A monotonic method for nonlinear optimal control problems with concave dependence on the state

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.

Keywords:

• monotonic algorithms
• nonlinear control
• optimal control
• non-convex optimisation

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

The operator A _x is then called the Fréchet differential (or Fréchet derivative) of f at x and is denoted by D _x f ≜ A _x . Let us also recall that given an open set Ω ⊂ ℝ^γ and a Hilbert space H ₁, the set L ^∞(Ω; H ₁) is the space of functions f from Ω with values in the Hilbert space H ₁ such that for almost all t ∈ Ω the norm is bounded by the same constant (the lowest of which is the L ^∞(Ω; H ₁) norm of f). One can likewise define L ²(Ω; H ₁):

When the derivatives of f are considered, the Sobolev spaces W ^1,∞ have to be introduced. We refer to Yosida (1995) and Adams and Fournier (2003) for further details.

2. For a space V we denote by V* its dual space.

3. For any operator M, we denote by M* its adjoint.