Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas dissymétrique

This paper generalizes the first author's preceding works concerning admissible functions on certain Fano manifolds [A. Ben Abdesselem, Lower bound of admissible functions on sphere, Bull. Sci. Math. 126 (2002) 675-680 [2]; A. Ben Abdesselem, Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique, Bull. Sci. Math. 130 (2006) 341-353 [3]]. Here, we study a larger class of functions which can be less symmetric than the ones studied before. When the sup of these functions is null, we prove that they admit a lower bound, giving precisely Tian invariant [G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C₁(M)>0, Invent. Math. 89 (1987) 225-246 [7]] (see also [T. Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés Kählériennes à la démonstration d'une inégalité, J. Funct. Anal. 57 (1984) 143-153 [1]]) on these manifolds.     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016     

A projective synchronization scheme based on fuzzy adaptive control for unknown multivariable chaotic systems

In this paper, a projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive controller is investigated for a projective synchronization of uncertain multivariable chaotic systems. The adaptive fuzzy-logic systems are used to approximate the unknown functions. A decomposition property of the control gain matrix is used in the controller design and the stability analysis. A Lyapunov approach is employed to derive the parameter adaptation laws and prove the boundedness of all signals of the closed-loop system as well as the exponential convergence of the synchronization errors to an adjustable region. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

A two variable refined plate theory for the bending analysis of functionally graded plates

Bending analysis of functionally graded plates using the two variable refined plate theory is presented in this paper.The number of unknown functions involved is reduced to merely four,as against five in other shear deformation theories. The variationally consistent theory presented here has, in many respects,strong similarity to the classical plate theory. It does not require shear correction factors,and gives rise to such transverse shear stress variation that the transverse shear stresses vary parabolically across the thickness and satisfy shear stress free surface conditions.Material properties of the plate are assumed to be graded in the thickness direction with their distributions following a simple power-law in terms of the volume fractions of the constituents.Governing equations are derived from the principle of virtual displacements, and a closed-form solution is found for a simply supported rectangular plate subjected to sinusoidal loading by using the Navier method.Numerical results obtained by the present theory are compared with available solutions,from which it can be concluded that the proposed theory is accurate and simple in analyzing the static bending behavior of functionally graded plates.     

Adaptive fuzzy control‐based projective synchronization of uncertain nonaffine chaotic systems

This article aims to introduce a projective synchronization approach based on adaptive fuzzy control for a class of perturbed uncertain multivariable nonaffine chaotic systems. The fuzzy‐logic systems are employed to approximate online the uncertain functions. A Lyapunov approach is used to design the parameter adaptation laws and to demonstrate the boundedness of all signals of the closed‐loop system as well as the convergence of the synchronization errors to bounded residual sets. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of the proposed synchronization system based on fuzzy adaptive controller. © 2014 Wiley Periodicals, Inc. Complexity 21: 180–192, 2015     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique

We prove that admissible functions for Fubini-Study metric on the complex projective space PmC of complex dimension m, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of PmC. A similar lower bound holds on some projective manifolds. This gives an optimal constant in a Hörmander type inequality on these manifolds, which allows us, using Tian's invariant, to establish the existence of Einstein-Kähler metrics on them.     

Complex Finsler structures on tensor products

We give a complex Finsler structure T on E₁ ⊗ E₂ where E₁ is a negative complex vector bundle, and where E ₂ ^* is generated by global sections. The construction of T uses the Finsler structure on E₁, obtained by Kobayashi in [4]. We prove that this structure is strongly pseudoconvex and we show that it has negative curvature.     

Estimation of Ricci tensor on certain Fano manifolds

We give time existence to some Monge-Ampère equations on certain Fano manifolds which cannot carry Einstein-K?hler metrics. These solutions allow us to obtain an estimation of Ricci tensors on our manifolds.

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Résumé. On met en évidence un intervalle de temps pour lequel des équations de Monge-Ampère sur certaines variétés de Fano, qui ne possèdent pas de métriques d'Einstein-K?hler, admettent des solutions. Ces solutions permettent d'obtenir des métriques dont on sait minorer le tenseur de Ricci.

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Received June 22, 1998; in final form January 25, 1999