{\mu \in \left\{\overline{1,\ell }\right\}:{\omega }_{\mu }\in T_1\cap T_2\},$$

$$S_3:=\{j\in \left\{\overline{1,s}\right\}:h_j\in T_1\cap T_2\},S_4:=\{j\in \left\{\overline{1,s}\right\}:h_j\in T_2\backslash T_1\}.$$

Set $T:=T_1\cup T_2$, $\theta :=\left|T\right|$. Set ${\sigma }_0:=0$. Let us denote the elements of the set T as ${\sigma }_1<{\sigma }_2<\dots <{\sigma }_{\theta }$.

Let

$$K_1:=\{\rho \in \left\{\overline{1,\theta }\right\}:\exists \mu \in S_1{\sigma }_{\rho }={\omega }_{\mu }\},$$

$$K_2:=\{\rho \in \left\{\overline{1,\theta }\right\}:\exists j\in S_3{\sigma }_{\rho }=h_j\},$$

$$K_3:=\{\rho \in \left\{\overline{1,\theta }\right\}:\exists j\in S_4{\sigma }_{\rho }=h_j\}.$$

Then, equalities (21) take the form

$$\begin{array}{cc}\hfill & {\gamma }_{i0}+\left(\sum _{\mu \in S_1}+\sum _{\mu \in S_2}\right){\gamma }_{i\mu }{e}^{-\lambda {\omega }_{\mu }}\hfill \\ \hfill & =a_{i0}-\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_0\right)+\left(\sum _{j\in S_3}+\sum _{j\in S_4}\right)a_{ij}{e}^{-\lambda h_j}\hfill \\ \hfill & -\left(\sum _{\rho \in K_1}+\sum _{\rho \in K_2}+\sum _{\rho \in K_3}\right)\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right)e^{-\lambda {\sigma }_{\rho }}.\hfill \end{array}$$

(23)

Denote $R:[-{\sigma }_{\theta },0]\to M_{m,k}\left(\mathbb{K}\right)$:

$$R\left(\tau \right):=\left\{\begin{array}{cc}{R}_1\left(\tau \right),\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ R_2\left(\tau \right),\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ \dots \dots ,\hfill & \\ R_{\theta }\left(\tau \right),\hfill & \tau \in [-{\sigma }_{\theta },-{\sigma }_{\theta -1}).\hfill \end{array}\right.$$

Set

$${\delta }_i\left(\tau \right):=\left\{\begin{array}{cc}{\delta }_{i1}\left(\tau \right),\hfill & \tau \in [-{\omega }_1,0],\hfill \\ {\delta }_{i2}\left(\tau \right),\hfill & \tau \in [-{\omega }_2,-{\omega }_1),\hfill \\ \dots \dots ,\hfill & \\ {\delta }_{i\ell }\left(\tau \right),\hfill & \tau \in [-{\omega }_{\ell },-{\omega }_{\ell -1}),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_{\theta },-{\omega }_{\ell }),\hfill \end{array}\right.$$

$$g_i\left(\tau \right):=\left\{\begin{array}{cc}{g}_{i1}\left(\tau \right),\hfill & \tau \in [-h_1,0],\hfill \\ g_{i2}\left(\tau \right),\hfill & \tau \in [-h_2,-h_1),\hfill \\ \dots \dots ,\hfill & \\ g_{is}\left(\tau \right),\hfill & \tau \in [-h_s,-h_{s-1}),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_{\theta },-h_s),\hfill \end{array}\right.$$

Then, equalities (22) take the form

$${\int }_{-{\sigma }_{\theta }}^0{\delta }_i\left(\tau \right)e^{\lambda \tau }d\tau ={\int }_{-{\sigma }_{\theta }}^0{g}_i\left(\tau \right)e^{\lambda \tau }d\tau -{\int }_{-{\sigma }_{\theta }}^0\mathrm{Sp}\left(C^{*}{J}^{i-1}BR\left(\tau \right)\right)e^{\lambda \tau }d\tau .$$

(24)

