Theorem 1
Suppose that (A1)–(A2) hold. System (1)–(2) is almost sure ISS if there exist constants
\(q_{i} ( i ) > 0\)
for any
\(r ( t ) = i \in S\), \(i,j = 1,2,\ldots,n\), such that
$$ \begin{aligned}[b] &{-} \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum _{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} + \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \\ &\quad\quad{}+ \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum _{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} + \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1 \\ &\quad{}< 0. \end{aligned} $$
(4)
Here
\(\Xi= \frac{\underline{\alpha} ( m - 2 )^{2}}{2\pi ^{2}}\ + \frac{2\underline{\alpha} \Lambda_{2}}{R_{\Omega}^{2}}\), \(\underline{\alpha} = \min \{ D_{il},i = 1,\ldots,n;l = 1,\ldots,m \} > 0\), and
π
is a radial bound of an open domain Ω.
Proof
If condition (4) holds, then we can choose a positive number ε (may be very small) such that, for \(i = 1,2,\ldots,n\),
$$ \begin{aligned}[b] & {-} \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum _{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \\ &\qquad{}+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert + \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} + \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1 + \varepsilon\\ &\quad{}< 0. \end{aligned} $$
(5)
Consider the following functions:
$$ \begin{aligned}[b] F_{i} ( y_{i} ) ={}& 2y_{i} - \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum _{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert + \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} k_{ji} ( s )e^{2y_{i}s}\,ds + \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} \\ &+ \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1. \end{aligned} $$
(6)
From (6) and (A2) we obtain that \(F_{i} ( 0 ) < - \varepsilon< 0\) and \(F_{i} ( y_{i} )\) is continuous for \(y_{i} \in [ 0, + \infty )\); moreover, \(F_{i} ( y_{i} ) \to+ \infty\) as \(y_{i} \to+ \infty\), and thus there exists constant \(\alpha_{i} \in ( 0, + \infty )\) such that
$$ \begin{aligned}[b] F_{i} ( \alpha_{i} ) ={}& 2 \alpha_{i} - \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum _{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert + \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i}^{2} \bigl\vert b_{ji} ( i ) \bigr\vert \int_{0}^{ + \infty} k_{ji} ( s )e^{2\alpha_{i}s}\,ds + \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j}}{q_{i}}L_{i} \\&+ \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1 \\={}& 0. \end{aligned} $$
(7)
Let \(\alpha= \min_{1 \le i \le n} \{ \alpha_{i} \}\). Clearly, \(\alpha> 0\), and we can get
$$ \begin{aligned}[b] F_{i} ( \alpha ) ={}& 2\alpha- \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert + \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} k_{ji} ( s )e^{2\alpha s}\,ds + \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j}}{q_{i}}L_{i} \\ &+ \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1 \\ \le{}&0. \end{aligned} $$
