A New Approach for Stabilization Criteria of n-Order Function Differential Equation by Distributed Control Function

Abstract

In the current paper, we demonstrate a new approach for an stabilization criteria for n-order functional-differential equation with distributed feedback control in the integral form. We present a correlation between the order of the functional-differential equation and degree of freedom of the distributed control function. We present two cases of distributed control function in the integral form. Such a case of stabilization control functions plays a very important role in physics, aeronautics, aerospace, ship navigation and traffic network control management. Structure of functional-differential equations is based on the symmetry properties.

1. Introduction

Consider the n-order differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^{n-1}{p}_i\left(t\right)x^{\left(i\right)}\left(t\right)+w_k\left(t\right)=g\left(t\right),k=1,2,$$

(1)

with distributed feedback control defined as

$$w_1\left(t\right)=\sum _{j=1}^k{\int }_0^t{K}_{1j}\left(t,s\right)x\left(s\right)ds,$$

(2)

or

$$w_2\left(t\right)=\sum _{j=1}^k{\int }_0^t{K}_{2j}\left(t,s\right)x^{\prime }\left(s\right)ds,$$

(3)

where $K_{1j}\left(t,s\right)$, $K_{2j}\left(t,s\right)$$1\le j\le k$ are integrable kernel functions, while $g\left(t\right)$, $p_i\left(t\right)$$1\le i\le n-1$ are continuous functions.

Stabilization of a general n-order differential equation with distributed feedback control is a very important problem.

The noise in the feedback control is the main reason for investigating mathematical models with distributed inputs. The reason is that it is impossible to handle our control on the value of $x\left(t_j\right)$ at a single time-point $t_j$ only. We need to average the process $x\left(t\right)$ in the close neighborhood of $t_j$ by integral term (2) or (3).

It is presented in [1] that models with distributed inputs can appear in population dynamics, in network control systems and in propellant rocket motors.

Only a few papers have been devoted to stabilization by distributed feedback control. The problems of linear systems with delayed control action, transformed into systems without delays, were discussed in [2]. Under an absolute continuity condition, the new system is an ordinary differential control equation. In the general case, the new system is a measure-differential control system.

Stabilization of linear systems with distributed input were presented in [3] and stabilization of non-linear systems with distributed input were presented in [4].

Asymptotic stability criteria of the zero solution of second-order linear delay differential equation were presented in [5], where in proving results Pontryagin’s theory of quasi-polynomials was used.

Results on boundedness of solutions were obtained in [6]. Stability of second order equations with damping terms were obtained in [7,8,9]. Stability of a third order differential equation is presented in [10]. There are various applications of models described by equations with distributed feedback control in aeronautics, aerospace, ship navigation and traffic network control management.

Stabilization of mathematical models, by distributed feedback control, play a very important role in medicine (see, for example, [11,12,13], where the mathematical model of testosterone regulation and model of hepatitis B virus were proposed).

The results of exponential stability of Equation (1) for $n=2,3$ were presented in [14,15]. In the current paper, we generalize these results: we present the new approach for stabilization impossibility of an n-order differential equation by distributed feedback control.

Let us introduce the kernel function that is defined in the following form

$$K_j\left(t,s\right)={\beta }_j{e}^{-{\alpha }_j\left(t-s\right)},{\alpha }_j,{\beta }_j\in \mathbb{R}{\alpha }_j>0,1\le j\le k$$

(4)

In Section 2, we demonstrate the stabilization impossibility by the distributed control function, that is defined by (2) in the case of $k\le n-1$.

In Section 3, we demonstrate the stabilization impossibility by the distributed control function, that is defined by (3) in the case of $k\le n-2$.

In Section 4, we demonstrate various examples for Section 2 and Section 3.

2. Stabilization Impossibility in the Case of $\mathit{k}\mathbf{=}\mathit{n}\mathbf{-}\mathbf{1}$ with Control Function (2)

Let us introduce the following integro-differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^{n-1}{\beta }_i{\int }_0^t{e}^{-{\alpha }_i\left(t-s\right)}x\left(s\right)ds=0,{\alpha }_i,{\beta }_i\in \mathbb{R},1\le i\le n-1.$$

(5)

Theorem 1.

The solution of integro-differential Equation (5) is exponentially unstable.

Proof. 

