| This work focuses on solving optimal control problems numerically by the method of adjoints. Two theorems illustrating the use of the adjoint equations for computing gradient values are presented for two different control problems. This first is control via initial conditions and the second problem is for a distributed control problem. These theorems give insight into why this approach works for minimizing cost functionals directly. The theory consistent approximations to control problems developed by Hager is then expanded to include Runge-Kutta consistent quadratures. It is shown that when using RK-consistent quadratures that the augmented order conditions of Hager also hold which results in a fully consistent discrete control problem. It is then shown that the SDIRK3 method of Hairer Wanner is appropriate for optimal control and it allows control problems with stiff differential constraints to be solved. Afterwards, the costate equations are derived for several extensions to the Runge-Kutta method but they are not susceptible to Hager's expanded consistency analysis. Finally, these methods are used to solve an atmospheric chemistry data assimilation inverse problem. |