Equalities (23) hold for all $i=\overline{1,n}$ iff for all $i=\overline{1,n}$ the following equalities hold:

$$\begin{array}{c}\begin{array}{cc}\hfill {\gamma }_{i0}=& a_{i0}-\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_0\right);\hfill \\ \hfill {\gamma }_{i\mu }=& -\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right),\mu \in S_1,\rho \in K_1,{\sigma }_{\rho }={\omega }_{\mu };\hfill \\ \hfill {\gamma }_{i\mu }=& a_{ij}-\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right),\mu \in S_2,j\in S_3,\rho \in K_2,{\sigma }_{\rho }=h_j={\omega }_{\mu };\hfill \\ \hfill 0=& a_{ij}-\mathrm{Sp}\left(C^{*}{J}^{i-1}B{Q}_{\rho }\right),j\in S_4,\rho \in K_3,{\sigma }_{\rho }=h_j.\hfill \end{array}\end{array}$$

(25)

Equalities (24) hold for all $i=\overline{1,n}$ iff for a.e. $\tau \in [-{\sigma }_{\theta },0]$ the following equalities hold:

$${\delta }_i\left(\tau \right)=g_i\left(\tau \right)-\mathrm{Sp}\left(C^{*}{J}^{i-1}BR\left(\tau \right)\right),i=\overline{1,n}.$$

(26)

Every $\rho$th system of (25) consists of n equations with $mk$ unknown entries of the matrix $Q_{\rho }$, $\rho =\overline{0,\theta }$. System (26) consists of n equations with $mk$ unknown entries of the matrix function $R\left(\tau \right)$, $\tau \in [{\sigma }_{\theta },0]$. Let us rewrite systems (25), (26) in the vector form. By definition of the mapping $\mathrm{vec}$, we have $\mathrm{Sp}\left(XY\right)={\left(\mathrm{vec}X\right)}^T·\left(\mathrm{vec}{Y}^T\right)$ for any $X\in M_{p,q}\left(\mathbb{K}\right)$, $Y\in M_{q,p}\left(\mathbb{K}\right)$. Let us apply this equality in system (25), for every $i=\overline{1,n}$, to the matrix $X=C^{*}{J}^{i-1}B$ and to the matrices $Y=Q_{\rho }$, $\rho =\overline{0,\theta }$, and in system (26), for every $i=\overline{1,n}$, to the matrix $X=C^{*}{J}^{i-1}B$ and to $Y=R\left(\tau \right)$. Let us construct the $mk\times n$-matrix

$$P=[\mathrm{vec}\left(C^{*}B\right),\mathrm{vec}\left(C^{*}JB\right),\dots ,\mathrm{vec}\left(C^{*}{J}^{n-1}B\right)].$$

(27)

Denote $v_{\rho }:=\mathrm{vec}\left(Q_{\rho }^T\right)\in {\mathbb{K}}^{mk}$, $\rho =\overline{0,\theta }$, $f\left(\tau \right):=\mathrm{vec}\left(R^T\left(\tau \right)\right)\in {\mathbb{K}}^{mk}$, $\tau \in [-{\sigma }_{\theta },0]$,

$$\begin{array}{cc}\hfill w_0:=& \mathrm{col}(a_{10}-{\gamma }_{10},\dots ,a_{n0}-{\gamma }_{n0})\in {\mathbb{K}}^n;\hfill \\ \hfill w_{\rho }:=& \mathrm{col}(-{\gamma }_{1\mu },\dots ,-{\gamma }_{n\mu })\in {\mathbb{K}}^n,\mu \in S_1,\rho \in K_1,{\sigma }_{\rho }={\omega }_{\mu };\hfill \\ \hfill w_{\rho }:=& \mathrm{col}(a_{1j}-{\gamma }_{1\mu },\dots ,a_{nj}-{\gamma }_{n\mu })\in {\mathbb{K}}^n,\mu \in S_2,j\in S_3,\rho \in K_2,{\sigma }_{\rho }=h_j={\omega }_{\mu };\hfill \\ \hfill w_{\rho }:=& \mathrm{col}(a_{1j},\dots ,a_{nj})\in {\mathbb{K}}^n,j\in S_4,\rho \in K_3,{\sigma }_{\rho }=h_j;\hfill \end{array}$$

$$\vartheta \left(\tau \right):=\mathrm{col}\left(g_1\left(\tau \right)-{\delta }_1\left(\tau \right),\dots ,g_n\left(\tau \right)-{\delta }_n\left(\tau \right)\right)\in {\mathbb{K}}^n.$$