(8)
Given \(\varphi\in L_{F_{0}}^{p}( ( - \infty,0 ] \times \Omega;R^{n} )\), fix the system mode \(i \in S\) arbitrarily. Let \(\phi_{j} ( t ) = t - \tau_{j} ( t )\). Since the derivative \(\dot{\phi}_{j} ( t ) = 1 - \dot{\tau}_{j} ( t ) \ge1 - \mu> 0\), \(\phi_{j} ( t )\) has an inverse function. We denote this inverse function by \(\phi_{j}^{ - 1} ( t )\). Construct the Lyapunov functional
$$\begin{aligned}[b] V \bigl( t,u ( t ),r ( t ) = i \bigr) ={}& \int_{\Omega} \sum_{i = 1}^{n} q_{i} ( i ) \Biggl[ e^{2\alpha t}u_{i} ( t )^{2} \\& + \frac{1}{1 - \mu} \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \int_{t - \tau_{j} ( t )}^{t} u_{j} ( s )^{2}e^{2\alpha ( s + \tau_{j} ( \phi _{j}^{ - 1} ( s ) ) )} \,ds\\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \int_{0}^{ + \infty} k_{ij} ( s ) \int_{t - s}^{t} g_{j} \bigl( u_{j} ( z ) \bigr)^{2} e^{2\alpha ( z + s )}\,dz\,ds \Biggr]\,dx.\end{aligned} $$
(9)
Along the solutions of model (1), we have
$$\begin{aligned} LV \bigl( t,u ( t ),r ( t ) = i \bigr) ={}& \lim_{\Delta\to0^{ +}} \frac{1}{\Delta} \bigl[ E \bigl\{ V \bigl( t + \Delta,u ( t + \Delta ),r ( t + \Delta ) \bigr)|u ( t ),r ( t ) = i \bigr\} \\ &- V \bigl( t,u ( t ),r ( t ) = i \bigr) \bigr] \\ ={}& \int_{\Omega} e^{2\alpha t}\sum_{i = 1}^{n} q_{i} ( i ) \Biggl\{ 2u_{i} ( t ) \Biggl[ \sum _{l = 1}^{m} \frac{\partial}{\partial x_{l}} \biggl( D_{il} \frac{\partial u_{i} ( t )}{\partial x_{l}} \biggr) - a_{i} ( i )u_{i} ( t ) \\ &+ \sum_{j = 1}^{n} w_{ij} ( i )g_{j} \bigl( u_{j} ( t ) \bigr) + \sum _{j = 1}^{n} h_{ij} ( i )g_{j} \bigl( u_{j} \bigl( t - \tau_{j} ( t ) \bigr) \bigr) \\ & + \sum_{j = 1}^{n} b_{ij} ( i ) \int_{ - \infty}^{t} k_{ij} ( t - s )g_{j} \bigl( u_{j} ( s ) \bigr)\,ds + v_{i} ( t ) \Biggr] + 2\alpha u_{i}^{2} ( t ) \\ &+ \sum_{j = 1}^{n} \frac{ \vert h_{ij} ( i ) \vert }{1 - \mu} L_{j} \bigl[ e^{2\alpha\tau} u_{j} ( t )^{2} - ( 1 - \mu )u_{j} \bigl( t - \tau_{j} ( t ) \bigr)^{2} \bigr] \\&+ e^{2\alpha t}\sum_{j = 1}^{n} \gamma_{ij}q_{i} ( j )u_{i} ( t )^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \biggl[ \int _{0}^{ + \infty} e^{2\alpha s}k_{ij} ( s )g_{j} \bigl( u_{j} ( t ) \bigr)^{2}\,ds \\& - \int_{0}^{ + \infty} k_{ij} ( s ) g_{j} \bigl( u_{j} ( t - s ) \bigr)^{2}\,ds \biggr] \Biggr\} \,dx \\ \le{}& \int_{\Omega} e^{2\alpha t}\sum_{i = 1}^{n} q_{i} ( i ) \Biggl\{ \Biggl[ 2u_{i} ( t )\sum _{l = 1}^{m} \frac{\partial}{\partial x_{l}} \biggl( D_{il} \frac{\partial u_{i} ( t )}{\partial x_{l}} \biggr) - 2a_{i} ( i )u_{i} ( t )^{2} \\ &+ 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i}u_{i} ( t )^{2} + 2\sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert \bigl\vert g_{j} \bigl( u_{j} ( t ) \bigr) \bigr\vert \\&+ 2 \bigl\vert u_{i} ( t ) \bigr\vert \sum _{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \bigl\vert u_{j} \bigl( t - \tau_{j} ( t ) \bigr) \bigr\vert \\ & + 2 \bigl\vert u_{i} ( t ) \bigr\vert \sum _{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \int_{ - \infty}^{t} k_{ij} ( t - s ) \bigl\vert g_{j} \bigl( u_{j} ( s ) \bigr) \bigr\vert \,ds + 2 \bigl\vert u_{i} ( t ) \bigr\vert v_{i} ( t ) \Biggr] \\ &+ 2\alpha u_{i} ( t )^{2}+ \sum_{j = 1}^{n} \frac{ \vert h_{ij} ( i ) \vert }{1 - \mu} L_{j} \bigl[ e^{2\alpha\tau} u_{j} ( t )^{2} - ( 1 - \mu )u_{j} \bigl( t - \tau_{j} ( t ) \bigr)^{2} \bigr] \\ &+ \sum_{j = 1}^{n} \gamma_{ij}q_{i} ( j )u_{i} ( t )^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \biggl[ \int _{0}^{ + \infty} k_{ij} ( s )e^{2\alpha s} \bigl\vert g_{j} \bigl( u_{j} ( t ) \bigr) \bigr\vert ^{2}\,ds \\& - \int_{0}^{ + \infty} k_{ij} ( s ) \bigl\vert g_{j} \bigl( u_{j} ( t - s ) \bigr) \bigr\vert ^{2} \,ds \biggr] \Biggr\} \,dx. \end{aligned}$$
(10)
From Young’s inequality and (A2), we obtain
$$ 2\sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert \bigl\vert g_{j} \bigl( u_{j} ( t ) \bigr) \bigr\vert \le\sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \bigl\vert g_{j} \bigl( u_{j} ( t ) \bigr) \bigr\vert ^{2} $$
(11)
and
$$ \begin{aligned}[b] &2 \bigl\vert u_{i} ( t ) \bigr\vert \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \int_{ - \infty}^{t} k_{ij} ( t - s ) \bigl\vert g_{j} \bigl( u_{j} ( s,x ) \bigr) \bigr\vert \,ds \\ &\quad\le\sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \int_{ - \infty}^{t} k_{ij} ( t - s ) \bigl\vert g_{j} \bigl( u_{j} ( s,x ) \bigr) \bigr\vert ^{2}\,ds \end{aligned} . $$
(12)
Applying the Green formula, the Dirichlet boundary condition, and Lemma 1, we have
$$ \begin{aligned}[b] 2 \int_{\Omega} \sum_{l = 1}^{m} u_{i} ( t )\frac{\partial}{\partial x_{l}} \biggl( D_{il} \frac{\partial u_{i} ( t )}{\partial x_{l}} \biggr)\,dx &= - 2\sum_{l = 1}^{m} \int_{\Omega} D_{il} \biggl( \frac{\partial u_{i} ( t )}{\partial x_{l}} \biggr)^{2}\,dx \\ &< - \biggl( \frac{\underline{\alpha} ( m - 2 )^{2}}{2\pi^{2}} + \frac{2\underline{\alpha} \Lambda_{2}}{R_{\Omega}^{2}} \biggr) \int _{\Omega} u_{i} ( t )^{2}\,dx\\& = - \Xi \int_{\Omega} u_{i} ( t )^{2}\,dx. \end{aligned} $$
(13)
By (11)–(13) and (A2) we derive
$$\begin{aligned} LV ( t,u,i ) \le{}& \int_{\Omega} e^{2\alpha t}\sum_{i = 1}^{n} q_{i} ( i ) \Biggl\{ \Biggl[ - \Xi u_{i} ( t )^{2} - 2a_{i} ( i )u_{i} ( t )^{2} \\ &+ 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i}u_{i} ( t )^{2} + \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert L_{j}^{2} \bigl\vert u_{j} ( t ) \bigr\vert ^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \bigl( \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + \bigl\vert u_{j} \bigl( t - \tau_{j} ( t ) \bigr) \bigr\vert ^{2} \bigr) + \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \bigl\vert u_{i} ( t ) \bigr\vert ^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \int_{ - \infty}^{t} k_{ij} ( t - s ) \bigl\vert g_{j} \bigl( u_{j} ( s ) \bigr) \bigr\vert ^{2}\,ds + \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + v_{i} ( t )^{2} \Biggr] \\& + \sum _{j = 1}^{n} \frac{ \vert h_{ij} ( i ) \vert }{1 - \mu} L_{j}e^{2\alpha \tau} \bigl\vert u_{j} ( t ) \bigr\vert ^{2} \\ &- \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} \bigl\vert u_{j} \bigl( t - \tau_{j} ( t ) \bigr) \bigr\vert ^{2} + \sum_{j = 1}^{n} \gamma_{ij}q_{i} ( j )u_{i} ( t )^{2} + 2\alpha u_{i} ( t )^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \biggl[ \int _{0}^{ + \infty} k_{ij} ( s )e^{2\alpha s} \bigl\vert g_{j} \bigl( u_{j} ( t ) \bigr) \bigr\vert ^{2}\,ds \\& - \int_{0}^{ + \infty} k_{ij} ( s ) \bigl\vert g_{j} \bigl( u_{j} ( t - s ) \bigr) \bigr\vert ^{2} \,ds \biggr] \Biggr\} \,dx \\ ={}& \int_{\Omega} e^{2\alpha t}\sum_{i = 1}^{n} q_{i} ( i ) \Biggl[ \Biggl( - \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert \\& + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} \\ &+ \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} + \sum _{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert + \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} k_{ji} ( s )e^{2\alpha s}\,ds \\ &+ \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} + 1 + 2 \alpha+ \sum_{j = 1}^{n} \gamma_{ij}q_{i} ( j ) \Biggr) \bigl\vert u_{i} ( t ) \bigr\vert ^{2} + v_{i} ( t )^{2} \Biggr] \,dx. \end{aligned}$$
(14)
It follows from Dynkin’s formula and (14) that
$$ \begin{aligned}[b] EV ( t,u,i ) \le{}& EV \bigl( 0,\varphi ( 0 ),i \bigr) + \Biggl\{ \int_{0}^{t} e^{2\alpha\xi} \sum _{i = 1}^{n} q_{i} ( i ) \Biggl[ \Biggl( - \Xi- 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} \\ &+ \sum_{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum _{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} + \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \\&+ \sum _{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} k_{ji} ( s )e^{2\alpha s}\,ds \\ &+ \sum_{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} + 1 + 2\alpha+ \sum_{j = 1}^{n} \gamma_{ij}q_{i} ( j ) \Biggr)E \bigl\Vert u_{i} ( \xi ) \bigr\Vert _{2}^{2} \Biggr]d\xi \Biggr\} \\ &+ \frac{n}{2\alpha} E \bigl\Vert v ( t ) \bigr\Vert _{\Omega}^{2} \bigl( e^{2\alpha t} - 1 \bigr) .\end{aligned} $$
(15)
Since
$$ V ( t,u,i ) \ge\sum_{i = 1}^{n} q_{i} ( i )e^{2\alpha t} \bigl\Vert u_{i} ( t ) \bigr\Vert _{2}^{2} \ge \min_{1 \le i \le n} \bigl\{ q_{i} ( i ) \bigr\} e^{2\alpha t}\sum_{i = 1}^{n} \bigl\Vert u_{i} ( t ) \bigr\Vert _{2}^{2},\quad t \ge0, $$
(16)
and