This equation can be reduced to the system of $2n-1$ first order differential equations (see Appendix A).

$$\left\{\begin{array}{c}{x}_1^{\prime }=-\sum _{j=n+1}^{2n-1}{x}_j\hfill \\ x_2^{\prime }=x_1\hfill \\ ...\hfill \\ x_n^{\prime }=x_{n-1}\hfill \\ x_{n+1}^{\prime }={\beta }_1{x}_n-{\alpha }_1{x}_{n+1}\hfill \\ ...\hfill \\ x_{2n-1}^{\prime }={\beta }_{n-1}{x}_n-{\alpha }_{n-1}{x}_{2n-1}\hfill \end{array}\right.$$

(6)

We obtain the following coefficient matrix $(\left(2n-1\right)\times \left(2n-1\right))$

$$M=\left(\begin{array}{ccccccccc}0& 0& 0& \cdots & 0& 0& -1& \cdots & -1\\ 1& 0& 0& \cdots & 0& 0& 0& \cdots & 0\\ 0& 1& 0& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0& {\beta }_1& -{\alpha }_1& \cdots & 0\\ \cdots & \cdots & \cdots & \ddots & ...& \cdots & \cdots & \ddots & \cdots \\ 0& 0& 0& \cdots & 0& {\beta }_{n-1}& 0& \cdots & -{\alpha }_{n-1}\end{array}\right)$$

Let us find the characteristic polynomial of matrix M

$$\begin{array}{ccc}\hfill det(I\lambda -M)& =& det\left(\begin{array}{cccccccc}\lambda & 0& \cdots & 0& 0& 1& \cdots & 1\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ 0& -1& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}& 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right)\hfill \\ & =& \lambda det\left(\begin{array}{cccccccc}\lambda & 0& \cdots & 0& 0& 0& \cdots & 0\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}& 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right)\hfill \\ & +& det\left(\begin{array}{cccccccc}0& 0& \cdots & 0& 0& 1& \cdots & 1\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}& 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right)\hfill \end{array}$$

This determinant can be represented by the following form (by blocks)

$$det(I\lambda -M)=\lambda det\left(\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right)+det\left(\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right)$$

where

$$M_1=\left(\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right),M_2=\left(\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right),$$

$$\begin{array}{ccc}\hfill {\left[A_1\right]}_{\left(n-1\right)\times \left(n-1\right)}& =& \left(\begin{array}{ccccc}\lambda & 0& \cdots & 0& 0\\ -1& \lambda & ...& 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& \lambda \end{array}\right),{\left[B_1\right]}_{\left(n-1\right)\times \left(n-1\right)}=\left(\begin{array}{ccc}0& \cdots & 0\\ 0& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 0\end{array}\right),\hfill \\ \hfill {\left[C_1\right]}_{\left(n-1\right)\times \left(n-1\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& -{\beta }_1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}\end{array}\right),{\left[D_1\right]}_{\left(n-1\right)\times \left(n-1\right)}=\left(\begin{array}{ccc}\lambda +{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right),\hfill \\ \hfill {\left[A_2\right]}_{\left(n-1\right)\times \left(n-1\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& 0\\ -1& \lambda & \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& .\cdots & -1& \lambda \end{array}\right),{\left[B_2\right]}_{\left(n-1\right)\times \left(n-1\right)}=\left(\begin{array}{ccc}1& \cdots & 1\\ 0& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 0\end{array}\right),\hfill \\ \hfill {\left[C_2\right]}_{\left(n-1\right)\times \left(n-1\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& -{\beta }_1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}\end{array}\right),{\left[D_2\right]}_{\left(n-1\right)\times \left(n-1\right)}=\left(\begin{array}{ccc}\lambda +{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right).\hfill \end{array}$$

