Related papers: On exponential stability of linear delay equations…
We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $ where $ 0\leq a(t)\leq A_0<1$, $0<b_0\leq b(t)\leq B$, using the…
This paper deals with the stability of linear periodic difference delay systems, where the value at time $t$ of a solution is a linear combination with periodic coefficients of its values at finitely many delayed instants…
The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(\Omega\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent…
Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional…
For the nonlinear second order Lienard-type equations with time-varying delays $$ \ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^l s_k(t,x(h_k(t)))=0, $$ global asymptotic stability conditions are obtained. The results are…
This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other…
In this paper we prove a variation of constants formula for a non autonomous and non homogeneous Cauchy problems whenever the linear part is not densely defined and is not a Hille-Yosida operator. By using this variation of constants…
We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also…
This paper provides a necessary and sufficient condition for guaranteeing exponential stability of the linear difference equation $x(t)=Ax(t-a)+Bx(t-b)$ where $a>0,b>0$ are constants and $A,B$ are $n\times n$ square matrices, in terms of a…
The main result of the paper is that the oscillation (non-oscillation) of the impulsive delay differential equation $\dot {x}(t)+\sum_{k=1}^m A_k(t)x[h_k(t)]=0,~~t\geq 0$, $x(\tau_j)=B_jx(\tau_j-0), \lim \tau_j = \infty$ is equivalent to…
It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ is in no relation to the sign of the real parts of the eigenvalues of…
It is well-known that the exponential stability of Integral Difference Equations and Delay Difference Equations, in the usual state space of continuous functions, is equivalent to the location of the roots of its associated characteristic…
A sufficient condition for asymptotic stability of the zero solution to an abstract nonlinear evolution problem is given. The governing equation is $\dot{u}=A(t)u+F(t,u),$ where $A(t)$ is a bounded linear operator in Hilbert space $H$ and…
A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for…
We consider a nonlinear non-autonomous system with time-varying delays $$ \dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which has a large number of applications in the theory of artificial neural networks. Via…
In this paper we investigate the uniform exponential stability of the system $\frac{dx(t)}{dt}=Ax(t)-\rho Bx(t), \; (\rho >0), $ where the unbounded operator $A$ is the infinitesimal generator of a linear $C_0-$semigroup of contractions…
This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx {{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance,…
We give an approach to exponential stability within the framework of evolutionary equations due to [R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci.,…
We study the stability of general $n$-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first…
For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying…