The goal of this article is to study the stability and the convergence of cell-centered finite volumes (FV) in a domain $\Omega= (0,1)\times(0,1)\subset R^2$ with non-uniform rectangular control volumes. The discrete FV derivatives are obtained using the Taylor Series Expansion Scheme (TSES), (see [4] and [10]), which is valid for any quadrilateral mesh. Instead of using compactness arguments, the convergence of the FV method is obtained by comparing the FV method to the associated finite differences (FD) scheme. As an application, using the FV discretizations, convergence results are proved for elliptic equations with Dirichlet boundary condition.

In this work, we study priori error estimates for the numerical approximation of an optimal control problem governed by the heat equation with certain control constraint and ending point state constraint. By making use of the classical space-time discretization scheme, namely, finite element method with the space variable and backward Euler discretization for the time variable, we first project the original optimal control problem into a semi-discrete control and state constrained optimal control problem governed by an ordinary differential equation, and then project the aforementioned semi-discrete problem into a fully discrete optimization problem with constraints. With the help of Pontryagin's maximum principle, we obtain, under a certain reasonable condition of Slater style, not only an error estimate between the optimal controls for the original problem and the semi-discrete problem, but also an error estimate between the solutions of the semi-discrete problem and the fully discrete problem, which leads to an error estimate between the solutions of the original problem and the fully discrete problem. By making use of the aforementioned result, we also establish an numerical approximation for the exactly null controllability of the internally controlled heat equation.

The modelling and controlling for complex dynamic systems which are too complicated to establish conventionally mathematical mechanism models require new methodology that can utilize the existing knowledge, human experience and historical data. Fuzzy cognitive maps (FCMs) are a very convenient, simple, and powerful tool for simulation and analysis of dynamic systems. Since human experts are subjective and can handle only relatively simple FCMs, there is an urgent need to develop methods for automated generation of FCM models using historical data. In this paper, a novel FCM, which is automatically generated from data and can be applied to on-line control, is developed by improving its constitution, introducing Least Square methods and using Hebbian Learning techniques. As an illustrative example, the simulations results of truck backer-upper control problem quantifies the performance of the proposed constructions of FCM and emphasizes its effectiveness and advantageous characteristics of the learning techniques and control ability.

In this paper, a discontinuous Galerkin finite element method with interior penalties for convection-diffusion optimal control problem is studied. A semi-discrete time DG scheme for this problem is presented. We analyze the stability of this scheme, and derive a priori and a posteriori error estimates for both the state and the control approximation.

In this paper we study the finite difference approximation of a hemivariational inequality of parabolic type arising from temperature control problem. Stability and convergence of the proposed method are analyzed. Numerical results are also presented to show the effectiveness and usefulness of the discretization scheme.

The main goal of this work is the proposal of an efficient space-time adaptive procedure for a cGdG approximation of an unsteady diffusion problem. We derive a suitable a posteriori error estimator where the contribution of the spatial and of the temporal discretization is kept distinct. In particular our interest is addressed to phenomena characterized by temporal multiscale as well as strong spatial directionalities. On the one hand we devise a sound criterion to update the time step, able to follow the evolution of the problem under investigation. On the other hand we exploit an anisotropic triangular adapted grid. The reliability and the efficiency of the proposed error estimator are assessed numerically.

An initial boundary value problem for a two-dimensional parabolic equation in two disconnected rectangles is investigated. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data (up to a logarithmic factor of the mesh-size), is obtained.

The non-dimensional form of Navier-Stokes equations for two dimensional mixing layer flow are solved using direct numerical simulation. The governing equations are discretized in streamwise and cross stream direction using a sixth order compact finite difference scheme and a mapped compact finite difference method, respectively. A tangent mapping of $y =\beta\tan(\pi \zeta/2)$ is used to relate the physical domain of $y$ to the computational domain of $\zeta$. The third order Runge-Kutta method is used for the time-advancement purpose. The convective outflow boundary condition is employed to create a non-reflective type boundary condition at the outlet. An inviscid (Stuart flow) and a completely viscous solution of the Navier-Stokes equations are used for verification of the numerical simulation. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation. The results of mixing layer simulation also indicate that the time traces of the velocity components are periodic. Results in self-similar coordinate were also investigated which indicate that the time-averaged statistics for velocity, vorticity, turbulence intensities and Reynolds stress distribution tend to collapse on top of each other at the flow downstream locations.