By Schur formulas [16] (see Appendix A)

$$det\left(M_1\right)=det\left(A_1\right)det\left(D_1-C_1{A}_1^{-1}{B}_1\right)=det\left(A_1\right)det\left(D_1\right)={\lambda }^{n-1}\prod _{i=1}^{n-1}\left(\lambda +{\alpha }_i\right)$$

$$det\left(M_2\right)=det\left(D_2\right)det\left(A_2-B_2{D}_2^{-1}{C}_2\right)=\prod _{j=1}^{n-1}\left(\lambda +{\alpha }_j\right)·\left(\sum _{i=1}^{n-1}\frac{{\beta }_i}{\lambda +{\alpha }_i}\right)=\sum _{i=1}^{n-1}{\beta }_i\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-1}\left(\lambda +{\alpha }_j\right)$$

$$det(I\lambda -M)=P_1\left(\lambda \right)+P_2\left(\lambda \right),$$

where

$$P_1\left(\lambda \right)={\lambda }^n\prod _{i=1}^{n-1}\left(\lambda +{\alpha }_i\right),P_2\left(\lambda \right)=\sum _{i=1}^{n-1}{\beta }_i\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-1}\left(\lambda +{\alpha }_j\right).$$

The degree of $P_1\left(\lambda \right)$ is $2n-1$, and degree of $P_2\left(\lambda \right)$ is $n-2$. We obtain that coefficient of expression ${\lambda }^{n-1}$ equal to zero, so by Hurwitz Criteria the solution of (5) is exponentially unstable (see Appendix A). □

Corollary 1.

The solution of the following integro-differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^k{\beta }_i{\int }_0^t{e}^{-{\alpha }_i\left(t-s\right)}x\left(s\right)ds=0,{\alpha }_i,{\beta }_i\in \mathbb{R},1\le k\le n-1.$$

for every $k\le n-1$ is exponentially unstable.

Proof. 

Set ${\beta }_{k+1},...,{\beta }_{n-1}=0$ in Theorem 1. □

3. Stabilization Impossibility in the Case of $\mathit{k}\mathbf{=}\mathit{n}\mathbf{-}\mathbf{2}$ with Control Function (3)

Let us introduce the following integro-differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^{n-2}{\beta }_i{\int }_0^t{e}^{-{\alpha }_i\left(t-s\right)}{x}^{\prime }\left(s\right)ds=0,{\alpha }_i,{\beta }_i\in \mathbb{R},1\le i\le n-1.$$

(7)

Theorem 2.

The solution of integro-differential Equation (7) is exponentially unstable.

Proof. 

This equation can be reduced to the system of $2n-1$ first order differential equations (see Appendix A).

$$\left\{\begin{array}{c}{x}_1^{\prime }=-\sum _{j=n}^{2n-3}{x}_j\hfill \\ x_2^{\prime }=x_1\hfill \\ ...\hfill \\ x_{n-1}^{\prime }=x_{n-2}\hfill \\ x_n^{\prime }={\beta }_1{x}_{n-1}-{\alpha }_1{x}_n\hfill \\ ...\hfill \\ x_{2n-3}^{\prime }={\beta }_{n-2}{x}_{n-1}-{\alpha }_{n-2}{x}_{2n-3}\hfill \end{array}\right.$$

(8)

We obtain the following coefficient matrix $(\left(2n-3\right)\times \left(2n-3\right))$

$$M=\left(\begin{array}{ccccccccc}0& 0& 0& \cdots & 0& 0& -1& \cdots & -1\\ 1& 0& 0& \cdots & 0& 0& 0& \cdots & 0\\ 0& 1& 0& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0& {\beta }_1& -{\alpha }_1& \cdots & 0\\ \cdots & \cdots & \cdots & \ddots & ...& \cdots & \cdots & \ddots & \cdots \\ 0& 0& 0& \cdots & 0& {\beta }_{n-2}& 0& \cdots & -{\alpha }_{n-2}\end{array}\right)$$

Let us find the characteristic polynomial of matrix M

$$\begin{array}{ccc}\hfill det(I\lambda -M)& =& det\left(\begin{array}{cccccccc}\lambda & 0& \cdots & 0& 0& 1& \cdots & 1\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ 0& -1& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-2}& 0& \cdots & \lambda +{\alpha }_{n-2}\end{array}\right)\hfill \\ & =& \lambda det\left(\begin{array}{cccccccc}\lambda & 0& \cdots & 0& 0& 0& \cdots & 0\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-2}& 0& \cdots & \lambda +{\alpha }_{n-2}\end{array}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& +& det\left(\begin{array}{cccccccc}0& 0& \cdots & 0& 0& 1& \cdots & 1\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-2}& 0& \cdots & \lambda +{\alpha }_{n-2}\end{array}\right)\hfill \end{array}$$

