A Computational Framework for the Numerical Solution of Optimal Control Problems Governed by Partial Differential Equations

A computational framework for the solution of optimal control problems with time-dependent partial differential equations (PDEs) is presented. The optimal control problem is transformed from a continuous time and space optimal control problem to a sparse nonlinear programming problem through state parameterization with Lagrange polynomials and discrete controls defined at Legendre-Gauss- Radau (LGR) points. The standard LGR collocation method is coupled with a modified Radau method to produce a collocation point on the typically noncollocated boundary. The newly collocated endpoint allows for a representation of the state derivative and control on the originally noncollocated boundary such that Neumann boundary conditions may be satisfied. Finally, the method developed in this paper is demonstrated on a viscous Burgers' tracking problem and the results are compared to an existing solution.