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.
254ELGINDY AND KARASÖZEN many OC applications, and this crucial objective becomes considerably more challenging when the dynamics is described by nonlinear partial differential equations so that the need for developing novel, highly accurate, and efficient techniques is evident. 2 Since the past two decades, OC problems (OCPs) of Burgers' equation has become one of the active topics in applied mathematics occurring in fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow, heat conduction, elasticity, some probabilistic models, etc. Much interest have been developed toward the analysis of such problems for many reasons, among them: (i) to impose the OC on the many phenomena described by Burgers' equation such as the modeling of gas dynamics, traffic flow, wave processes in acoustics and hydrodynamics, etc. (ii) Such a problem is considered a first step toward developing methods for intricate flow control such as the Navier-Stokes equation that is hard to deal with. (iii) The need to analyze the inverse design of Burgers' equation in long time horizons. In particular, given a final desired target, the goal is to identify the initial datum that leads to it, along the Burgers' dynamics. This forms an ill-posed backward problem that requires a highly proper discretization scheme in the numerical approximation of the equation to obtain an accurate approximation of the OCP. 3 (iv) The problem of variational data assimilation for a nonlinear evolution model can be formulated as an OCP governed by Burgers' equation, thus providing a means to develop a diagnostics to check Gauss-verifiability of the optimal solution. 4 Moreover, OCPs of...
