Stabilization Method for nth-Order ODE by Distributed Control Function

Abstract

The stabilization of solutions by distributed feedback control functions for second- and third-order ordinary differential equations (ODEs) has been presented in earlier studies. The present paper extends these results to the stabilization of n-th order ODEs using a distributed control function expressed in integral form with first-order derivatives. The problem of stabilizing n-th order ODE solutions by distributed control functions is significantly more complex and nontrivial. This work introduces a method for selecting the parameter set within the distributed control function. Furthermore, the connection between palindromic polynomials, log-concavity, and stability with respect to initial conditions (Lyapunov stability) in n-th order ODEs with distributed feedback control functions is established. We use the symmetry property of palindromic polynomials.

1. Introduction

Stabilization and exponential stability are very challenging, well-researched problems. Stability analysis determines whether small perturbations to a system will cause it to return to equilibrium (stability), diverge from it (instability), or exhibit sustained oscillations and other nontrivial dynamics. Addressing the stability problem in differential equations is essential for understanding their long-term behavior. The main direction in the development of integro-differential equations is applications (see, for example, [1,2]). In recent years, the study of stability has also expanded to include systems with time delays, stochastic influences, and spatial dimensions, introducing even more challenges and complexity to the analysis.

In their paper [3], the authors proposed an approach for the global exponential stability study of a general class of switched systems described by time-varying nonlinear functional differential equations. Various results and techniques, such as the Bohl–Perron theorem and matrix measures, are applied to obtain new explicit exponential stability conditions for systems of functional differential equations in [4]. The article [5] presents the existence, uniqueness, and exponential stability results for mild solutions to fractional neutral stochastic differential systems. The paper [6] investigates the exponential stability of nonlinear time-varying differential equations. The results extend existing stability criteria and provide new insights into the behavior of nonlinear time-varying systems. In [7], the authors study the exponential stability of impulsive functional differential equations with finite or infinite delays. The paper [8] addresses both the pth moment and the exponential stability of stochastic functional differential equations with delay. The author introduces a new type of Lyapunov functional and utilizes stochastic analysis techniques to establish sufficient conditions for exponential stability. The paper [9] presents new explicit exponential stability conditions for non-autonomous linear functional differential equations characterized by oscillatory and potentially discontinuous coefficients and kernels. In the paper [10], the authors study the exponential stability of impulsive switched nonlinear time-delay systems with delayed impulses. The paper [11] addresses the global exponential stability of the impulsive functional differential system with delays using Lyapunov functions and the Razumikhin technique. Some new delay-independent criteria of global exponential stability are obtained. The paper [12] systematically studies the Hurwitz stability of polynomials in combinatorics. A criterion for the Hurwitz stability of the Turán expressions of recursive polynomials is obtained, implying the q-log-convexity or q-log-concavity of the original polynomials. Hurwitz stability criteria are then given for the recursive polynomials. The paper [13] focuses on the roots of palindromic polynomials. The result has applications in the theory of systems.

The stabilization of solutions by distributed feedback control functions for second-order ODEs, along with the corresponding domain of stability, was presented in [14]. In the paper [15], the results of solution stabilization by the distributed feedback control function of a third-order ODE were presented. The present paper extends these findings to the stabilization of n-th order ODEs. Compared to the second- and third-order cases, research on the stabilization of n-th order ODE solutions by distributed control functions is substantially more complex and nontrivial. A key aspect of this work is the original application of the reduction method proposed in [16], followed by the use of palindromic polynomials (their symmetry properties). In addition, the method described in [17] is used. In this article, we discuss the connection between palindromic polynomials and log-concavity for the stability with respect to the initial condition (Lyapunov’s stability) of functional differential equations with distributed control functions in integral form with first-order derivatives. We propose a method of parameter choosing in the distributed control function in the stabilization process.

2. Stabilization Problem Statement

Consider the n-order ordinary differential equation

$$x^{\left(n\right)}\left(t\right)+\sum _{i=0}^{n-1}{p}_i\left(t\right)x^{\left(i\right)}\left(t\right)=f\left(t\right),$$

(1)

where $f\left(t\right), p_i\left(t\right)$ $1\le i\le n-1$ are continuous functions.

