Abstract
Initially introduced in the framework of quantum control, the so-called monotonic algorithms have demonstrated very good numerical performance when dealing with bilinear optimal control problems. This article presents a unified formulation that can be applied to more general nonlinear settings compatible with the hypothesis detailed below. In this framework, we show that the well-posedness of the general algorithm is related to a nonlinear evolution equation. We prove the existence of the solution to the evolution equation and give important properties of the optimal control functional. Finally we show how the algorithm works for selected models from the literature. We also compare the algorithm with the gradient algorithm.
Acknowledgements
We thank the anonymous referees for their helpful comments. This work is partially supported by the French ANR programs OTARIE (grant ANR-07-BLAN-0235) C-QUID (grant ANR-06-BLAN-0052-04) and by a CNRS-NFS PICS grant. G. Turinici acknowledges partial support by INRIA Rocquencourt (MicMac and OMQP).
Notes
1. Recall that, given H 1 and H 2 two Banach spaces and U ⊂ H 1 an open subset of H 1, a function f : U → H 2 is said to be Fréchet differentiable at x ∈ U if there exists a continuous linear operator A x ∈ ℒ(H 1, H 2) such that
2. For a space V we denote by V* its dual space.
3. For any operator M, we denote by M* its adjoint.