Symmetrization for Fractional Elliptic and Parabolic Equations and an Isoperimetric Application(Dedicated to Haim Brezis,great master of analysis,on his 70th birthday)

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.

Nash and Stackelberg Differential Games

A large class of stochastic differential games for several players is considered in this paper.The class includes Nash differential games as well as Stackelberg differential games.A mix is possible.The existence of feedback strategies under general conditions is proved.The limitations concern the functionals in which the state and the controls appear separately.This is also true for the state equations.The controls appear in a quadratic form for the payoff and linearly in the state equation.The most serious restriction is the dimension of the state equation,which cannot exceed 2.The reason comes from PDE(partial differential equations) techniques used in studying the system of Bellman equations obtained by Dynamic Programming arguments.In the authors' previous work in 2002,there is not such a restriction,but there are serious restrictions on the structure of the Hamiltonians,which are violated in the applications dealt with in this article.

VARIATIONAL DISCRETIZATION FOR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH CONTROL CONSTRAINTS

This paper studies variational discretization for the optimal control problem governed by parabolic equations with control constraints. First of all, the authors derive a priori error estimates where｜｜｜u - Uh｜｜｜L∞（J;L2（Ω）） = O（h2 ＋ k）. It is much better than a priori error estimates of standard finite element and backward Euler method where ｜｜｜u- Uh｜｜｜L∞（J;L2（Ω）） = O（h ＋ k）. Secondly, the authors obtain a posteriori error estimates of residual type. Finally, the authors present some numerical algorithms for the optimal control problem and do some numerical experiments to illustrate their theoretical results.

Bang-Bang Principle of Time Optimal Controls and Null Controllability of Fractional Order Parabolic Equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.

INFLUENCE OF OCEAN INTERNAL WAVE ON PROPAGATION OF UNDER-WATER RADIATION NOISE

The underwater acoustic field influenced by a selected ocean internal wavewas computed using the Parabolic Equation (PE) method and split-step difference algorithm in thispaper. Acoustic field is formed by sound source with different frequency covering the range ofradiation noise of ships and submarines. Owing to the adoption of complex variables, sparse matrix,Gaussian source and analysis on the grid size, numerical results are achieved smoothly. The resultsshow that internal wave''s influence on underwater sound can''t be neglected, especially for highersound frequency. So it'' s necessary to take internal wave into account in identifying radiationnoise of ships and submarines, namely for sound intensity, transmission loss and spectra shape.