Sentence examples for obtained exponential from inspiring English sources

The obtained exponential decay rate and guaranteed cost performance level critically depend on both the lengths and activation frequencies of the varying sampling periods, and it is revealed that nonuniformly sampled control systems can remain to be stable even in the appearance of a very large sampling period, provided that the activation frequency of the large sampling period is small enough.

On the other hand, Kim and Kim [10], Nooghabi and Azarnoosh [11], and Xing et al. [12] obtained exponential inequalities for negatively associated random variables.

The main advantages of the obtained exponential stability conditions include: (I) In [9, 14, 15, 18, 22], all of those results require that the time sequence {t k } satisfies inf k ∈ ℤ + { t k - t k - 1 } > τ δ, δ > 1.

Moreover, when u0 = 0 on Γ1, they obtained exponential or polynomial decay of solutions, even if the relaxation function g does not converge to 0 at ∞. Later, Messaoudi and Soufyane [16] generalized this result to the case of a system of Timoshenko type.

The obtained exponential growth fitting demonstrates clearly the good relation between Sc and the relative area under the TD-GC/MS chromatographic signal showed in Fig. 15 of the combined effect of the abundant organic compounds adsorbed on the rum filter.

Messaoudi et al. [24] considered the following wave equation with strong time delay: begin{aligned} u_{tt } -Delta u -mu_{1} Delta u_{t} - mu_{2} Delta u_{t} (t-tau) =0 end{aligned} (1.13) and proved the well-posedness under the condition (vert mu _{2} vert leq mu_{1}) and obtained exponential decay of energy under the condition (vert mu_{2} vert < mu_{1} ).

This allows an efficient analysis via the Wirtinger inequality, which is extended here to obtain exponential stability.

Then, the criteria for stability and stabilization of linear switched systems are obtained via a newly constructed switching strategy, which allows us to obtain exponential stability results.

The proposed hp-adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom, and results in significant savings in computational cost.

We state error analysis for the proposed methods, which show that these methods obtain exponential convergence order.

To obtain exponential stability of system (1), we first establish the following integral inequality with impulses and infinite delays.