Then, one can rewrite systems (25), (26) in the vector form

$$\begin{array}{c}{P}^T{v}_{\rho }=w_{\rho },\rho =\overline{0,\theta },\end{array}$$

(28)

$$\begin{array}{c}{P}^{T}f\left(\tau \right)=\vartheta \left(\tau \right)\mathrm{a}.\mathrm{e}.\tau \in [-{\sigma }_{\theta },0].\end{array}$$

(29)

System (11), (12) is arbitrary coefficient assignable by feedback (13) if and only if system (28), (29) is solvable with respect to $v_{\rho }$, $\rho =\overline{0,\theta }$, and $f\left(\tau \right)$, $\tau \in [-{\sigma }_{\theta },0]$, for any $0={\omega }_0<{\omega }_1<\dots <{\omega }_{\ell }$, any numbers ${\gamma }_{i\mu }\in \mathbb{K}$, $\mu =\overline{0,\ell }$, and any integrable functions ${\delta }_{i\xi }:[-{\omega }_{\xi },-{\omega }_{\xi -1}]\to \mathbb{K}$, $\xi =\overline{1,\ell }$, $i=\overline{1,n}$. The condition of linear independency of the matrices (16) is necessary and sufficient for solvability of system (28), (29). In that case, system (28), (29) has the particular solution

$$\begin{array}{cc}\hfill v_{\rho }& =P{\left(P^{T}P\right)}^{-1}{w}_{\rho },\rho =\overline{0,\theta },\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill f\left(\tau \right)& =P{\left(P^{T}P\right)}^{-1}\vartheta \left(\tau \right),\tau \in [-{\sigma }_{\theta },0].\hfill \end{array}$$

(31)

The required matrices $Q_{\rho }$, $\rho =\overline{0,\theta }$, and $R\left(\tau \right)$, $\tau \in [-{\sigma }_{\theta },0]$, can be found from the equalities

$$Q_{\rho }={\left({\mathrm{vec}}^{-1}{v}_{\rho }\right)}^T,\rho =\overline{0,\theta },R\left(\tau \right)={\left({\mathrm{vec}}^{-1}f\left(\tau \right)\right)}^T.$$

□

3. Corollaries

If the characteristic function of the closed-loop system (14) turns into a polynomial, then the spectrum $\mathrm{\Lambda }$ of system (14) is finite. We say that system (11), (12) is arbitrary finite spectrum assignable by linear static output feedback (13) if for any ${\zeta }_i\in \mathbb{K}$, $i=\overline{1,n}$, there exist an integer $\theta \ge 0$, numbers $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$, constant matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, and integrable matrix functions $R_{\varkappa }:[-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}]\to M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$, such that the characteristic function (15) of the closed-loop system (14) satisfies the equality

$$\phi \left(\lambda \right)={\lambda }^n+{\zeta }_1{\lambda }^{n-1}+\dots +{\zeta }_n.$$

Corollary 1.

System (11), (12) is arbitrary finite spectrum assignable by linear static output feedback (13) if and only if the matrices (16) are linearly independent.

The proof of Corollary 1 follows from the proof of Theorem 2: the problem under consideration is equivalent to solvability of system (28), (29), where $\ell =0$, ${\gamma }_{i0}={\zeta }_i$ ($i=\overline{1,n}$), $T_1=⌀$, $T_2=\{h_1,\dots ,h_s\}$, $S_1=S_2=S_3=⌀$, $S_4=\left\{\overline{1,s}\right\}$, $\theta =s$, ${\sigma }_{\rho }=h_{\rho }$ ($\rho =\overline{1,s}$), $K_1=K_2=⌀$, $K_3=S_4$.

Corollary 2.

System (11), (12) is exponentially stabilizable with an arbitrary pregiven decay rate by linear static output feedback (13) if the matrices (16) are linearly independent.

Corollary 2 follows from Corollary 1.