This determinant can be represented by the following form (by blocks)

$$det(I\lambda -M)=\lambda det\left(\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right)+det\left(\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right)$$

where

$$M_1=\left(\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right),M_2=\left(\begin{array}{cc}{A}_2& B_2\\ C_2& D_2\end{array}\right),$$

$$\begin{array}{ccc}\hfill {\left[A_1\right]}_{\left(n-2\right)\times \left(n-2\right)}& =& \left(\begin{array}{ccccc}\lambda & 0& \cdots & 0& 0\\ -1& \lambda & ...& 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& \lambda \end{array}\right),{\left[B_1\right]}_{\left(n-2\right)\times \left(n-2\right)}=\left(\begin{array}{ccc}0& \cdots & 0\\ 0& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 0\end{array}\right),\hfill \\ \hfill {\left[C_1\right]}_{\left(n-2\right)\times \left(n-2\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& -{\beta }_1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-2}\end{array}\right),{\left[D_1\right]}_{\left(n-2\right)\times \left(n-2\right)}=\left(\begin{array}{ccc}\lambda +{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \lambda +{\alpha }_{n-2}\end{array}\right),\hfill \\ \hfill {\left[A_2\right]}_{\left(n-2\right)\times \left(n-2\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& 0\\ -1& \lambda & \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& .\cdots & -1& \lambda \end{array}\right),{\left[B_2\right]}_{\left(n-2\right)\times \left(n-2\right)}=\left(\begin{array}{ccc}1& \cdots & 1\\ 0& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 0\end{array}\right),\hfill \\ \hfill {\left[C_2\right]}_{\left(n-2\right)\times \left(n-2\right)}& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& -{\beta }_1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-2}\end{array}\right),{\left[D_2\right]}_{\left(n-2\right)\times \left(n-2\right)}=\left(\begin{array}{ccc}\lambda +{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \lambda +{\alpha }_{n-2}\end{array}\right).\hfill \end{array}$$

By Schur formulas [16] (see Appendix A)

$$det\left(M_1\right)=det\left(A_1\right)det\left(D_1-C_1{A}_1^{-1}{B}_1\right)=det\left(A_1\right)det\left(D_1\right)={\lambda }^{n-2}\prod _{i=1}^{n-2}\left(\lambda +{\alpha }_i\right)$$

$$det\left(M_2\right)=det\left(D_2\right)det\left(A_2-B_2{D}_2^{-1}{C}_2\right)=\prod _{j=1}^{n-2}\left(\lambda +{\alpha }_j\right)·\left(\sum _{i=1}^{n-2}\frac{{\beta }_i}{\lambda +{\alpha }_i}\right)=\sum _{i=1}^{n-2}{\beta }_i\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-2}\left(\lambda +{\alpha }_j\right)$$

$$det(I\lambda -M)=P_1\left(\lambda \right)+P_2\left(\lambda \right),$$

where

$$P_1\left(\lambda \right)={\lambda }^{n-1}\prod _{i=1}^{n-2}\left(\lambda +{\alpha }_i\right),P_2\left(\lambda \right)=\sum _{i=1}^{n-2}{\beta }_i\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-2}\left(\lambda +{\alpha }_j\right).$$

The degree of $P_1\left(\lambda \right)$ is $2n-3$, and degree of $P_2\left(\lambda \right)$ is $n-3$. We obtain that coefficient of expression ${\lambda }^{n-2}$ equal to zero, so by Hurwitz Criteria the solution of (7) is exponentially unstable (see Appendix A). □

Corollary 2.

The solution of the following integro-differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^k{\beta }_i{\int }_0^t{e}^{-{\alpha }_i\left(t-s\right)}x\left(s\right)ds=0,{\alpha }_i,{\beta }_i\in \mathbb{R},1\le k\le n-1,$$

for every $k\le n-2$ is exponentially unstable.

Proof. 

Set ${\beta }_{k+1},...,{\beta }_{n-2}=0$ in Theorem 2. □

4. Examples

Example 1.

$$x^{″}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}x\left(s\right)ds=0$$

By reducing this integro-differential equation to the system of ODE, we obtain

$$\left\{\begin{array}{c}{x}_1^{\prime }=-x_3\hfill \\ x_2^{\prime }=x_1\hfill \\ x_3^{\prime }={\beta }_1{x}_2-{\alpha }_1{x}_3\hfill \end{array}\right..$$

The coefficient matrix is as follows

$$A_1=\left(\begin{array}{ccc}0& 0& -1\\ 1& 0& 0\\ 0& {\beta }_1& -{\alpha }_1\end{array}\right)$$

We obtain the characteristic polynomial of this matrix

$$P_1\left(\lambda \right)={\lambda }^3+{\alpha }_1{\lambda }^2+{\beta }_1$$

The coefficient of λ is equal to zero, so we obtain by the Hurwitz criteria that the equation is exponentially unstable.

