Distributed optimal control of viscous Burgers' equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method

Elgindy, Kareem T.
Karasözen, Bülent
We developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite-time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well-conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite-dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces "an auxiliary control function" that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points.


Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints
Yücel, Hamdullah (2013-08-30)
We study the numerical solution of control constrained optimal control problems governed by a system of convection diffusion equations with nonlinear reaction terms, arising from chemical processes. Control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches.
Optimal boundary control of the unsteady Burgers equation with simultaneous space-time discretization
Karasözen, Bülent (2014-07-01)
The optimality system for boundary controlled unsteady Burgers equation is transformed after linearization into a biharmonic equation in the space-time domain. It is then discretized in space and time simultaneously, so that standard finite element software can be easily implemented. Numerical experiments with and without control constraint problems confirm the applicability of this approach. Copyright (C) 2013 John Wiley & Sons, Ltd.
Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation
UZUNCA, MURAT; Karasözen, Bülent; Sariaydin-Filibelioglu, Ayse (2015-09-18)
We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with nondivergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.
Distributed Optimal Control of Diffusion Convection Reaction Equations Using Discontinuous Galerkin Methods
Yücel, Hamdullah; Karasözen, Bülent (2011-09-09)
We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the discretize-then-optimize (DO) and the optimize-then-discretize (OD) approach lead to the same discrete systems and in both approaches the observed L 2 convergence for states and controls...
Moving mesh discontinuous Galerkin methods for PDEs with traveling waves
UZUNCA, MURAT; Karasözen, Bülent; Kucukseyhan, T. (2017-01-01)
In this paper, a moving mesh discontinuous Galerkin (dG) method is developed for nonlinear partial differential equations (PDEs) with traveling wave solutions. The moving mesh strategy for one dimensional PDEs is based on the rezoning approach which decouples the solution of the PDE from the moving mesh equation. We show that the dG moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal, arc-length and curvature monitor functions. Numerical results reveal the effi...
Citation Formats
K. T. Elgindy and B. Karasözen, “Distributed optimal control of viscous Burgers’ equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method,” OPTIMAL CONTROL APPLICATIONS & METHODS, pp. 253–277, 2020, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/31516.