This paper deals with the problem of stochastic stability for a class of neutral distributed parameter systems with Markovian jump. In this model, we only need to know the absolute maximum of the state transition probability on the principal diagonal line; other transition rates can be completely unknown. Based on calculating the weak infinitesimal generator and combining Poincare inequality and Green formula, a stochastic stability criterion is given in terms of a set of linear matrix inequalities (LMIs) by the Schur complement lemma. Because of the existence of the neutral term, we need to construct Lyapunov functionals showing more complexity to handle the cross terms involving the Laplace operator. Finally, a numerical example is provided to support the validity of the mathematical results.
Ethylene-Propylene Side by Side (ES ) fibre remains an important challenge in hydrophilicity. In this study, we obtained folded nonwoven fabrics (FNF) by the through air-bonding process. The fiber with the fold structure was prepared by regulating the treatment temperature, treatment time, and cooling rate. The results showed that the melting point of the shell layer should be controlled at 135°C. The fold structure was most obvious at 135°C, 2.5 min, and rapid cooling. Compared with the common sample, liquid absorption capacity, thickness, and the diffusion area of the FNF increased by 64%, 0.3 mm, and 41%, respectively, and the rewet decreased by 35%. The contact angle of FNF was 140°, which was 30°larger than that of common nonwovens. Except for the first time, the penetration time for the two other times was better than that of ordinary materials.
In this paper, we establish the existence of a solitary wave in a KdV-mKdV equation with dissipative perturbation by applying the geometric singular perturbation technique and Melnikov function. The distance of the stable manifold and unstable manifold is computed to show the existence of the homoclinic loop for the related ordinary differential equation systems on the slow manifold, which implies the existence of a solitary wave for the KdV-mKdV equation with dissipative perturbation.
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