Example 2.

$$x^{‴}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}x\left(s\right)ds+{\beta }_2{\int }_0^t{e}^{-{\alpha }_2\left(t-s\right)}x\left(s\right)ds=0$$

By reducing this integro-differential equation to the system of ODE, we obtain

$$\left\{\begin{array}{c}{x}_1^{\prime }=-x_4-x_5\hfill \\ x_2^{\prime }=x_1\hfill \\ x_3^{\prime }=x_2\hfill \\ x_4^{\prime }={\beta }_1{x}_3-{\alpha }_1{x}_4\hfill \\ x_5^{\prime }={\beta }_2{x}_3-{\alpha }_2{x}_5\hfill \end{array}\right..$$

The coefficient matrix is as follows

$$A_2=\left(\begin{array}{ccccc}0& 0& 0& -1& -1\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& {\beta }_1& -{\alpha }_1& 0\\ 0& 0& {\beta }_2& 0& -{\alpha }_2\end{array}\right)$$

We obtain the characteristic polynomial of this matrix

$$P_2\left(\lambda \right)={\lambda }^5+\left({\alpha }_1+{\alpha }_2\right){\lambda }^4+{\alpha }_1{\alpha }_2{\lambda }^3+\left({\beta }_1+{\beta }_2\right)\lambda +{\beta }_1{\alpha }_2+{\beta }_2{\alpha }_1$$

The coefficient of ${\lambda }^2$ is equal to zero, so we obtain by the Hurwitz criteria that the equation is exponentially unstable.

Example 3.

$$x^{⁗}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}x\left(s\right)ds+{\beta }_2{\int }_0^t{e}^{-{\alpha }_2\left(t-s\right)}x\left(s\right)ds+{\beta }_3{\int }_0^t{e}^{-{\alpha }_3\left(t-s\right)}x\left(s\right)ds=0$$

By reducing this integro-differential equation to the system of ODE, we obtain

$$\left\{\begin{array}{c}{x}_1^{\prime }=-x_5-x_6-x_7\hfill \\ x_2^{\prime }=x_1\hfill \\ x_3^{\prime }=x_2\hfill \\ x_4^{\prime }=x_3\hfill \\ x_5^{\prime }={\beta }_1{x}_4-{\alpha }_1{x}_5\hfill \\ x_6^{\prime }={\beta }_2{x}_4-{\alpha }_2{x}_6\hfill \\ x_7^{\prime }={\beta }_3{x}_4-{\alpha }_3{x}_7\hfill \end{array}\right..$$

The coefficient matrix is as follows

$$A_3=\left(\begin{array}{ccccccc}0& 0& 0& 0& -1& -1& -1\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& {\beta }_1& -{\alpha }_1& 0& 0\\ 0& 0& 0& {\beta }_2& 0& -{\alpha }_2& 0\\ 0& 0& 0& {\beta }_3& 0& 0& -{\alpha }_3\end{array}\right)$$

We obtain the characteristic polynomial of this matrix

$$\begin{array}{ccc}\hfill P_3\left(\lambda \right)& =& {\lambda }^7+\left({\alpha }_1+{\alpha }_2+{\alpha }_3\right){\lambda }^6+\left(\left({\alpha }_2+{\alpha }_3\right){\alpha }_1+{\alpha }_2{\alpha }_3\right){\lambda }^5+{\alpha }_1{\alpha }_2{\alpha }_3{\lambda }^4+\left({\beta }_1+{\beta }_2+{\beta }_3\right){\lambda }^2\hfill \\ & & +\left(\left({\beta }_2+{\beta }_3\right){\alpha }_1+\left({\beta }_1+{\beta }_3\right){\alpha }_2+\left({\beta }_1+{\beta }_2\right){\alpha }_3\right)\lambda +\left({\alpha }_2{\beta }_3+{\alpha }_3{\beta }_2\right){\alpha }_1+{\alpha }_2{\alpha }_3{\beta }_1\hfill \end{array}$$

The coefficient of ${\lambda }^3$ is equal to zero, so we obtain by the Hurwitz criteria that the equation is exponentially unstable.

