High Order Gegenbauer Integral Discontinous Gelerkin Methods: Advances in Computational Non-Linear PDE Based Optimal Control Theory

Optimal control problems arise in a wide variety of scientific areas and applications, and their numerical solutions are highly indispensable in many cases. We introduce some novel numerical strategies based on the highly successful discontinuous Galerkin (DG) methods to furnish stable and fully exponentially convergent numerical approximations of minimizers for optimal control problems governed by scalar PDEs in one space dimension. The proposed methods combine the superior advantages possessed by the family of DG methods with the wellconditioning of integral operators furnished through the use of the integral weak form of the dynamical system equations, the spectral accuracy provided by the latest technology of optimal Gegenbauer barycentric quadratures, and the useful linearization strategy provided by the powerful Cole-Hopf transformation in a fashion that allows us to take advantage of the strengths of these four methodologies. We focus on the 1-d viscous Burgers equation as an important case study occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow, heat conduction, elasticity, some probabilistic models, etc. To handle the formation of shocks and steep gradients in the limit, as the diffusion or viscosity tends to zero, we extend the proposed techniques within the framework of efficiently parallelizable adaptive integral DG spectral element methods. The current project investigations can be very useful when dealing with more complicated problems such as optimal control of the Navier-Stokes equations.