Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation

In this paper, we propose a modified nonlocal boundary value problem method for an homogeneous biharmonic equation in a rectangular domain. We show that the considered problem is ill-posed in the sense of Hadamard, i.e. the solution does not depend continuously on the given data. Convergence estimates for the regularized solution are obtained under a priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.     

Adaptive Fourier Series Neural Network PID Controller

In this work, an Adaptive Neural Networks PID controller structure, called Adaptive Fourier Series Neural Networks PID controller (AFSNNPID), is developed. The main objective is to obtain a simple controller for nonlinear systems that can be tuned online to reject perturbations effect and compensate the system parameters variation. Due to its simple architecture and very attractive proprieties, the Fourier Series Neural Network (FSNN) is used to online adjust the parameters of the PID controller. Furthermore, using the delta-rule algorithm, the adaptation dynamics of the FSNN is globally stable. The design procedure of the proposed controller and the stability analysis of the closed loop system using the small gain theorem are given. To assess the effectiveness of the proposed control scheme, the control of a 3-DOF robot arm manipulator is considered and a comparative study, using the adaptive neural network PID controller and the particle swarm optimization based PID controller, is carried out. The obtained results, through the experimental study, indicate that the AFSNNPID controller presents better control performance than the other controllers.