$$\begin{aligned} V \bigl( 0,\varphi ( 0 ),0 \bigr) ={}& \int_{\Omega} \sum_{i = 1}^{n} q_{i} ( i ) \Biggl[ \varphi_{i} ( 0 )^{2} + \frac{1}{1 - \mu} \sum_{j = 1}^{n} \bigl\vert h_{ij} ( 0 ) \bigr\vert L_{j} \int_{ - \tau_{j} ( 0 )}^{0} u_{j}^{2} ( s,x )e^{2\alpha ( s + \tau_{j} ( \psi_{j}^{ - 1} ( s ) ) )} \,ds \\ &+ \sum_{j = 1}^{n} \bigl\vert b_{ij} ( 0 ) \bigr\vert \int_{0}^{ + \infty} k_{ij} ( s ) \int_{ - s}^{0} g_{j} \bigl( u_{j} ( z,x ) \bigr)^{2} e^{2\alpha ( z + s )}\,dz\,ds \Biggr]\,dx \\ \le{}&\max_{1 \le i \le n} \bigl\{ q_{i} ( 0 ) \bigr\} \sum _{i = 1}^{n} \Biggl\{ \bigl\Vert \varphi_{i} ( 0 ) \bigr\Vert _{2}^{2} \\ &+ \sum _{j = 1}^{n} \bigl\vert b_{ij} ( 0 ) \bigr\vert L_{j}^{2} \int_{0}^{ + \infty} k_{ij} ( s ) \biggl[ \int_{ - s}^{0} \bigl\Vert u_{j} ( z,x ) \bigr\Vert _{2}^{2}e^{2\alpha ( z + s )}\,dz \biggr]\,ds \\ &+ \frac{1}{1 - \mu} \sum_{j = 1}^{n} \bigl\vert h_{ij} ( 0 ) \bigr\vert L_{j} \int_{ - \tau}^{0} \bigl\Vert u_{i} ( s ) \bigr\Vert _{2}^{2}e^{2\alpha ( s + \tau _{j} ( \psi_{j}^{ - 1} ( s ) ) )} \,ds \Biggr\} \\ \le{}&\max_{1 \le i \le n} \bigl\{ q_{i} ( 0 ) \bigr\} \Biggl\{ 1 + \max_{1 \le i \le n} \Biggl\{ \sum_{j = 1}^{n} \bigl\vert b_{ji} ( 0 ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} se^{2\alpha s}k_{ji} ( s )\,ds \Biggr\} \\ & + \frac{\tau e^{2\alpha\tau}}{1 - \mu} \sum _{j = 1}^{n} \bigl\vert h_{ij} ( 0 ) \bigr\vert L_{j} \Biggr\} \Vert \varphi_{i} \Vert _{2}^{2}, \end{aligned}$$
(17)
combining (4) and (15)–(17), we derive
$$ \begin{aligned}[b] E \bigl[ \bigl\Vert u ( t ) \bigr\Vert _{2} \bigr] \le \biggl( \frac{\max_{1 \le i \le n} \{ q_{i} ( 0 ) \}}{\min_{1 \le i \le n} \{ q_{i} ( i ) \}} \biggr)^{1 / 2}e^{ - \alpha t} \Biggl\{ 1 + \max_{1 \le i \le n} \Biggl\{ \sum _{j = 1}^{n} \bigl\vert b_{ji} ( 0 ) \bigr\vert L_{i}^{2} \int_{0}^{ + \infty} se^{2\alpha s}k_{ji} ( s )\,ds \Biggr\} \\ + \frac{\tau e^{2\alpha\tau}}{1 - \mu} \sum_{j = 1}^{n} \bigl\vert h_{ij} ( 0 ) \bigr\vert L_{j} \Biggr\} ^{1 / 2}E \bigl[ \Vert \varphi \Vert \bigr] + \biggl( \frac{n}{2\alpha\min_{1 \le i \le n} \{ q_{i} ( i ) \}} \biggr)^{1 / 2}E \bigl[ \bigl\Vert v ( t ) \bigr\Vert _{\Omega} \bigr]. \end{aligned} $$
(18)
Hence, from (3) we get that system (1)–(2) is almost sure ISS. This completes the proof of Theorem 1. □
Remark 4
In this paper, we concern with the Markovian jump RDNNs with Dirichlet boundary conditions. The results are expressed by a set of inequalities. These conditions are easy to verify, and our results play an important role in the design and applications of almost sure ISS. It is worth mentioning that the effect of reaction–diffusion terms is considered by the Hardy–Poincaré inequality. In Theorem 1, the Hardy–Poincaré inequality is used firstly. Moreover, we can see a very interesting fact that as long as diffusion coefficients \(D_{il}\) in system (1) are large enough, (4) always be satisfied. This shows that a large enough diffusion can always make system (1)–(2) almost sure ISS.
