Restricted second order information for the solution of optimal control problems using control vector parameterization

Abstract

Abstract A new scheme using a Truncated Newton algorithm with and exact Hessian-search direction vector product is presented for the solution of optimal control problems. The derivation of formulae for second order parametric sensitivity analysis of differential-algebraic equations is presented, following earlier published work [V.S. Vassiliadis, E. Balsa-Canto, J.R. Banga, Second order sensitivities of general dynamic systems with application to optimal control problems. Chem. Eng. Sci. 54 (17) (1999) 3851–3860]. An original result in this work is the derivation of Hessian matrix-vector product forms which are shown to have the same computational complexity as the evaluation of first order sensitivities. This result for optimal control Hessian-vector products using control vector parameterization is shown to be a very effective way to solve optimal control problems. It is also noted that this work introduces the use of suitable Truncated Newton solvers which can exploit the exact vector products in using conjugate gradient iterations to converge the Newton equations. Such a solver is the TN algorithm of Nash [(S.G. Nash-Newton type minimization via the Lanczos method. SIAM J. Num. Anal. 21, (1984) 770–778)]. Because no full Hessian update is necessary it is demonstrated that the resulting optimal control solver performs very well for a very large number of degrees of freedom, limited only by the necessity for many right-hand-side calculations in the first and second order sensitivity equations (the Hessian vector product). It is also demonstrated by several case studies that the scheme is capable of starting far from the solution and yet arrive there in almost invariant performance.