Finite-Time Synchronization of Complex Multilinks Networks with Perturbations and Time-Varying Delay Based on Nonlinear Adaptive Controller

This paper utilizes nonlinear adaptive feedback controller to make the complex multilinks networks with perturbations and time-varying delays achieve the finite-time synchronization. By designing nonlinear controllers, we use suitable Lyapunov functions and sufficient conditions to guarantee the finite-time synchronization between the drive system and the response system in terms of adaptive control. Several novel and useful finite-time synchronization criteria are accurately derived based on linear matrix inequality, Kronecker product, inequality analytical technique, and finite-time stability theory. Finally, numerical examples are given to demonstrate the validity and the effectiveness of our theoretical results.

The Proofs of Product Inequalities in Vector Spaces

In this paper, we introduce the proofs of product inequalities:u v ≤ u + v , for all u, v ∈ [0, 2], and u + v ≤ u v , for allu, v ∈ [2, ∞). The first product inequality u v ≤ u + v holds forany two vectors in the interval [0, 1] in Holder’s space and also valid anytwo vectors in the interval [1, 2] in the Euclidean space. On the otherhand, the second product inequality u + v ≤ u v ∀u, v ∈ [2, ∞)only in Euclidean space. By applying the first product inequality to theL p spaces, we observed that if f : Ω → [0, 1], and g : Ω → R, thenf p g p ≤ f p + g p . Also, if f, g : Ω → R, then f p + g p ≤f p g p .

Piecewise straightening and Lipschitz simplicial volume

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above.

Note on a Difference-Product Inequality

L. J. Mordell has recently considered (1) the squared modulus of a complex difference-product, namelyunder the conditionsand also under the quite different conditionHe proves that under (2) the maximum of δ is nn, and is attained when and only when the zr are vertices of a regular n-gon on the unit circle.