Exponential stability criterion for interval neural networks with discrete and distributed delays

Delay-dependent LMI-based robust stability criterion for discrete and distributed time-delays Markovian jumping reaction–diffusion CGNNs under Neumann boundary value

“…Next, note that Homomorphic mapping, M-matrix and H-matrix methods are not easy to obtain the linear matrix inequalities (LMIs) criterion conditions (see [14]) Finally, note that some variational methods have been applied to obtain the stability criterion of reaction-diffusion dynamic systems (see, e.g. [2,[9][10][11][12][13][14]), and in these works, the role of diffusion terms for exponentially stabilizing reaction-diffusion CGNN has remained ignored.…”

“…For example, by using Lyapunov functional, modified stability condition for neural networks with discrete and distributed delays has been obtained in [15]; by constructing general Lyapunov functional and convex combination approach, the stability criterion for neural networks with mixed delays has been obtained in [16]; by partitioning the time delay and using Jensen integral inequalities, the stability condition on delayed neural networks with both discrete and distributed delays has been obtained in [17]. In 2015, Homomorphic mapping theory and M-matrix method were employed to obtain a stability criterion ( [14,Theorem 3.3]) for the following neural networks ( [14, (22)]) with discrete and distributed time-delays, z(t) = −Cz(t) + Ag(z(t)) + Bg(z(t − τ(t))) + D t t−σ g(z(s))ds.…”

Existence of Exponentialp-Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

“…In this paper, we also need to consider the equilibrium solution of (4) defined on [ 0 , +∞). Different from [22,23], we consider the nonconstant equilibrium solution = * ( ) = ( * 1 ( ), * 2 ( ), . .…”

“…Motivated by [27], we proposed some conditions on activation functions to set up existence criterion for the equilibrium solution of system (4). In [22,23], the constant equilibrium solution = * for all ∈ [ 0 , +∞) was obtained by homomorphic mapping theory and matrix theory, or matrix theory and homotopy invariance theorem, where * = ( * 1 , * 2 , . .…”

“…Our methods employed in this paper are different from those of previous related literature. For example, homomorphic mapping theory was employed to obtain the existence of equilibrium of ordinary differential equations in [22]; topological degree theory was used to obtain the existence of equilibrium for fuzzy ordinary differential equations in [23] and of equilibrium for reaction-diffusion partial differential equations in [17]. In this paper, Ekeland variational principle is originally proposed to solve the existence of nonconstant equilibrium for nonlinear diffusion equations.…”