Positive solutions and exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays

Abstract

This paper is concerned with positive solutions and global exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays. By utilizing the comparison principle via differential inequalities, we first explore conditions on damping coefficients and self-excitation coefficients to ensure that, with nonnegative connection weights and inputs, all state trajectories of the system initiating in an admissible set of initial conditions are always nonnegative. Then, based on the method of using homeomorphisms, we derive conditions in terms of linear programming problems via M-matrices for the existence, uniqueness, and global exponential stability of a positive equilibrium of the system. Two examples with numerical simulations are given to illustrate the effectiveness of the obtained results.

Appendices

Appendix 1: Proof of Lemma 3

(Necessity) Let $\eta _i>0$, $\xi _i>0$, $i\in [n]$, satisfying $D_{\alpha }\succ 0$ and $D_{\beta }\succ 0$. Then, we have

$$\begin{aligned} \xi _i(a_i-\xi _i)-b_i>0,\quad i\in [n]. \end{aligned}$$

Observe that $\xi _i(a_i-\xi _i)-b_i=\frac{1}{4}\left( a_i^2-4b_i\right) -\left( \xi _i-\frac{1}{2}a_i\right) ^2$. Thus,

$$\begin{aligned} a_i^2-4b_i>\left( 2\xi _i-a_i\right) ^2\ge 0,\quad i\in [n]. \end{aligned}$$

(Sufficiency) Let condition (11) hold. Then, the constants $\xi _i^l=\frac{a_i-\sqrt{a_i^2-4b_i}}{2}$ and $\xi _i^u=\frac{a_i+\sqrt{a_i^2-4b_i}}{2}$ are well defined, $\xi _i^l>0$, and $\xi _i^u<a$. In addition, $\xi _i(a_i-\xi _i)-b_i=(\xi _i^u-\xi _i)(\xi _i-\xi _i^l)>0$ for all $\xi _i\in (\xi _i^l,\xi _i^u)$. Therefore,

$$\begin{aligned} {\left\{ \begin{array}{ll} 0<\xi _i<a_i,\\ -\xi _i^2+a_i\xi _i-b_i>0,\quad i\in [n], \end{array}\right. } \end{aligned}$$

(29)

for any $\xi _i\in (\xi _i^l,\xi _i^u)$. It follows from (29) that $D_{\alpha }\succ 0$ and $D_{\beta }\succ 0$ for any $\eta _i>0$ and $\xi _i\in (\xi _i^l,\xi _i^u)$. The proof is completed.

Appendix 2: Proof of Theorem 1

For any initial condition in ${\mathscr {A}}_T$, it suffices to prove that the corresponding solution $z(t)=(x^{\top }(t),y^{\top }(t))^{\top }$ of (5) is positive. To this end, we note at first that if $y_i(t)\ge 0$, $t\in [0,t_f)$, for some $t_f>0$, then from the first equation in (5), we have

$$\begin{aligned} x_i(t)&=e^{-\xi _it}\Big (\phi _i(0)+\eta _i\int _0^te^{\xi _is}y_i(s)ds\Big )\\&\ge \phi _i(0)e^{-\xi _it}\ge 0,\quad \forall t\in [0,t_f). \end{aligned}$$

Thus, it is only necessary to show that $y(t)=(y_i(t))\succeq 0$ for all $t\ge 0$.

For a given $\epsilon >0$, let $z_{\epsilon }(t)=(x^{\top }_{\epsilon }(t),y^{\top }_{\epsilon }(t))^{\top }$ be solution (5) with initial functions $\phi _i(.)$ and $\psi _{i\epsilon }(.)=\psi _i(.)+\epsilon$. Since $\phi (s)=(\phi _i(s))\succeq 0$ and $\psi _{\epsilon }(s)=(\psi _{i\epsilon }(s))\succeq \epsilon {\mathbf {1}}_n$ for all $s\in [-\tau ^+,0]$, where ${\mathbf {1}}_n$ denotes the vector in ${\mathbb {R}}^n$ with all entries equal one, it follows from (5) that $y_{\epsilon }(t)=(y_{i\epsilon }(t))\succ 0$, $t\in [0,t_f)$, for some small $t_f>0$. By virtue of the contrary argument method, we assume that there exist a $t_1>0$ and an index $i\in [n]$ such that

$$\begin{aligned} y_{i\epsilon }(t_1)=0,\quad y_{i\epsilon }(t)>0,\quad t\in [0,t_1), \end{aligned}$$

