Sentence examples for established exponential from inspiring English sources

They established exponential stability.

In this case we previously established exponential stability of the beam-joint system.

They established exponential stability of solutions under the condition begin{aligned} g'(t)leq-cg(t), quadforall tgeq0, end{aligned} (1.5) for some (c>0).

An established exponential model that combined the soil erosion, topography, and average annual rainfall in the region to estimate SOC density with varying soil depth was modified by simulated rainfall experiments and 137Cs (Caesium-137) tracer soil erosion techniques.

Muñoz Rivera [3] considered equations for linear isotropic viscoelastic solids of integral type, and established exponential decay and polynomial decay in a bounded domain and in the whole space, respectively.

In this work, they investigated the following equation: u_{tt}-Delta u-mu_{1}Delta u_{t}- mu_{2}Delta u_{t} t-tau)=0, and proved the well-posedness under the assumption (|mu_{2} t-tau_{1}) and established exponential decay of energy under the assumption (|mu_{2}|

Theorem 1.2 establishes exponential decay.

We establish exponential stability of the algorithms using two-time-scale averaging theory.

In the section, we will establish exponential stability of system (1) by some integral inequalities.

We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise.

We establish exponential stability of nonlinear time-varying impulsive systems by employing Lyapunov functions with discontinuity at the impulse times.