Consider system (11), (12) containing only lumped delays, i.e., suppose that

$$g_{i\eta }\left(\tau \right)\equiv 0,\tau \in [-h_{\eta },-h_{\eta -1}],i=\overline{1,n},\eta =\overline{1,s}.$$

(32)

Let the controller (13) also contains only lumped delays, i.e.,

$$R_{\varkappa }\left(\tau \right)\equiv 0,\tau \in [-{\sigma }_{\varkappa },-{\sigma }_{\varkappa -1}],\varkappa =\overline{1,\theta }.$$

(33)

In this case, the closed-loop system (14) does not contain distributed delays. Under conditions (32) and (33), the statements of the problem is as follows: system (11), (12) is said to be arbitrary coefficient assignable by static output feedback (13) if, for any integer $\ell \ge 0$, for any given $0={\omega }_0<{\omega }_1<\dots <{\omega }_{\ell }$, and for any numbers ${\gamma }_{i\mu }\in \mathbb{K}$, $i=\overline{1,n}$, $\mu =\overline{0,\ell }$, there exist an integer $\theta \ge 0$, numbers $0={\sigma }_0<{\sigma }_1<\dots <{\sigma }_{\theta }$, and constant matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, such that the characteristic function (15) of the closed-loop system (14) satisfies the equality

$$\phi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n\sum _{\mu =0}^{\ell }{\gamma }_{i\mu }{\lambda }^{n-i}{e}^{-\lambda {\omega }_{\mu }}$$

Corollary 3.

Under conditions (32) and (33), system (11), (12) is arbitrary coefficient assignable by linear static output feedback (13) if and only if the matrices (16) are linearly independent.

The proof of Corollary 3 repeats the proof of Theorem 2, under conditions (32), (33) and condition ${\delta }_i\left(\tau \right)\equiv 0$, $\tau \in [-{\omega }_{\ell },0]$, $i=\overline{1,n}$. In fact, this proof was carried out in [44] (Theorem 2). So, Theorem 2 is an extension of [44] (Theorem 2) from systems with only lumped delays to systems with lumped and distributed delays.

Suppose that the delays in system (11), (12) and in feedback (13) are commensurate, i.e., for some $h>0$,

$$\begin{array}{c}{h}_j=jh,j=\overline{0,s},\end{array}$$

(34)

$$\begin{array}{c}{\sigma }_{\rho }=\rho h,\rho =\overline{0,\theta }.\end{array}$$

(35)

In this case, the closed-loop system (14) contains only commensurate delays. Under conditions (34) and (35), the statements of the problem is as follows: system (11), (12) is said to be arbitrary coefficient assignable by static output feedback (13) if, for any integer $\ell \ge 0$, for any numbers ${\gamma }_{i\mu }\in \mathbb{K}$, $i=\overline{1,n}$, $\mu =\overline{0,\ell }$, and for any integrable functions ${\delta }_{i\xi }:[-\xi h,-(\xi -1)h]\to \mathbb{K}$, $i=\overline{1,n}$, $\xi =\overline{1,\ell }$, there exist an integer $\theta \ge 0$, constant matrices $Q_{\rho }\in M_{m,k}\left(\mathbb{K}\right)$, $\rho =\overline{0,\theta }$, and integrable matrix functions $R_{\varkappa }:[-\varkappa h,-(\varkappa -1)h]\to M_{m,k}\left(\mathbb{K}\right)$, $\varkappa =\overline{1,\theta }$, such that the characteristic function (15) of the closed-loop system (14) satisfies the equality

$$\phi \left(\lambda \right)={\lambda }^n+\sum _{i=1}^n{\lambda }^{n-i}\left(\sum _{\mu =0}^{\ell }{\gamma }_{i\mu }{e}^{-\lambda \mu h}+\sum _{\xi =1}^{\ell }{\int }_{-\xi h}^{-(\xi -1)h}{\delta }_{i\xi }\left(\tau \right)e^{\lambda \tau }d\tau \right).$$

(36)

Corollary 4.

Under conditions (34) and (35), system (11), (12) is arbitrary coefficient assignable by linear static output feedback (13) if and only if the matrices (16) are linearly independent.

The proof of Corollary 4 repeats the proof of Theorem 2, under conditions (34), (35) and condition ${\omega }_{\mu }=\mu h$, $\mu =\overline{0,\ell }$. In fact, this proof was carried out in [46] (Theorem 1). So, Theorem 2 is an extension of [46] (Theorem 1) from systems with commensurate delays to systems with non-commensurate delays.

Remark 2.