Remark 5
If we do not consider Markov jump parameters, that is, the Markov chain \(\{ r ( t ),t \ge0 \}\) only takes a unique value 1 (i.e., \(S = \{ 1 \}\)), then, for simplicity, we write \(a_{i} ( 1 ) = a_{i}\), \(w_{ij} ( 1 ) = w_{ij}\), \(h_{ij} ( 1 ) = h_{ij}\), \(b_{ij} ( 1 ) = b_{ij}\). Then system (1) will be reduced to the following deterministic delayed RDNNs:
$$\begin{aligned}[b] \frac{\partial u_{i} ( t,x )}{\partial t} ={}& \sum_{l = 1}^{m} \frac{\partial}{\partial x_{l}} \biggl( D_{il}\frac{\partial u_{i} ( t,x )}{\partial x_{l}} \biggr) - a_{i}u_{i} ( t,x ) + \sum_{j = 1}^{n} w_{ij}g_{j} \bigl( u_{j} ( t,x ) \bigr)\\ &+ \sum _{j = 1}^{n} h_{ij}g_{j} \bigl( u_{j} \bigl( t - \tau_{j} ( t ),x \bigr) \bigr) + \sum_{j = 1}^{n} b_{ij} \int_{ - \infty}^{t} k_{ij} ( t - s )g_{j} \bigl( u_{j} ( s,x ) \bigr)\,ds\\ & + v_{i} ( t ),\quad t \ge0,x \in\Omega. \end{aligned} $$
(19)
It is worth pointing out that particular cases of system (19) were studied in [19, 23].
The next theorem shows that the equilibrium solution of system (19) is ISS. The proof of Theorem 2 is similar to that in Theorem 1, and thus we omit it.
Theorem 2
Suppose that (A1)–(A2) hold. System (19) and (2) is ISS if there exist constants
\(q_{i} > 0\)
for any
\(i,j = 1,2,\ldots,n\)
such that
$$ \begin{aligned}[b] &{-} \Xi- 2a_{i} + 2 \vert w_{ii} \vert L_{i} + \sum_{j = 1,j \ne i}^{n} \vert w_{ij} \vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j}}{q_{i}} \vert w_{ji} \vert L_{i}^{2} + \sum_{j = 1}^{n} \vert h_{ij} \vert L_{j} + \sum_{j = 1}^{n} \vert b_{ij} \vert \\ &\quad{}+ \sum_{j = 1}^{n} \frac{q_{j}}{q_{i}} \vert b_{ji} \vert L_{i}^{2} + \sum _{j = 1}^{n} \frac{ \vert h_{ji} \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j}}{q_{i}}L_{i} + 1 < 0. \end{aligned} $$
(20)
Remark 6
Theorem 1 reduces to almost sure exponential stability condition for delayed RDNNs with Markovian jump parameters if \(v ( t ) = 0\). Similarly, Theorem 2 becomes an exponential stability condition for delayed RDNNs when \(v ( t ) = 0\). In [34], the authors employed the Lyapunov direct method to consider the almost sure stability of Itô stochastic reaction–diffusion systems with Brownian motion defined in a complete probability space, including asymptotic stability in probability and almost sure exponential stability. In addition, the stability criteria in [34] are independent on reaction–diffusion coefficients and the regional feature. Compared with [34], this paper studies the ISS analysis for a class of RDNNs with mixed delays and Markovian jump parameters. Furthermore, the given ISS criteria are true to Dirichlet boundary conditions and concerned with the regional feature, the reaction–diffusion coefficients, and the first eigenvalue of the Dirichlet Laplacian.
Some famous NN models are particular cases of model (1). In system (1)–(2), ignoring the role of reaction–diffusion, system (1) reduces to the following delayed NNs:
$$ \begin{aligned}[b] &\begin{aligned}du_{i} ( t ) = {}&\Biggl[ - a_{i} \bigl( r ( t ) \bigr)u_{i} ( t ) + \sum _{j = 1}^{n} w_{ij} \bigl( r ( t ) \bigr)g_{j} \bigl( u_{j} ( t ) \bigr) + \sum _{j = 1}^{n} h_{ij} \bigl( r ( t ) \bigr)g_{j} \bigl( u_{j} \bigl( t - \tau_{j} ( t ) \bigr) \bigr) \\& + \sum_{j = 1}^{n} b_{ij} \bigl( r ( t ) \bigr) \int_{ - \infty}^{t} k_{ij} ( t - s )g_{j} \bigl( u_{j} ( s ) \bigr)\,ds + v_{i} ( t ) \Biggr]\,dt,\quad t \ge0,\end{aligned} \\ &u_{i} ( s ) = \varphi_{i} ( s ),\quad s \in ( - \infty,0 ].\end{aligned} $$
(21)
As a consequence of Theorems 1 and 2, we get the following results.
Corollary 1
Assume that (A1) and (A2) are satisfied. System (21) is almost sure ISS if there exist constants
\(q_{i} ( i ) > 0\)
for any
\(r ( t ) = i \in S\), \(i,j = 1,2,\ldots,n\), such that
$$ \begin{aligned}[b] &{-} 2a_{i} ( i ) + 2 \bigl\vert w_{ii} ( i ) \bigr\vert L_{i} + \sum _{j = 1,j \ne i}^{n} \bigl\vert w_{ij} ( i ) \bigr\vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert w_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum_{j = 1}^{n} \bigl\vert h_{ij} ( i ) \bigr\vert L_{j} + \sum_{j = 1}^{n} \bigl\vert b_{ij} ( i ) \bigr\vert \\ &\qquad{}+ \sum_{j = 1}^{n} \frac{q_{j} ( i )}{q_{i} ( i )} \bigl\vert b_{ji} ( i ) \bigr\vert L_{i}^{2} + \sum _{j = 1}^{n} \frac{ \vert h_{ji} ( i ) \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j} ( i )}{q_{i} ( i )}L_{i} + \sum _{j = 1}^{n} \gamma_{ij}q_{i} ( j ) + 1\\&\quad < 0. \end{aligned} $$
(22)
Corollary 2
Assume that (A1) and (A2) are satisfied. System (21) is ISS if there exist constants
\(q_{i} > 0\)
for any
\(i,j = 1,2,\ldots,n\)
such that
$$ \begin{aligned}[b] &{-} 2a_{i} + 2 \vert w_{ii} \vert L_{i} + \sum_{j = 1,j \ne i}^{n} \vert w_{ij} \vert + \sum_{j = 1,j \ne i}^{n} \frac{q_{j}}{q_{i}} \vert w_{ji} \vert L_{i}^{2} + \sum_{j = 1}^{n} \vert h_{ij} \vert L_{j} + \sum_{j = 1}^{n} \vert b_{ij} \vert \\ &\quad+ \sum_{j = 1}^{n} \frac{q_{j}}{q_{i}} \vert b_{ji} \vert L_{i}^{2} + \sum _{j = 1}^{n} \frac{ \vert h_{ji} \vert }{1 - \mu} e^{2\alpha\tau} \frac{q_{j}}{q_{i}}L_{i} + 1 < 0. \end{aligned} $$
(23)
Remark 7
Our model in (21) is more general than some well-studied NNs. When \(b_{ij} = 0\), the model in (21) reduces the model studied in [15]. The authors in [15] present criteria for the ISS of NNs with time-varying delays. Corollary 2 in this paper is much less conservative than those in [15]. Moreover, our results depend on Markovian jump parameters and can be easily checked by simple computation. To the best of authors’ knowledge, up to now, little work is reported on almost sure ISS of NNs with Markovian jump parameters and mixed time-varying delays.