Let us introduce the distributed feedback control function in integral form:

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

(2)

where the kernel function is defined in the following form:

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

(3)

Remark 1.

The kernels in the form (3) look like some particular case of possible kernels. But in [18], it was demonstrated that a very wide class of kernels can be represented in the form of an infinite number of kernels in the form (3). In all applications which we know of at the moment, only finite sums are used.

Definition 1.

Consider the equation

$$x^{\left(n\right)}+\sum _{i=0}^{n-1}{p}_i\left(t\right)x^{\left(i\right)}\left(t\right)=f\left(t\right),t\in [t_0,\infty ).$$

The equation is exponentially stable if there exist positive numbers γ and N such that

$$max\left\{\right|x\left(t\right)|,|x^{\prime }\left(t\right)|,...,|x^{(n-1)}\left(t\right)\left|\right\}\le N{e}^{-\gamma (t-t_0)}max\left\{\right|x\left(t_0\right)|,|x^{\prime }\left(t_0\right)|,...,|x^{(n-1)}\left(t_0\right)\left|\right\},$$

for $t\in [t_0,\infty )$.

In [19], we presented the results of the solution stabilization impossibility of Equation (1) by the distributed control function (2), where the number of integral components $k<n-1$ was insufficient for stabilization.

In the current paper, we analyze the case when the number of integral components in a distributed control function (2) is $k=n-1$. We present the method of choosing the set of $2n-2$ parameters in a distributed control function, using palindromic polynomials and log-concavity. A stabilization method is proposed.

3. Stabilization Method of $\mathit{n}$-Order Ordinary Differential Equation by Control Function in the Form (2) in the Case of $\mathit{k}=\mathit{n}-\mathbf{1}$

Let us introduce the following ordinary differential equation:

$$x^{\left(n\right)}\left(t\right)+w\left(t\right)=0.$$

(4)

If we set

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

then Equation (4) becomes an integro-differential one.

Denoting $x_{n-1}\left(t\right)=x^{\prime }\left(t\right)$, $x_{n-2}\left(t\right)=x_{n-1}^{\prime }\left(t\right)=x^{″}\left(t\right),\dots ,x_1\left(t\right)=x^{(n-1)}\left(t\right)$ and $x_j\left(t\right)={\beta }_{j-n+1}{\int }_0^t{e}^{-{\alpha }_{j-n+1}(t-s)}{x}^{\prime }\left(s\right)ds$, where $n\le j\le 2n-2$ and using the reduction method, we pass from integro-differential Equation (4) to an ordinary differential system with $2n-2$ equations.

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

(5)

Remark 2.

The solution of the first-order ODE system (5) and the solution’s first-order derivative of the integro-differential Equation (4) coincide (see [17]).

Theorem 1.

The characteristic polynomial of the coefficient matrix of system (5) is

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

(6)

Proof. 

The coefficient matrix $\left(\right(2n-2)\times (2n-2\left)\right)$ of system (5) is

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

(7)

Let us denote

$$det(\lambda I-A)=\left(\begin{array}{cccccccc}\lambda & 0& \cdots & 0& 0& 1& \cdots & 1\\ -1& \lambda & \cdots & 0& 0& 0& \cdots & 0\\ 0& -1& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1& \lambda & 0& \cdots & 0\\ 0& 0& \cdots & 0& -{\beta }_1& \lambda +{\alpha }_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}& 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right)=det\left(\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right).$$

Here, the $\left(\right(n-1)\times (n-1\left)\right)$ matrices are

$$\begin{array}{ccc}\hfill A_1& =& \left(\begin{array}{ccccc}\lambda & 0& \cdots & 0& 0\\ -1& \lambda & \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& \lambda \end{array}\right), A_2=\left(\begin{array}{ccc}1& \cdots & 1\\ 0& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 0\end{array}\right),\hfill \\ \hfill A_3& =& \left(\begin{array}{ccccc}0& 0& \cdots & 0& -{\beta }_1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& -{\beta }_{n-1}\end{array}\right), A_4=\left(\begin{array}{ccc}\lambda +{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \lambda +{\alpha }_{n-1}\end{array}\right).\hfill \end{array}$$

