Stability Results for the Kdv Equation with Time-Varying Delay

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

“…Now, using the variable norm theory of Kato [58][59][60], we will show the existence and uniqueness of the above abstract system with time dependent operator. Similar types of analysis has been done in many works; see [48,51,53,61] for instances. The main idea of the existence-uniqueness theory is to show that the triplet {, , ((0)} with  = {(t) ∶ t ∈ [0, T]} for some fixed T > 0 forms a constant domain system.…”

“…Now, using the variable norm theory of Kato [58–60], we will show the existence and uniqueness of the above abstract system with time dependent operator. Similar types of analysis has been done in many works; see [48, 51, 53, 61] for instances. The main idea of the existence‐uniqueness theory is to show that the triplet $$$ \left\{\mathcal{A},\mathcal{H},\mathcal{D}\right(\mathcal{A}(0)\Big\} $$$ with $$$ \mathcal{A}=\left\{\mathcal{A}(t):t\in \left[0,T\right]\right\} $$$ for some fixed $$$ T>0 $$$ forms a constant domain system.…”

“…Internal feedback stabilization with boundary delay for wave equation has been established [52]. Recently, in [53], Parada et al have utilized the above mentioned works to produce stability analysis of KdV equation with interior and boundary time varying delay. Time-dependent delay phenomenon for second-order evolution equation can be found in [54].…”

Delay-Independent Stability of Stochastic Systems with Time-Varying Delay and Lévy Noise

“…There are many different types of delays, for instance distributed delay, time-varying delay, constant delay, etc. Due to time delays often leading to instability, it has been extensively studied for decades that stability for stochastic differential delay systems [9,10]. Generally, stability criteria are usually divided into time-delay dependent stability and time-delay independent stability [11,12].…”