General Stability Result Research Articles

Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay and a very general class of memory kernel functions. Under some suitable conditions and by introducing a suitable Lyapunov functional, we state and prove explicit and general stability results of the energy. We give also some applications to illustrate our results. This work generalizes the previous results without time delay term to those with time-varying delay.

A novel average dwell time strategy for stability analysis of discrete-time switched systems by T–S fuzzy modeling

In this paper, the stability problem of discrete-time switched nonlinear systems is considered. The fast and slow Φ-dependent average dwell time switchings are first presented. By developing a multiple Lyapunov functions approach, the general stability results are derived with a designed switching law, where fast and slow switchings are applied to deal with unstable and stable modes, respectively. A Takagi–Sugeno (T–S) fuzzy model is used for approximating discrete-time switched nonlinear systems, and stability conditions of the switched T–S fuzzy model are also built under fast and slow Φ-dependent average dwell time switchings. Finally, a numerical example is given to demonstrate the validity and correctness of our proposed method.

Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history

<abstract><p>The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.</p></abstract>

General stability result of swelling porous elastic soils with a viscoelastic damping

We consider a swelling porous-elastic system with a single memory term as the only damping source. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils. Unexpectedly, using the multiplier method, we establish a general decay result irrespective of the wave speeds of the system.

Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator g allowing a general growth in the state variable y and a sub-quadratic growth in the state variable z, when the terminal condition satisfies a sub-exponential moment integrability condition, which is weaker than the usual exp(μL)-integrability and stronger than Lp(p>1)-integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under some additional assumptions and establish a general stability result for the unbounded solutions. Finally, we derive the nonlinear Feynman–Kac formula in this context.

A general and optimal stability result for a laminated beam

We consider a thermoelastic laminated beam with a finite memory acting on the effective rotation angle. We establish an explicit and optimal stability estimate for the solution energy with minimal conditions on the relaxation function, from which the exponential and polynomial stability are just particular cases. This new result improves substantially many earlier results in the literature.

Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with L2-kernels combined with a general stability result. Most importantly, we establish weak uniqueness using a duality argument on the Fourier–Laplace transform via a deterministic Riccati–Volterra integral equation. We illustrate the applicability of our results on Hawkes processes and a class of hyper-rough Volterra Heston models with a Hurst index H∈(−1/2,1/2].

A general stability result for the semilinear viscoelastic wave model under localized effects

Our main goal in the present work is to address an integro-differential model under localized viscoelastic and frictional effects arising in the Boltzmann theory of viscoelasticity. More precisely, we consider a general version in the history context of the pioneer localized viscoelastic problem approached by Cavalcanti and Oquendo (2003) in the null history scenario, and more recently by Cavalcanti et al. (2018) in the history framework. By means of a new observability inequality, we prove a general stability result to the model under a weaker assumption on the localized frictional damping and a slower condition on the decreasing memory kernel (of polynomial type) than the previously mentioned works. To achieve such stability results, we still work in a general setting by removing the assumption on complementary damping mechanisms and show, in some reasonable situations concerning the density coefficient, that the localized viscoelastic effect is enough to ensure the general stability (of polynomial type) to the problem.

General stability result for a viscoelastic plate equation with past history and general kernel

In this paper, we consider the following plate problem:utt−σΔutt+Δ2u−∫0+∞k(s)Δ2u(t−s)ds=0, and we show that the stability of this problem holds for a much larger class of kernels. More precisely, we consider the kernel k:[0,+∞)→(0,+∞) satisfiesk′(t)≤−ξ(t)Ψ(k(t)),t≥0, where ξ and Ψ are functions satisfying some specific properties. Under this very general assumption on the behavior of k at infinity, we establish a relation between the decay rate of the solution and the growth of k at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumption on the history data, usually made in the literature.

A general stability result for second order stochastic quasilinear evolution equations with memory

The goal of this paper is to discuss an initial boundary value problem for the stochastic quasilinear viscoelastic evolution equation with memory driven by additive noise. We prove the existence of global solution and asymptotic stability of the solution using some properties of the convex functions. Moreover, our result is established without imposing restrictive assumptions on the behavior of the relaxation function at infinity.