Let us indicate to differences between systems with commensurate and non-commensurate delays. In systems with commensurate delays, one needs, for a given triplet

$${\mathcal{T}}_1=\left(\ell ,\{{\gamma }_{i\mu },i=\overline{1,n},\mu =\overline{0,\ell }\},\{{\delta }_{i\xi }(·),i=\overline{1,n},\xi =\overline{1,\ell }\}\right),$$

to construct a triplet

$${\mathcal{T}}_2=\left(\theta ,\{Q_{\rho },\rho =\overline{0,\theta }\},\{R_{\varkappa }(·),\varkappa =\overline{1,\theta }\}\right)$$

ensuring (36), while in systems with non-commensurate delays, one needs, for a given quadruple

$${\mathfrak{Q}}_1=\left(\ell ,\{{\omega }_{\mu },\mu =\overline{0,\ell }\},\{{\gamma }_{i\mu },i=\overline{1,n},\mu =\overline{0,\ell }\},\{{\delta }_{i\xi }(·),i=\overline{1,n},\xi =\overline{1,\ell }\}\right),$$

to construct a quadruple

$${\mathfrak{Q}}_2=\left(\theta ,\{{\sigma }_{\rho },\rho =\overline{0,\theta }\},\{Q_{\rho },\rho =\overline{0,\theta }\},\{R_{\varkappa }(·),\varkappa =\overline{1,\theta }\}\right),$$

ensuring the equality from Definition 3. Thus, the problem statements are different. Corollary 4, for systems with commensurate delays, was proved in [46] (Theorem 1). Here we prove a more general result. Difference and difficulty here, with respect to [46], is in choosing the required numbers ${\sigma }_{\rho }$. The proof given in [46] does not provide an algorithm for constructing the indicated numbers ${\sigma }_{\rho }$ and the corresponding gain coefficients $Q_{\rho }$ and $R_{\varkappa }(·)$. Here, we overcome these difficulties. It provides the novelty of the results obtained.

4. Modeling Example

Let $h_1=1$, $h_2=\sqrt{2}$. Consider the system

$$\begin{array}{cc}\hfill & x^{‴}\left(t\right)+x^{″}(t-h_1)+x^{″}(t-h_2)+x^{\prime }\left(t\right)+x\left(t\right)\hfill \\ \hfill & +x(t-h_1)-x(t-h_2)+{\int }_{-h_1}^0{x}^{″}(t+\tau )\cos \tau d\tau \hfill \\ \hfill & -2{\int }_{-h_2}^{-h_1}{x}^{\prime }(t+\tau )\sin \tau d\tau -{\int }_{-h_1}^{0}x(t+\tau )\cos 2\tau d\tau \hfill \\ \hfill & =u_1^{\prime }\left(t\right)+u_1\left(t\right)+u_2^{\prime }\left(t\right),\hfill \end{array}$$

(37)

$$y_1\left(t\right)=-x\left(t\right)-x^{\prime }\left(t\right),y_2\left(t\right)=x^{\prime }\left(t\right),$$

(38)

$x\in \mathbb{R}$, $u=\mathrm{col}(u_1,u_2)\in {\mathbb{R}}^2$, $y=\mathrm{col}(y_1,y_2)\in {\mathbb{R}}^2$. System (37), (38) has the form (11), (12) where $n=3$, $m=2$, $k=2$, $p=2$, $s=2$;

$$\begin{array}{c}{a}_{10}=0,a_{11}=1,a_{12}=1,a_{20}=1,a_{21}=0,a_{22}=0,\\ a_{30}=1,a_{31}=1,a_{32}=-1;\\ g_{11}\left(\tau \right)=\cos \tau ,g_{12}\left(\tau \right)=0,g_{21}\left(\tau \right)=0,g_{22}\left(\tau \right)=-2\sin \tau ,\\ g_{31}\left(\tau \right)=-\cos 2\tau ,g_{32}\left(\tau \right)=0;\\ b_{21}=1,b_{22}=1,b_{31}=1,b_{32}=0;\\ c_{11}=-1,c_{21}=-1,c_{12}=0,c_{22}=1.\end{array}$$

From system (37), (38), construct the matrices B, C: we obtain $B=\left[\begin{array}{cc}0& 0\\ 1& 1\\ 1& 0\end{array}\right]$, $C=\left[\begin{array}{cc}-1& 0\\ -1& 1\\ 0& 0\end{array}\right]$. Hence,

$$C^{*}B=\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right],C^{*}JB=\left[\begin{array}{cc}-2& -1\\ 1& 0\end{array}\right],C^{*}{J}^{2}B=\left[\begin{array}{cc}-1& 0\\ 0& 0\end{array}\right].$$

