State Of Partial Differential Equation Research Articles

3D Nonparametric Neural Identification

This paper presents the state identification study of 3D partial differential equations (PDEs) using the differential neural networks (DNNs) approximation. There are so many physical situations in applied mathematics and engineering that can be described by PDEs; these models possess the disadvantage of having many sources of uncertainties around their mathematical representation. Moreover, to find the exact solutions of those uncertain PDEs is not a trivial task especially if the PDE is described in two or more dimensions. Given the continuous nature and the temporal evolution of these systems, differential neural networks are an attractive option as nonparametric identifiers capable of estimating a 3D distributed model. The adaptive laws for weights ensure the “practical stability” of the DNN trajectories to the parabolic three-dimensional (3D) PDE states. To verify the qualitative behavior of the suggested methodology, here a nonparametric modeling problem for a distributed parameter plant is analyzed.

Model predictive control of Kuramoto–Sivashinsky equation with state and input constraints

This work focuses on the model predictive control design methodology that successfully accounts for the state and input constraints applied in the context of highly dissipative Kuramoto–Sivashinsky (KS) partial differential equation (PDE) describing stability of a thin film thickness in the two-phase annular flow in vertical pipes. The evolution of a linear dissipative KSE PDE state is given by an abstract evolution equation in an appropriate functional space. The proposed constrained predictive control law utilizes a low order modal representation in the optimization functional, while higher modes are included only in the PDE state constraints. Simulation results demonstrate a successful application of the proposed predictive control technique that achieves optimal stabilization of a spatially-uniform unstable steady state of Kuramoto–Sivashinsky equation in the presence of input and state constraints.