(30)

and $y_{\epsilon }(t)\succeq 0$ for all $t\in [0,t_1]$. Then, by multiplying with $e^{\alpha _it}$, it follows from (5) that

$$\begin{aligned} \frac{d}{dt}&\left( y_{i\epsilon }(t)e^{\alpha _it}\right) =\beta _ix_{i\epsilon }(t)e^{\alpha _it} +\eta _i^{-1}\sum _{j=1}^nc_{ij}e^{\alpha _it}f_j(x_{j\epsilon }(t))\nonumber \\&+\eta _i^{-1}\Big [\sum _{j=1}^nd_{ij}e^{\alpha _it}f_j(x_{j\epsilon }(t-\tau _j(t)))+I_ie^{\alpha _it}\Big ]. \end{aligned}$$

(31)

It follows from (31) and $D_{\beta }e^{D_{\alpha }t}x_{\epsilon }(t)\succeq 0$, $t\in [0,t_1]$, that

$$\begin{aligned} \begin{aligned} D_{\eta }&y_{\epsilon }(t)\succeq e^{-D_{\alpha }t}D_{\eta }\psi _{\epsilon }(0) +C\int _0^te^{D_{\alpha }(s-t)}f(x_{\epsilon }(s))ds\\&+D\int _0^te^{D_{\alpha }(s-t)}{\hat{f}}({\hat{x}}_{\epsilon }(s))ds +D_{\alpha }^{-1}\big (E_n-e^{-D_{\alpha }t}\big )I, \end{aligned} \end{aligned}$$

(32)

where ${\hat{f}}({\hat{x}}_{\epsilon }(s))=(f_j(x_{j\epsilon }(s-\tau _j(s)))$ and $E_n$ is the identity matrix in ${\mathbb {R}}^{n\times n}$.

Since the vector fields $F_1(x)=Cf(x)$ and $F_2(x)=Df(x)$ are order-preserving, $x_{\epsilon }(t)\succeq 0$ and ${\hat{x}}_{\epsilon }(t)\succeq 0$ for $t\in [0,t_1]$, from (32), we have

$$\begin{aligned} y_{\epsilon }(t)\succeq e^{-D_{\alpha }t}\psi _{\epsilon }(0)+D_{\alpha \eta }^{-1}\big (E_n-e^{-D_{\alpha }t}\big )I, \end{aligned}$$

(33)

where $D_{\alpha \eta }=D_{\alpha }D_{\eta }$. Let $t\uparrow t_1$ in (33), and note also that $E_n-e^{-D_{\alpha }t_1}\succ 0$, we readily obtain

$$\begin{aligned} y_{\epsilon }(t_1)\succeq \epsilon e^{-D_{\alpha }t_1}{\mathbf {1}}_n\succ 0, \end{aligned}$$

which gives a contradiction with (30). Therefore, $x_{\epsilon }(t)\succeq 0$ and $y_{\epsilon }(t)\succ 0$ for all $t\ge 0$. Let $\epsilon \downarrow 0$, we then obtain $z(t)=\lim _{\epsilon \rightarrow 0}z_{\epsilon }(t)\succeq 0$. The proof is completed.

Appendix 3: Proof of Theorem 2

We define the following mappings

$$\begin{aligned} {\mathscr {H}}_1(\chi )&=-D_{\xi }x+D_{\eta }y,\\ {\mathscr {H}}_2(\chi )&=(D_{\alpha \xi }-B)x-D_{\alpha \eta }y+(C+D)f(x)+I. \end{aligned}$$