We obtain

$$\begin{array}{cc}\hfill det\left(A_1-A_2{A}_4^{-1}{A}_3\right)& =det\left(\begin{array}{ccccc}\lambda & 0& \cdots & 0& {\sum }_{i=1}^{n-1}{\displaystyle \frac{{\beta }_i}{\lambda +{\alpha }_i}}\\ -1& \lambda & \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& \lambda \end{array}\right)\hfill \\ & =& {\left(-1\right)}^n\sum _{i=1}^{n-1}{\displaystyle \frac{{\beta }_i}{\lambda +{\alpha }_i}}·det\left(\begin{array}{cccc}-1& \lambda & \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1\end{array}\right)\hfill \\ & & +{\left(-1\right)}^{2n-2}\lambda det\left(\begin{array}{ccccc}\lambda & 0& \cdots & 0& 0\\ -1& \lambda & \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& \lambda \end{array}\right)=\sum _{i=1}^{n-1}{\displaystyle \frac{{\beta }_i}{\lambda +{\alpha }_i}}+{\lambda }^{n-1}.\hfill \end{array}$$

Using the Schur formula [20], we obtain the characteristic polynomial of matrix (7):

$$P\left(\lambda \right)=det(\lambda I-A)=det\left(A_4\right)det(A_1-A_2{A}_4^{-1}{A}_3)=\prod _{j=1}^{n-1}\left(\lambda +{\alpha }_j\right)\left({\lambda }^{n-1}+\sum _{i=1}^{n-1}{\displaystyle \frac{{\beta }_i}{\lambda +{\alpha }_i}}\right)$$

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

□

Definition 2.

The polynomial $Q\left(x\right)=x^n+b_{n-1}{x}^{n-1}+b_{n-2}{x}^{n-2}+\dots +b_{1}x+1$ is called a palindrome polynomial (or palyndrome) if $b_k=b_{n-k}$, $k=0,1,\dots ,n.$

Theorem 2.

Let ${\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}$ where all ${\alpha }_i,1\le i\le n-1$ are different. If characteristic polynomial (6) of ODE system (5) is a palindrome polynomial, then parameters ${\beta }_i\in \mathbb{R}$ $1\le i\le n$ can be represented via ${\alpha }_1,\dots ,{\alpha }_n$ and this representation is unique.

Proof. 

If the characteristic polynomial (6) is a palindrome, it can be represented in the following form:

$$\begin{array}{c}P\left(\lambda \right)={\lambda }^{2n-2}+{\lambda }^{2n-3}{\sum }_{j=1}^{n-1}{\alpha }_j+{\lambda }^{2n-4}{\sum }_{\begin{array}{c}j,k=1\\ j\ne k\end{array}}^{n-1}{\alpha }_j{\alpha }_k+\dots +{\lambda }^n{\prod }_{\begin{array}{c}k,j=1\\ j\ne i\end{array}}^{n-1}{\alpha }_j{\alpha }_k+\\ {\lambda }^{n-1}{\prod }_{j=1}^{n-1}{\alpha }_j+{\lambda }^{n-2}{\sum }_{i=1}^{n-1}{\beta }_i+\dots +\lambda {\sum }_{i=1}^{n-1}{\beta }_i{\sum }_{\begin{array}{c}k=1\\ k\ne i\end{array}}^{n-1}{\prod }_{\begin{array}{c}j=1\\ j\ne k\\ j\ne i\end{array}}^{n-1}{\alpha }_j+{\sum }_{i=1}^{n-1}{\beta }_i{\prod }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-1}{\alpha }_j\end{array}$$