(39)

Construct the matrix (27):

$$\begin{array}{c}P=\left[\begin{array}{ccc}-1& -2& -1\\ -1& -1& 0\\ 1& 1& 0\\ 1& 0& 0\end{array}\right].\end{array}$$

We have $\mathrm{rank}P=3=n$, hence, the matrices (39) are linearly independent. Therefore, by Theorem 2, system (37), (38) is arbitrary coefficient assignable by linear static output feedback (13). Let us construct such a controller. Suppose, for example, that $\ell =2$, ${\omega }_1=1$, ${\omega }_2=\sqrt{3}$, and

$$\begin{array}{cc}\hfill & \phi \left(\lambda \right)={\lambda }^3+{\lambda }^2\left(2+e^{-\lambda {\omega }_2}-{\int }_{-{\omega }_1}^0{e}^{\lambda \tau }(\sin \tau -\cos \tau )d\tau \right)\hfill \\ \hfill & +\lambda \left(1+2e^{-\lambda {\omega }_1}-{\int }_{-{\omega }_2}^{-{\omega }_1}{e}^{\lambda \tau }\sin 2\tau d\tau \right)+e^{-\lambda {\omega }_1}.\hfill \end{array}$$

(40)

Then

$$\begin{array}{c}{T}_1=\{{\omega }_1,{\omega }_2\}=\{1,\sqrt{3}\},T_2=\{h_1,h_2\}=\{1,\sqrt{2}\};\\ {\gamma }_{10}=2,{\gamma }_{11}=0,{\gamma }_{12}=1,{\gamma }_{20}=1,{\gamma }_{21}=2,\\ {\gamma }_{22}=0,{\gamma }_{30}=0,{\gamma }_{31}=1,{\gamma }_{32}=0;\\ {\delta }_{11}\left(\tau \right)=\cos \tau -\sin \tau ,{\delta }_{12}\left(\tau \right)=0,{\delta }_{21}\left(\tau \right)=0,\\ {\delta }_{22}\left(\tau \right)=-\sin 2\tau ,{\delta }_{31}\left(\tau \right)=0,{\delta }_{32}\left(\tau \right)=0.\end{array}$$

From the proof of Theorem 2, we obtain

$$\begin{array}{c}{S}_1=\left\{2\right\},S_2=\left\{1\right\},S_3=\left\{1\right\},S_4=\left\{2\right\},\\ T=\{h_1,h_2,{\omega }_2\}=\{1,\sqrt{2},\sqrt{3}\},\theta =3,\\ {\sigma }_1=h_1={\omega }_1=1,{\sigma }_2=h_2=\sqrt{2},{\sigma }_3={\omega }_2=\sqrt{3},\\ K_1=\left\{3\right\},K_2=\left\{1\right\},K_3=\left\{2\right\}.\end{array}$$

We have

$$\begin{array}{cc}\hfill g_1\left(\tau \right)=& \left\{\begin{array}{cc}\cos \tau ,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ 0,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2),\hfill \end{array}\right.\hfill \\ \hfill g_2\left(\tau \right)=& \left\{\begin{array}{cc}0,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ -2\sin \tau ,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2),\hfill \end{array}\right.\hfill \\ \hfill g_3\left(\tau \right)=& \left\{\begin{array}{cc}-\cos 2\tau ,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ 0,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2),\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\delta }_1\left(\tau \right)& =\left\{\begin{array}{cc}\cos \tau -\sin \tau ,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ 0,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2),\hfill \end{array}\right.\hfill \\ \hfill {\delta }_2\left(\tau \right)& =\left\{\begin{array}{cc}0,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ -\sin 2\tau ,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ -\sin 2\tau ,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2),\hfill \end{array}\right.\hfill \\ \hfill {\delta }_3\left(\tau \right)& =\left\{\begin{array}{cc}0,\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ 0,\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ 0,\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2).\hfill \end{array}\right.\hfill \end{array}$$