For any vectors $\chi _1=\begin{bmatrix}x_1\\ y_1\end{bmatrix}$ and $\chi _2=\begin{bmatrix}x_2\\ y_2\end{bmatrix}$ in ${\mathbb {R}}^{2n}$, we have

$$\begin{aligned} {\mathscr {H}}_1(\chi _1)-{\mathscr {H}}_1(\chi _2)=-D_{\xi }(x_1-x_2)+D_{\eta }(y_1-y_2). \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} {\mathscr {S}}(x_1-x_2)&[{\mathscr {H}}_1(\chi _1)-{\mathscr {H}}_1(\chi _2)]=-D_{\xi }|x_1-x_2|\\&+{\mathscr {S}}(x_1-x_2)D_{\eta }(y_1-y_2)\\&\preceq -D_{\xi }|x_1-x_2|+D_{\eta }|y_1-y_2|, \end{aligned} \end{aligned}$$

(34)

where ${\mathscr {S}}(x_1-x_2)={{\mathrm{diag}}}\{{{\mathrm{sgn}}}(x_{1i}-x_{2i})\}$.

By assumption (A1), and $D_{\alpha \xi }-B=D_{\beta }D_{\eta }\succ 0$, similar to (34), we also have

$$\begin{aligned} \begin{aligned} {\mathscr {S}}(y_1&-y_2)[{\mathscr {H}}_2(\chi _1)-{\mathscr {H}}_2(\chi _2)]\preceq -D_{\alpha \eta }|y_1-y_2|\\&+(D_{\alpha \xi }-B)|x_1-x_2|+(C+D)|f(x_1)-f(x_2)|\\ \preceq \;&-D_{\alpha \eta }|y_1-y_2|+(D_{\alpha \xi }-B)|x_1-x_2|\\&+(C+D)L_f|x_1-x_2|, \end{aligned} \end{aligned}$$

(35)

where $L_f={{\mathrm{diag}}}\{l_i^f\}$. From (34) and (35), we have

$$\begin{aligned} \begin{aligned} {{\mathrm{diag}}}\{{\mathscr {S}}(x_1-x_2),{\mathscr {S}}(y_1-y_2)\}&[{\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)]\\&\preceq -{\mathscr {M}}|\chi _1-\chi _2|, \end{aligned} \end{aligned}$$

(36)

where ${\mathscr {M}}=\begin{pmatrix}D_{\xi }&-D_{\eta }\\ B-D_{\alpha \xi }-(C+D)L_f&D_{\alpha \eta }\end{pmatrix}$. On the other hand, it is clear that

$$\begin{aligned} \begin{aligned} -{{\mathrm{diag}}}\{{\mathscr {S}}(x_1-x_2),&{\mathscr {S}}(y_1-y_2)\}[{\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)]\\&\preceq |{\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)|. \end{aligned} \end{aligned}$$

(37)

Thus, combining (36) and (37) gives

$$\begin{aligned} |{\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)|\succeq {\mathscr {M}}|\chi _1-\chi _2|. \end{aligned}$$

(38)

Let $\chi _0$ be a positive vector satisfying (14). Then, from (38), we have

$$\begin{aligned} \chi _0^{\top }|{\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)|\succeq \chi _0^{\top }{\mathscr {M}}|\chi _1-\chi _2|. \end{aligned}$$

(39)

If ${\mathscr {H}}(\chi _1)-{\mathscr {H}}(\chi _2)=0$ then, by (14) and (39), $\chi _1=\chi _2$. This shows that the mapping ${\mathscr {H}}(.)$ is injective in ${\mathbb {R}}^{2n}$. In addition to this, let $\chi _2=0$, from (39), we have

$$\begin{aligned} \Vert {\mathscr {H}}(\chi )\Vert _{\infty }\ge \frac{1}{\Vert \chi _0\Vert _{\infty }}\chi _0^{\top }{\mathscr {M}}|\chi |-\Vert {\mathscr {H}}(0)\Vert _{\infty }, \end{aligned}$$

which ensures that $\Vert {\mathscr {H}}(\chi _k)\Vert _{\infty }\rightarrow \infty$ for any sequence $\{\chi _k\}$ in ${\mathbb {R}}^{2n}$ satisfying $\Vert \chi _k\Vert _{\infty }\rightarrow \infty$. Thus, the continuous mapping ${\mathscr {H}}(.)$ is proper. By Lemma 2, ${\mathscr {H}}(.)$ is a homeomorphism onto ${\mathbb {R}}^{2n}$. Consequently, the equation ${\mathscr {H}}(\chi )=0$ has a unique solution $\chi _*\in {\mathbb {R}}^{2n}$, which is a unique EP of (5).