Variables ${\beta }_1,\dots ,{\beta }_{n-1}\in \mathbb{R}$ are a solution of the following linear non-homogeneous system:

$$\left\{\begin{array}{c}{\displaystyle {\sum }_{i=1}^{n-1}}{\beta }_i={\prod }_{\begin{array}{c}k,j=1\\ j\ne i\end{array}}^{n-1}{\alpha }_j{\alpha }_k\hfill \\ ⋮\hfill \\ {\displaystyle {\sum }_{i=1}^{n-1}}{\beta }_i{\displaystyle {\sum }_{\begin{array}{c}k=1\\ k\ne i\end{array}}^{n-1}}{\displaystyle {\prod }_{\begin{array}{c}j=1\\ j\ne k\\ j\ne i\end{array}}^{n-1}}{\alpha }_j={\displaystyle {\sum }_{j=1}^{n-1}}{\alpha }_j\hfill \\ {\displaystyle {\sum }_{i=1}^{n-1}}{\beta }_i{\prod }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n-1}{\alpha }_j=1\hfill \end{array}\right..$$

(8)

The coefficient matrix of (8) is

$$A=\left(\begin{array}{cccc}1& 1& \cdots & 1\\ ⋮& ⋮& \cdots & ⋮\\ {\displaystyle {\sum }_{\begin{array}{c}k=2\end{array}}^{n-1}}{\displaystyle {\prod }_{\begin{array}{c}j=2\\ j\ne k\end{array}}^{n-1}}{\alpha }_j& {\displaystyle {\sum }_{\begin{array}{c}k=1\\ k\ne 2\end{array}}^{n-1}}{\displaystyle {\prod }_{\begin{array}{c}j=1\\ j\ne k\\ j\ne 2\end{array}}^{n-1}}{\alpha }_j& \cdots & {\displaystyle {\sum }_{\begin{array}{c}k=1\\ k\ne n-1\end{array}}^{n-1}}{\displaystyle {\prod }_{\begin{array}{c}j=1\\ j\ne k\\ j\ne n-1\end{array}}^{n-1}}{\alpha }_j\\ {\prod }_{\begin{array}{c}j=1\\ j\ne 1\end{array}}^{n-1}{\alpha }_j& {\prod }_{\begin{array}{c}j=1\\ j\ne 2\end{array}}^{n-1}{\alpha }_j& \cdots & {\prod }_{\begin{array}{c}j=1\\ j\ne n-1\end{array}}^{n-1}{\alpha }_j\end{array}\right).$$

Let us note that $det\left(A\right)\ne 0,$ because ${\alpha }_1\ne {\alpha }_2\ne ...\ne {\alpha }_{n-1}$ and all ${\alpha }_i,1\le i\le n-1$ are positive. Thus, system (8) has a unique solution. □

Definition 3.

A sequence $\left(b_n\right)$ is said to be log-concave if $b_n^2\ge b_{n-1}{b}_{n+1}$ for all $n\in \mathbb{N}$. If all terms of the sequence are positive, then log-concavity implies unimodality.

Definition 4.

Let $r\ge 1$. A sequence $\left(b_n\right)$ is r-factor log-concave if $b_n^2\ge r{b}_{n-1}{b}_{n+1}$ for all $n\in \mathbb{N}$ and r-factor strong log-concave if $b_n^2>r{b}_{n-1}{b}_{n+1}$ for all $n\in \mathbb{N}$.

Theorem 3

([21]). Let $Q\left(x\right)$ be a polynomial with positive coefficients and with a degree larger than 5. If the coefficients of $Q\left(x\right)$ are $r_0$-factor strongly log-concave, where $r_0\approx 1.466$, then all the roots of $Q\left(x\right)$ have negative real parts.

Theorem 4.

If $Q\left(x\right)=b_{2n-2}{x}^{2n-2}+b_{2n-1}{x}^{2n-1}+\dots +b_{1}x+b_0$ is a palindrome polynomial of even degree, such that ${\left\{b_i\right\}}_{i=n-1}^{2n-2}$ is r-factor strongly log-concave and the following inequality is fulfilled $b_n^2>r_0{b}_{n+1}^2$, then all coefficients of the polynomial are r-factor strongly log-concave.

