Some Remark on Optimal Stochastic Control with Partial Information

Abstract

We are interested in the control problem of a partially observable diffusion process, which is initialized at a fixed point of ? ⁿ , and we want to characterize the associated value function. To resort to the theory of viscosity solutions depends on the possibility to translate such a problem on Hilbert spaces like L ²(? ⁿ ), and so it can not be used here. Nevertheless, a result of N. Bouleau and F. Hirsch allows us to introduce a broadened problem which fulfills the condition. The fact remains to link these two control problems.     

Stability of Taylor–Couette and Dean flow: A semi-analytical study

An alternative method is presented for solving the eigenvalue problem that governs the stability of Taylor–Couette and Dean flow. The eigenvalue problems defined by the two-point boundary value problems are converted into initial value problems by applying unit disturbance method developed by Harris and Reid [27] in 1964. Thereafter, the initial value problems are solved by differential transform method in series and the eigenvalues are computed by shooting technique. Critical wave number and Taylor number for Taylor–Couette flow are computed for a wide range of rotation ratio (μ), −4 ? μ ? 1 (first mode) and −2 ? μ ? 1 (second mode). The radial eigenfunction and cell patterns are presented for μ = −1, 0, 1. Also, we have computed critical wave number and Dean number successfully.     

Analysis of a conducting fluid in a thin annulus with rotating insulated walls under radial magnetic effect

This work considers an electrically conducting fluid filled between two concentric cylindrical walls relatively close to each other. A theoretical solution for the steady Taylor–Couette flow between these two electrically insulated rotating cylinders under the influence of a radial magnetic field is provided in this work. By solving the appropriate set of governing equations simultaneously, the profiles of fluid tangential velocity component and induced magnetic field were obtained as complicated functions involving the modified Bessel functions of the first and second kinds of the first-order in terms of radial coordinates and Hartmann number. A computational study was also performed to validate the present theoretical solution. The analytical and computational results are identical when Ha = 1 while these results only slightly deviate from each other as Ha increases. Current results show that, the presence of the external magnetic field causes the flow close to the slower cylinder to accelerate while that close to the faster cylinder to decelerate. This has clearly implied the fact that an external magnetic field tends to make the velocity distribution across the inner and outer cylinders more uniform.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

Traffic flow modelling with junctions

Motivated by the modelling of a roundabout, we are led to study the traffic on a road with points of entry and exit. In this note, we would like to describe the modellisation of a junction and solve the Riemann problem for such a model. More precisely, between each point of discontinuity we use a multi-class extension of the LWR model to describe the evolution of the density of the vehicles, the ‘multi-class’ approach being used in order to distinguish the vehicles after their origin and destination. Then, we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles. In the case of the one-T road we obtain a result of existence and uniqueness. This first step allows us to obtain a similar result for the n-T road. We describe these results and also some properties of the obtained solutions, in order to see how long this model is valid.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

Steady compressible Navier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R². We show existence if a solution that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of the constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Thorough analysis of the Oseen system in 2D exterior domains

We construct L_p‐estimates for the inhomogeneous Oseen system studied in a two‐dimensional exterior domain Ω with inhomogeneous slip boundary conditions. The kernel of the paper is a result for the half space ?. Analysis of this model system shows us a parabolic character of the studied problem, resulting as an appearance of the wake region behind the obstacle. Main tools are given by the Fourier analysis to obtain the maximal regularity estimates. The results imply the solvability for the Navier–Stokes system for small velocity at infinity. Copyright © 2009 John Wiley & Sons, Ltd.     

An optimal‐order estimate for MMOC‐MFEM approximations to porous medium flow

Mathematical models used to describe porous medium flow lead to coupled systems of time‐dependent partial differential equations. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid‐orientation effect. The MMOC‐MFEM time‐stepping procedure, in which the modified method of characteristics (MMOC) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this article we prove an optimal‐order error estimate for a family of MMOC‐MFEM approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009