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Distributed optimal control of viscous Burgers' equation via a high‐order, linearization, integral, nodal discontinuous Gegenbauer‐Galerkin method
Author(s) -
Elgindy Kareem T.,
Karasözen Bülent
Publication year - 2019
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2541
Subject(s) - mathematics , optimal control , discretization , linearization , burgers' equation , mathematical analysis , nonlinear system , mathematical optimization , control theory (sociology) , partial differential equation , computer science , physics , quantum mechanics , control (management) , artificial intelligence
Summary 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.