Proof. 

Because $Q\left(x\right)$ is a palindrome polynomial, $b_j=b_{2n-2-j}$ for all $0\le j\le n-2$. We obtain that $b_j^2=b_{2n-2-j}^2>r{b}_{2n-2-j-1}{b}_{2n-2-j+1}$ and so $b_j^2>r{b}_{j-1}{b}_{j+1}$ for all $1\le j\le n-2$. In addition, $b_n^2>r_0{b}_{n+1}^2$, which completes the proof. □

Theorem 5.

Let ${\alpha }_1,...,{\alpha }_n\in \mathbb{N}$ and all ${\alpha }_i,1\le i\le n-1$ be different. If the characteristic polynomial (6) of the ODE system (5) is a palindrome polynomial, and the coefficients ${\left\{c_i\right\}}_{i=1}^n$ are $r_0$-strongly log-concave, where $r_0\approx 1.466$ and the inequality $c_{n-1}^2>r_0{c}_n^2$ is fulfilled, then the solution of the ODE system (5) is exponentially stable and the solution of integro-differential Equation (4) is stable according to Lyapunov.

Proof. 

The proof follows from Theorems 3 and 4 and from the definition of solution exponential stability. □

4. Examples

Example 1.

In the case of a differential equation of the third order and a distributed control function with two integral terms, with the first-order derivative

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

(9)

for the “solution-vector” starting with the coordinate $x_1\left(t\right)=x^{\prime }\left(t\right)$, the following system can be written using the idea of [17]:

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

(10)

The characteristic polynomial of the ODE system (10), according to Formula (6), is the following:

$$\begin{array}{c}\hfill P\left(\lambda \right)={\lambda }^4+{\lambda }^3({\alpha }_1+{\alpha }_2)+{\lambda }^2{\alpha }_1{\alpha }_2+\alpha ({\beta }_1+{\beta }_2)+({\beta }_1{\alpha }_2+{\beta }_2{\alpha }_1)\end{array}$$

Let us choose ${\alpha }_1=2, {\alpha }_2=4$, so the polynomial coefficients $b_4=1, b_3={\alpha }_1+{\alpha }_2=6, b_2={\alpha }_1{\alpha }_2=8$ will be $r_0$-factor strongly log-concave, where $r_0\approx 1.466$, and the inequality $b_2^2>r_0{b}_3^2$ is fulfilled. For building a palindromic polynomial, let us solve the following system of equations:

$$\left\{\begin{array}{c}{\beta }_1+{\beta }_2={\alpha }_1+{\alpha }_2\hfill \\ {\beta }_1{\alpha }_2+{\beta }_2{\alpha }_1=1.\hfill \end{array}\right.$$

The solution is ${\beta }_1=-5.5, {\beta }_2=11.5$. For these values of parameters, we obtain the following characteristic polynomial $P\left(\lambda \right)={\lambda }^4+6{\lambda }^3+8{\lambda }^2+6\lambda +1,$ with the following roots:

$$\begin{array}{c}\hfill {\lambda }_1\approx -4.51033793029575,{\lambda }_2\approx -0.221712877273129,\\ \hfill {\lambda }_{3,4}\approx -0.{633974596215561}_{-}^{+}0.773353872010295i.\end{array}$$

For the parameter values ${\alpha }_1=2, {\alpha }_2=4, {\beta }_1=-5.5, {\beta }_2=11.5$, according to Theorem 5, the solution of the ODE system (10) is exponentially stable and the solution of integro-differential Equation (9) is stable according to Lyapunov (Figure 1).

Example 2.

