Adjoint concepts for the optimal control of Burgers equation

Abstract

Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradientbased optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.

Boundary Control for an Arterial System

“…This will be pursued elsewhere. In the sequel we turn to a pseudo spectral discretization of the optimal control problem [15], [19]. The discretization uses the same framework as the numerical scheme already discussed, using Legendre polynomials and Legendre Gauss nodes.…”

“…In fact, there seem to be no studies addressing the heart rate variability based on the detailed spatial description of the pressure and flow patterns in the aorta. More broadly, theory and applications of optimization and control in spatial networks have been extensively developed in literature, and several numerical approaches have been successfully applied to telecommunications, transportation or supply networks ( [5], [6], [15]). …”

Contemporary review of techniques for the solution of nonlinear Burgers equation

“…One type of stabilization method is Galerkin least squares (GLS) is given in [188][189][190]. The numerical treatment of the boundary control of Burgers equation was investigated in [191,192]. Besides, the well known viscosity independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied by Ohkitani and Dowker [193].…”

Necessary optimality conditions for optimal distributed and (Neumann) boundary control of Burgers equation

“…Vedantham in [18] develops a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the Burgers equation. Adjoint techniques are studied in [14] for the optimal control of the Burgers equation with Neumann boundary control. By the optimal control techniques, Leredde et al in [13] carry out the investigation for the Burgers equation and find the best parameters of the model which ensure the closest simulation to the observed values.…”