Next, we have

$$\begin{array}{cc}\hfill w_0& =\mathrm{col}(a_{10}-{\gamma }_{10},a_{20}-{\gamma }_{20},a_{30}-{\gamma }_{30})=(-2,0,1),\hfill \\ \hfill w_1& =\mathrm{col}(a_{11}-{\gamma }_{11},a_{21}-{\gamma }_{21},a_{31}-{\gamma }_{31})=(1,-2,0),\hfill \\ \hfill w_2& =\mathrm{col}(a_{12},a_{22},a_{32})=(1,0,-1),\hfill \\ \hfill w_3& =\mathrm{col}(-{\gamma }_{12},-{\gamma }_{22},-{\gamma }_{32})=(-1,0,0),\hfill \\ \hfill \vartheta \left(\tau \right)& =\mathrm{col}\left(g_1\left(\tau \right)-{\delta }_1\left(\tau \right),g_2\left(\tau \right)-{\delta }_2\left(\tau \right),g_3\left(\tau \right)-{\delta }_3\left(\tau \right)\right)\hfill \\ \hfill =& \left\{\begin{array}{cc}\mathrm{col}(\sin \tau ,0,-\cos 2\tau ),\hfill & \tau \in [-{\sigma }_1,0],\hfill \\ \mathrm{col}(0,-2\sin \tau +\sin 2\tau ,0),\hfill & \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ \mathrm{col}(0,\sin 2\tau ,0),\hfill & \tau \in [-{\sigma }_3,-{\sigma }_2).\hfill \end{array}\right.\hfill \end{array}$$

Calculating $v_0$, $v_1$, $v_2$, $v_3$, and $f\left(\tau \right)$ by formulas (30), (31), we obtain

$$\begin{array}{cccc}\hfill v_0& =\mathrm{col}(-1,1,-1,-1),\hfill & \hfill v_1& =\mathrm{col}(0,1,-1,3),\hfill \\ \hfill v_2& =\mathrm{col}(1,-1,1,0),\hfill & \hfill v_3& =\mathrm{col}(0,0,0,-1);\hfill \end{array}$$

$$f\left(\tau \right)=\left\{\begin{array}{c}\mathrm{col}(\cos 2\tau ,-\cos 2\tau ,\cos 2\tau ,\sin \tau -\cos 2\tau ),\hfill \\ \tau \in [-{\sigma }_1,0],\hfill \\ \mathrm{col}(0,\sin \tau -\sin \tau \cos \tau ,\hfill \\ -\sin \tau +\sin \tau \cos \tau ,2\sin \tau -\sin 2\tau ),\hfill \\ \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ \mathrm{col}(0,-\sin \tau \cos \tau ,\sin \tau \cos \tau ,-\sin 2\tau ),\hfill \\ \tau \in [-{\sigma }_3,-{\sigma }_2).\hfill \end{array}\right.$$

From this, it follows that

$$\begin{array}{cccc}\hfill Q_0& =\left[\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right],\hfill & \hfill Q_1& =\left[\begin{array}{cc}0& -1\\ 1& 3\end{array}\right],\hfill \\ \hfill Q_2& =\left[\begin{array}{cc}1& 1\\ -1& 0\end{array}\right],\hfill & \hfill Q_3& =\left[\begin{array}{cc}0& 0\\ 0& -1\end{array}\right].\hfill \end{array}$$

$$R\left(\tau \right)=\left\{\begin{array}{c}\left[\begin{array}{cc}\cos 2\tau & \cos 2\tau \\ -\cos 2\tau & \sin \tau -\cos 2\tau \end{array}\right],\hfill \\ \tau \in [-{\sigma }_1,0],\hfill \\ \left[\begin{array}{cc}0& -\sin \tau +\sin \tau \cos \tau \\ \sin \tau -\sin \tau \cos \tau & 2\sin \tau -\sin 2\tau \end{array}\right],\hfill \\ \tau \in [-{\sigma }_2,-{\sigma }_1),\hfill \\ \left[\begin{array}{cc}0& \sin \tau \cos \tau \\ -\sin \tau \cos \tau & -\sin 2\tau \end{array}\right],\hfill \\ \tau \in [-{\sigma }_3,-{\sigma }_2).\hfill \end{array}\right.$$

