An all-at-once approach for the optimal control of the unsteady Burgers equation

Abstract

a b s t r a c tWe apply an all-at-once method for the optimal control of the unsteady Burgers equation. The nonlinear Burgers equation is discretized in time using the semi-implicit discretization and the resulting first order optimality conditions are solved iteratively by Newton's method. The discretize then optimize approach is used, because it leads to a symmetric indefinite saddle point problem. Numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all-at-once approach with semi-implicit time discretization.

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

“…where 2 ( , ) = ( + 1) 1 ( ) + 1, ∀( , ) ∈  2 . Imposing Equations (9), (11), and (12) transforms the integro-differential Equation (8) into the following linear integral equation:…”

“…10 The implicit Euler and Crank-Nicholson methods were considered for solving the adjoint equations arising from the OC of Burgers' equation in the work of Noack and Walther. 9 In the work of Yılmaz and Karasözen, 11 the all-at-once method was used for the OC of the unsteady Burgers' equation, where the linearized optimality system at each time step leads to a symmetric indefinite saddle point problem. Model predictive control or receding horizon control strategies were also frequently applied for solving OCPs of unsteady Burgers' equation 12,13 ; they consist of suboptimal control problems of the infinite horizon problem by solving a series of finite horizon problems over increasing time intervals.…”

Preconditioning optimal control of the unsteady Burgers equations withH1regularized term

“…m > 0 is the viscosity and b > 0 is the regularization parameter. For a given function f 2 L 2 ðQ Þ, an initial condition y 0 2 H 1 0 ðXÞ and a control u 2 L 2 ðQ Þ, the existence and uniqueness conditions for the weak solution of the unsteady Burgers equations have already been studied in the literature, see [22] for more details.…”

“…In recent years, the development of the numerical methods for optimal control by PDEs has many contributions, see [1,2,14,11,12,23,24] and the references therein. It has been pointed out that the nonlinear Burgers equations can be discretized in time using the semi-implicit discretization and the resulting first-order optimality conditions are solved iteratively by Newton's method [22]. All-at-once method can be applied mostly to optimal control problems governed by PDEs and with coupled nonlinear diffusion-reaction equations [20].…”

“…The control conditions are given by the viscous Burgers equations with homogeneous Dirichlet boundary conditions which can be stated as follows ( [22]):…”