A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to C0C^0 IP methods

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers reliable and efficient a posteriori error estimators. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posed ness of the problem. Subsequently, applications of $C^0$ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings. Finally, we also discuss the variational discontinuous discretization method (without discretizing the control) and its corresponding error estimates.Comment: 23 pages, 5 figures, 1 tabl

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

The fishery, biology and stock assessment of jew fish resources of India

Sciaenids are one of the major component of the demersal trawl The total catch of this resource during 1990-94 period was 1.50,142 t contributing 8.86% to the demersal catch of India. A number of species are found in different states of India. Of which biological and stock assessment studies were made on eleven important species. Crustaceans and fish appear to be the chief food in Juvenile and adult stage respectively. Most of the species have a protracted spawning season. Among all the species studied the largest asymptotic length was estimated for O.ruber from Tuticorin and the smallest for J. sina from Cochin. The highest Z of 7.59 was recorded for K. axillaris from Chennai and the lowest was for O. cuvierifrom Mumbai. The average exploitation rate (E) and the Lc/ Lao was 0.62 and 0.53 respectively. The present yield is 91.222 t and the MSY is 1.42,613 t for all the species taken together. The exploitation rate for almost all the stocks in the states appears to be more than the optimum leve

Prawn, fish and molluscan seed resources along the Kerala and Tamilnadu coasts

The study detailed about the occurrence and quantitative abundance of prawn, fish and molluscan seed resources, their spatial, seasonal and diurnal variations, abundance in relation to lunar periodicities, influence of environmental features and pollution on them and areas suitable for brackishwater culture along the Kerala and Tamilnadu coasts

Convergence of the boundary control for the wave equation in domains with holes of critical size

In this paper, we consider the homogenization of the exact controllability problem for the wave equation in periodically perforated domain with holes of critical size. We show that the boundary control converges to the boundary control of the homogenized system under the assumption that the perforations are uniformly away from the boundary

Homogenization of a hyperbolic equation with highly contrasting diffusivity coefficients

International audienc

Homogenization of a nonlinear degenerate parabolic differential equation

In this article, we study the homogenization of the nonlinear degenerate parabolic equation $$partial_t b({x /varepsilon},u_varepsilon) - mathop{ m div} a({x /varepsilon},{t /varepsilon}, u_varepsilon,abla u_varepsilon)=f(x,t),$$ with mixed boundary conditions(Neumann and Dirichlet) and obtain the limit equation as $varepsilon o 0$. We also prove corrector results to improve the weak convergence of $abla u_varepsilon$ to strong convergence

Darcy-type law associated to an optimal control problem

The aim of this paper is to study the asymptotic behaviour (homogenization) of an optimal control problem in a periodically perforated domain with Dirichlet condition on the boundary of the holes. The optimal control problem considered here is governed by the Stokes system. The holes are assumed to be of the same order as that of the period. The homogenized limit of the Stokes system as well as its adjoint system arising from the optimal control problem is obtained. The convergence of the optimal control and cost functional is obtained on some specific control sets