The controller (13)

$$\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{u}_1\left(t\right)\\ u_2\left(t\right)\end{array}\right]=Q_0\left[\begin{array}{c}{y}_1\left(t\right)\\ y_2\left(t\right)\end{array}\right]+Q_1\left[\begin{array}{c}{y}_1(t-{\sigma }_1)\\ y_2(t-{\sigma }_1)\end{array}\right]\\ +Q_2\left[\begin{array}{c}{y}_1(t-{\sigma }_2)\\ y_2(t-{\sigma }_2)\end{array}\right]+Q_3\left[\begin{array}{c}{y}_1(t-{\sigma }_3)\\ y_2(t-{\sigma }_3)\end{array}\right]+{\int }_{-{\sigma }_3}^{0}R\left(\tau \right)y(t+\tau )d\tau \end{array}\end{array}$$

(41)

has the components

$$\begin{array}{cc}\hfill & u_1\left(t\right)=-x^{\prime }(t-h_1)+x\left(t\right)-x(t-h_2)-{\int }_{-h_1}^{0}x(t+\tau )\cos 2\tau d\tau \hfill \\ \hfill & +{\int }_{-h_2}^{-h_1}{x}^{\prime }(t+\tau )(\sin \tau \cos \tau -\sin \tau )d\tau +{\int }_{-{\omega }_2}^{-h_2}{x}^{\prime }(t+\tau )\sin \tau \cos \tau d\tau ,\hfill \\ \hfill & u_2\left(t\right)=-2x^{\prime }\left(t\right)+2x^{\prime }(t-h_1)+x^{\prime }(t-h_2)-x^{\prime }(t-{\omega }_2)\hfill \\ \hfill & -x\left(t\right)-x(t-h_1)+x(t-h_2)+{\int }_{-h_1}^{0}x(t+\tau )\cos 2\tau d\tau +{\int }_{-h_1}^0{x}^{\prime }(t+\tau )\sin \tau d\tau \hfill \\ \hfill & +{\int }_{-h_2}^{-h_1}x(t+\tau )(\sin \tau \cos \tau -\sin \tau )d\tau +{\int }_{-h_2}^{-h_1}{x}^{\prime }(t+\tau )(\sin \tau -\sin \tau \cos \tau )d\tau \hfill \\ \hfill & +{\int }_{-{\omega }_2}^{-h_2}x(t+\tau )\sin \tau \cos \tau d\tau -{\int }_{-{\omega }_2}^{-h_2}{x}^{\prime }(t+\tau )\sin \tau \cos \tau d\tau .\hfill \end{array}$$

System (37), (38) closed-loop by feedback (41) takes the form

$$\begin{array}{c}\begin{array}{c}{x}^{‴}\left(t\right)+2x^{″}\left(t\right)+x^{″}(t-{\omega }_2)+x^{\prime }\left(t\right)+2x^{\prime }(t-{\omega }_1)\\ +x(t-{\omega }_1)-{\int }_{-{\omega }_1}^0{x}^{″}(t+\tau )(\sin \tau -\cos \tau )d\tau -{\int }_{-{\omega }_2}^{-{\omega }_1}{x}^{\prime }(t+\tau )\sin 2\tau d\tau =0.\end{array}\end{array}$$

(42)

The characteristic function of system (42) is equal to (40). In particular, system (42) (with ${\omega }_1=1$, ${\omega }_2=\sqrt{3}$) is exponentially stable. This is confirmed by Figure 1 and Figure 2. Figure 1 shows the spectrum of system (42) with ${\omega }_1=1$, ${\omega }_2=\sqrt{3}$. The spectrum is in the left half-plane. Figure 2 shows solutions of system (42) with ${\omega }_1=1$, ${\omega }_2=\sqrt{3}$, with the initial functions $x\left(\tau \right)=1$ (blue plot), $x\left(\tau \right)=\tau$ (violet plot), $x\left(\tau \right)={\tau }^2$ (green plot) for all $\tau \in [-\sqrt{2},0]$. Solutions tend to zero exponentially.

5. Conclusions

Necessary and sufficient conditions were obtained for the problem of arbitrary coefficient assignment for the characteristic function of the closed-loop system by static output feedback for a linear differential equation with non-commensurate lumped and distributed delays. The obtained results extend the earlier corresponding results for systems with commensurate delays and for systems with only lumped delays. Corollaries on arbitrary finite spectrum assignment and on stabilization were stated. We provided an example illustrating our results. In future works, we expect to extend these results to control systems defined by non-scalar systems of differential equations. Moreover, this approach could be applied to problems of stabilization by static output feedback for linear quasi-differential equations and for nonlinear differential equations with delays based on linear approximation.