Example 4.

$$x^{‴}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}{x}^{\prime }\left(s\right)ds=0$$

By reducing this integro-differential equation to the system of ODE, we obtain

$$\left\{\begin{array}{c}{x}_1^{\prime }=-x_3\hfill \\ x_2^{\prime }=x_1\hfill \\ x_3^{\prime }={\beta }_1{x}_2-{\alpha }_1{x}_3\hfill \end{array}\right..$$

The coefficient matrix is as follows

$$A_4=\left(\begin{array}{ccc}0& 0& -1\\ 1& 0& 0\\ 0& {\beta }_1& -{\alpha }_1\end{array}\right)$$

We obtain the characteristic polynomial of this matrix

$$P_4\left(\lambda \right)={\lambda }^3+{\alpha }_1{\lambda }^2+{\beta }_1$$

The coefficient of λ is equal to zero, so we obtain by the Hurwitz criteria that the equation is exponentially unstable.

Example 5.

$$x^{⁗}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}{x}^{\prime }\left(s\right)ds+{\beta }_2{\int }_0^t{e}^{-{\alpha }_2\left(t-s\right)}{x}^{\prime }\left(s\right)ds=0$$

By reducing this integro-differential equation to the system of ODE, we obtain

$$\left\{\begin{array}{c}{x}_1^{\prime }=-x_4-x_5\hfill \\ x_2^{\prime }=x_1\hfill \\ x_3^{\prime }=x_2\hfill \\ x_4^{\prime }={\beta }_1{x}_3-{\alpha }_1{x}_4\hfill \\ x_5^{\prime }={\beta }_2{x}_3-{\alpha }_2{x}_5\hfill \end{array}\right..$$

The coefficient matrix is as follows

$$A_5=\left(\begin{array}{ccccc}0& 0& 0& -1& -1\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& {\beta }_1& -{\alpha }_1& 0\\ 0& 0& {\beta }_2& 0& -{\alpha }_2\end{array}\right)$$

We obtain the characteristic polynomial of this matrix

$$P_5\left(\lambda \right)={\lambda }^5+\left({\alpha }_1+{\alpha }_2\right){\lambda }^4+{\alpha }_1{\alpha }_2{\lambda }^3+\left({\beta }_1+{\beta }_2\right)\lambda +{\beta }_1{\alpha }_2+{\beta }_2{\alpha }_1$$

The coefficient of ${\lambda }^2$ is equal to zero, so we obtain by the Hurwitz criteria that the equation is exponentially unstable.

5. Conclusions

In the current paper, we described the case of the impossibility of stabilization by feedback control (2) and (3) in the case of an n-order functional-differential equation. We considered control functions in integral form

$$w_1\left(t\right)=\sum _{j=1}^k{\int }_0^t{K}_{1j}\left(t,s\right)x\left(s\right)ds=\sum _{j=1}^k{w}_{1j}\left(t\right)$$

where $w_{1j}\left(t\right)={\int }_0^t{K}_{1j}\left(t,s\right)x\left(s\right)ds$ and

$$w_2\left(t\right)=\sum _{j=1}^k{\int }_0^t{K}_{2j}\left(t,s\right)x^{\prime }\left(s\right)ds=\sum _{j=1}^k{w}_{2j}\left(t\right),$$

where $w_{2j}\left(t\right)={\int }_0^t{K}_{2j}\left(t,s\right)x\left(s\right)ds.$

The vectors $\left(w_{11},...,w_{1k}\right)$ and $\left(w_{21},...,w_{2k}\right)$ define the feedback control of Equations (2) and (3). Here, vectors dimensions define the degree of freedom of the feedback control function.

We prove that it is impossible to achieve stabilization of the differential Equation (1) by the feedback control if the degree of freedom is less or equal to $n-1$ in the case of feedback control in the form $w_1\left(t\right)$ and it is impossible to achieve stabilization of the differential Equation (1) by the feedback control if the degree of freedom is less or equal to $n-2$ in the case of feedback control in the form $w_2\left(t\right)$.