In this paper we study the convergence of the Galerkin approximation method applied to the Generalized Hamilton-Jacobi-Bellman (GHJB) equation over a compact set containing the origin. The GHJB equation gives the cost of an arbitrary control law and can be used to improve the performance of this control. The GHJB equation can also be used to successively approximate the Hamilton-Jacobi-Bellman ...

Instead of writing the Galerkin approximation as seeking the stationary point of some functional over a finite-dimensional subspace of the original function space, we write it as the initial, infinite-dimensional variational problem with an explicit constraint that models the fact that we are actually searching in a subspace. By formulating this as a constrained variational problem using a Lagr...

In this paper we describe a novel approach for designing nonlinear H 1 control laws. These robust controllers are obtained by numerically approximating the solution to the Hamilton-Jacobi-Isaacs (HJI) equation and result in nonlinear control laws in feedback form. The approximation is accomplished via a two-step successive Galerkin approximation scheme. An application of the technique to the co...

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for the all proposed...

We investigate the strong approximation of stochastic parabolic partial differential equations with additive noise. We introduce post-processing in the context of a standard Galerkin approximation, although other spatial discretizations are possible. In time, we follow [20] and use an exponential integrator. We prove strong error estimates and discuss the best number of postprocessing terms to ...

The moving least-square technique is used to construct shape function in the Element Free Galerkin Method at present, but sometimes the algebra equations system obtained from the moving least-square approximation is ill-conditioned and the shape function needs large quantity of inverse operation. In this paper, the weighted orthogonal functions are used as basis ones, the application in the cal...

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove existence uniqueness of solution then derive optimality system. For numerical approximation, we use Galerkin spectral methods, which apply method discretization random variables employ spectral-Galerkin appr...

In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for ellipti...

A class of explicit discontinuous Galerkin schemes is described which time approximation is based on a predictor corrector formulation. The approximation at the new time level is obtained in one step with use of the information from the direct neighbors only. This allows to introduce a local time-stepping for unsteady simulations with the property that every grid cell runs with its own optimal ...

Abstract: In this paper a type of inverse problem belongs to the class of parabolic equations is considered. The approximation of the problem is based on the Ritz-Galerkin method. This approximation provides greater flexibility in which to impose initial and boundary conditions. The results of numerical experiments are compared with analytical solutions and perspicuous examples are included to ...