In the case of a differential equation of the fourth order and a distributed control function with three integral terms with the first-order derivative

$$x^{\left(4\right)}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}{x}^{\prime }\left(s\right)ds+{\beta }_2{\int }_0^t{e}^{-{\alpha }_2\left(t-s\right)}{x}^{\prime }\left(s\right)ds+{\beta }_3{\int }_0^t{e}^{-{\alpha }_3\left(t-s\right)}{x}^{\prime }\left(s\right)ds=0,$$

(11)

for the “solution-vector” starting with the coordinate $x_1\left(t\right)=x^{\prime }\left(t\right)$, the following system can be written using the idea of [17]:

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

(12)

The characteristic polynomial of the ODE system (12), according to Formula (6), is the following:

$$\begin{array}{c}\hfill P\left(\lambda \right)={\lambda }^6+{\lambda }^5({\alpha }_1+{\alpha }_2+{\alpha }_3)+{\lambda }^4({\alpha }_1{\alpha }_2+{\alpha }_1{\alpha }_3+{\alpha }_2{\alpha }_3)+{\lambda }^3{\alpha }_1{\alpha }_2{\alpha }_3+\\ \hfill {\lambda }^2({\beta }_1+{\beta }_2+{\beta }_3)+\lambda ({\beta }_1({\alpha }_2+{\alpha }_3)+{\beta }_2({\alpha }_1+{\alpha }_3)+{\beta }_3({\alpha }_1+{\alpha }_2))+{\beta }_1{\alpha }_2{\alpha }_3+{\beta }_2{\alpha }_1{\alpha }_3+{\beta }_3{\alpha }_1{\alpha }_2.\end{array}$$

Let us choose ${\alpha }_1=3, {\alpha }_2=5, {\alpha }_3=6$, so the polynomial coefficients $b_6=1, b_5=14, b_4=63, b_3=90$ will be $r_0$-factor strongly log-concave, where $r_0\approx 1.466$, and the inequality $b_3^2>r_0{b}_4^2$ is fulfilled. For building a palindromic polynomial, let us solve the following system of equations:

$$\left\{\begin{array}{c}{\beta }_1+{\beta }_2+{\beta }_3=63\hfill \\ 11{\beta }_1+9{\beta }_2+8{\beta }_3=14\hfill \\ 30{\beta }_1+18{\beta }_2+15{\beta }_3=1\hfill \end{array}\right..$$

The solution is ${\beta }_1={\displaystyle \frac{263}{3}}, {\beta }_2=-753, {\beta }_3={\displaystyle \frac{2185}{3}}$. For these values of parameters, we obtain the following characteristic polynomial $P\left(\lambda \right)={\lambda }^6+14{\lambda }^5+63{\lambda }^4+90{\lambda }^3+63{\lambda }^2+14\lambda +1,$ with the following roots:

$$\begin{array}{c}\hfill {\lambda }_{1,2}\approx -6.{09283030351902}_{-}^{+}1.50918209621927i,\\ \hfill {\lambda }_{3,4}\approx -0.{752530164368208}_{-}^{+}0.658557781607626i,\\ \hfill {\lambda }_{5,6}\approx -0.{154639532112771}_{-}^{+}0.0383039082998136i.\end{array}$$

For the parameter values ${\alpha }_1=3, {\alpha }_2=5, {\alpha }_3=6, {\beta }_1={\displaystyle \frac{263}{3}}, {\beta }_2=-753, {\beta }_3={\displaystyle \frac{2185}{3}}$, according to Theorem 5, the solution of the ODE system (12) is exponentially stable and the solution of integro-differential Equation (11) is stable according to Lyapunov (Figure 2).

Example 3.

In the case of a differential equation of the fourth order and a distributed control function with three integral terms with the first-order derivative

$$x^{\left(4\right)}\left(t\right)+{\beta }_1{\int }_0^t{e}^{-{\alpha }_1\left(t-s\right)}{x}^{\prime }\left(s\right)ds+{\beta }_2{\int }_0^t{e}^{-{\alpha }_2\left(t-s\right)}{x}^{\prime }\left(s\right)ds+{\beta }_3{\int }_0^t{e}^{-{\alpha }_3\left(t-s\right)}{x}^{\prime }\left(s\right)ds=0,$$

(13)

for the “solution-vector” starting with the coordinate $x_1\left(t\right)=x^{\prime }\left(t\right)$, the following system can be written using the idea of [17]:

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

(14)