Stability analysis of partial differential variational inequalities in Banach spaces

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

Stability of geometric flows on the circle

In this paper, we prove a general stability result for higher-order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle (possibly not the one it started close to), and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance of the flows to which our result can be applied. We will also present general families of flows for which our stability result applies.

Stabilization for wave equation of variable coefficients with Balakrishnan–Taylor damping and source term

In this paper, we consider the wave equation of variable coefficients with Balakrishnan–Taylor damping, time-varying delay and source term. By establishing some Lyapunov functionals which are equivalent to the energy of the system and using the potential well method to deal with the source term, we prove a general stability result.

On superiority and weak stability of population states in evolutionary games

We revisit some concepts of superiority and weak stability of population states in evolutionary games with continuous strategy space. We prove a general stability result for replicator trajectories by introducing the concept of superiority with respect to a given closed set. Some important results in the literature regarding weak stability turn out to be special cases of our main result.

General Stability Result of Abstract Thermoelastic System with Infinite Memory

In this paper we deal with an abstract thermoelastic system with infinite memory. Under suitable conditions on the order of the coupling, the memory kernel function and the initial values, we give the well-posedness and the general decay rate of the abstract thermoelastic system by semigroup theory and perturbed energy functional technique. Our results allow a wider thermoelastic systems, and the usual exponential and polynomial decay rates are only special cases.

General decay of a von Karman plate equation with memory on the boundary

We investigate the influence of boundary dissipation on the decay property of the solutions for a von Karman plate equation with a memory condition on one part of the boundary. Dropping the condition u0=0 on one part of the boundary, we show a general stability result for the equation via setting modified energy functionals which are equivalent to the energy of the equation and using some properties of convex functions. This result allows a wider class of relaxation functions and improve earlier results of Mustafa and Abusharkh (2015) and Park and Park (2005).

General stability of memory-type thermoelastic Timoshenko beam acting on shear force

In this paper, we consider a linear thermoelastic Timoshenko system with memory effects where the thermoelastic coupling is acting on shear force under Neumann–Dirichlet–Dirichlet boundary conditions. The same system with fully Dirichlet boundary conditions was considered by Messaoudi and Fareh (Nonlinear Anal TMA 74(18):6895–6906, 2011, Acta Math Sci 33(1):23–40, 2013), but they obtained a general stability result which depends on the speeds of wave propagation. In our case, we obtained a general stability result irrespective of the wave speeds of the system.

Asymptotic stability of a viscoelastic problem with Balakrishnan–Taylor damping and time-varying delay

A viscoelastic problem with Balakrishnan–Taylordamping and time-varying delay of the form utt−(a+b‖∇u‖2+σ(∇u,∇ut))Δu+∫0tg(t−s)Δu(s)ds+μ1f1(ut(x,t))+μ2f2(ut(x,t−τ(t)))=0is considered. We prove a general stability result for the equation without the condition μ2>0 by establishing some Lyapunov functionals which are equivalent to the energy of the equation instead of multiplier technique and using some properties of convex functions.

Extended Rearrangement Inequalities and Applications to Some Quantitative Stability Results

In this paper, we prove a new functional inequality of Hardy-Littlewood type for generalized rearrangements of functions. We then show how this inequality provides {\em quantitative} stability results of steady states to evolution systems that essentially preserve the rearrangements and some suitable energy functional, under minimal regularity assumptions on the perturbations. In particular, this inequality yields a {\em quantitative} stability result of a large class of steady state solutions to the Vlasov-Poisson systems, and more precisely we derive a quantitative control of the $L^1$ norm of the perturbation by the relative Hamiltonian (the energy functional) and rearrangements. A general non linear stability result has been obtained in \cite{LMR} in the gravitational context, however the proof relied in a crucial way on compactness arguments which by construction provides no quantitative control of the perturbation. Our functional inequality is also applied to the context of 2D-Euler system and also provides quantitative stability results of a large class of steady-states to this system in a natural energy space.

On functions that are approximate fixed points almost everywhere and Ulam’s type stability

In this paper, we investigate functions that are approximate fixed points of some (possibly nonlinear) operators almost everywhere, with respect to some ideals of sets. We prove that (under suitable assumptions) there exist fixed points of the operators that are “near” those functions. The results are applied to obtain some general stability results of Ulam’s type almost everywhere; in particular, for the polynomial functional equation.