The characteristic polynomial of the ODE system (14), according to Formula (6), is the following:

$$\begin{array}{c}\hfill P\left(\lambda \right)={\lambda }^6+{\lambda }^5({\alpha }_1+{\alpha }_2+{\alpha }_3)+{\lambda }^4({\alpha }_1{\alpha }_2+{\alpha }_1{\alpha }_3+{\alpha }_2{\alpha }_3)+{\lambda }^3{\alpha }_1{\alpha }_2{\alpha }_3+\\ \hfill {\lambda }^2({\beta }_1+{\beta }_2+{\beta }_3)+\lambda ({\beta }_1({\alpha }_2+{\alpha }_3)+{\beta }_2({\alpha }_1+{\alpha }_3)+{\beta }_3({\alpha }_1+{\alpha }_2))+{\beta }_1{\alpha }_2{\alpha }_3+{\beta }_2{\alpha }_1{\alpha }_3+{\beta }_3{\alpha }_1{\alpha }_2.\end{array}$$

Let us choose ${\alpha }_1=2, {\alpha }_2=7, {\alpha }_3=9$, so the polynomial coefficients $b_6=1, b_5=18, b_4=95, b_3=126$ will be $r_0$-factor strongly log-concave, where $r_0\approx 1.466$, and the inequality $b_3^2>r_0{b}_4^2$ is fulfilled. For building a palindromic polynomial, let us solve the following system of equations:

$$\left\{\begin{array}{c}{\beta }_1+{\beta }_2+{\beta }_3=95\hfill \\ 16{\beta }_1+11{\beta }_2+9{\beta }_3=18\hfill \\ 63{\beta }_1+18{\beta }_2+14{\beta }_3=1\hfill \end{array}\right..$$

The solution is ${\beta }_1={\displaystyle \frac{69}{7}}, {\beta }_2=-453, {\beta }_3={\displaystyle \frac{3767}{7}}$. For these values of parameters, we obtain the following characteristic polynomial $P\left(\lambda \right)={\lambda }^6+14{\lambda }^5+63{\lambda }^4+90{\lambda }^3+63{\lambda }^2+14\lambda +1,$ with the following roots:

$$\begin{array}{c}\hfill {\lambda }_{1,2}\approx -8.{2437}_{-}^{+}0.8845i,\\ \hfill {\lambda }_{3,4}\approx -0.{63641}_{-}^{+}0.77135i,\\ \hfill {\lambda }_{5,6}\approx -0.{119925}_{-}^{+}0.012867i.\end{array}$$

For the parameter values ${\alpha }_1=2,{\alpha }_2=7,{\alpha }_3=9,{\beta }_1={\displaystyle \frac{69}{7}},{\beta }_2=-453,{\beta }_3={\displaystyle \frac{3767}{7}}$, according to Theorem 5, the solution of the ODE system (14) is exponentially stable and the solution of integro-differential Equation (13) is stable according to Lyapunov (Figure 3).

5. Conclusions

In the current article, we present the solution stabilization method for n-order ordinary differential equations by a distributed feedback control function in the integral form (2) with kernel (3).

In [19], we presented the solution impossibility of the stabilization of n-order ordinary differential equations by a distributed feedback control function in the integral form (2) with kernel (3), where the number of terms $k<n-1$. The number of integral terms that were less than $n-1$ was insufficient for stabilization. In the current article, we found the optimal form of the distributed control function for the stabilization of n-order ODEs. We present a method of choosing parameters set in the distributed control function, based on log-concavity and palindromic polynomials.

In the current paper, we analyze the case when the number of integral components in a distributed feedback control function (2) is $k=n-1$. The current article presents a solution stabilization method based on choosing the $2n-2$ set of parameters in a distributed feedback control function, using palindromic polynomials and strong log-concavity. Examples of choosing this parameter are presented.

In the current article, a stabilization method is presented that applies to a specific type of ODE equation, and sufficient conditions are given. Using the distributed control function, we obtain the Lyapunov stability of n-th order ODEs. In the future, it can be researched whether we can obtain the exponential stability, which is much stronger and does not depend on the initial condition.