Section: New Results
Trajectory optimization
Second-order Conditions for State Constrained Optimal Control
Participants : F. Bonnans, A. Hermant.
We have studied state-constrained optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a theory of second-order conditions without gap, in which the difference between second-order necessary or sufficient conditions is only a change of an inequality into a strict inequality. This allows us to characterize the second-order quadratic growth condition, using the second-order information.
Analysis of the shooting algorithm
Participants : F. Bonnans, A. Hermant.
We have studied the shooting
algorithm for optimal control problems
with a scalar control and a regular
scalar state constraint.
Additional conditions are displayed,
under which the so-called
alternative formulation is equivalent to
Pontryagin's minimum principle.
The shooting algorithm appears to be well-posed (invertible Jacobian),
iff (i) the no-gap second order sufficient optimality condition holds,
and (ii) when the constraint is of order q
3 ,
there is no boundary arc.
Stability and sensitivity results without strict complementarity at touch points
are derived using Robinson's strong regularity theory,
under a minimal second-order sufficient condition.
The directional derivatives of the control and state
are obtained as solutions of a linear quadratic problem.
The result is published in [23] .
Structural stability of Pontryaguine extremals for first-order state constraints
Participants : F. Bonnans, A. Hermant.
Assuming well posedness of a first-order state constraint and weak second-order optimality conditions (equivalent to uniform quadratic growth) we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. This is the first result of this type. It follows that the shooting algorithm (properly adapted to the possible structural transformations) is well-posed in this case again.
The result is published in [23] .
Multidimensional singular arcs
Participants : F. Bonnans, P. Martinon, E. Trélat (Univ. Orléans), J. Laurent-Varin (Direction des lanceurs, CNES Evry).
We just started in November 2006 a study of the multidimensional singular arc that can occur in the atmospheric flight of a launcher. The physical reason for not having a bang-bang control (despite the fact that the hamiltonian function is affine w.r.t. the control, is that aerodynamic forces may make a high speed ineffective. Our preliminary results suggest that we have an effective means for computing such extremals.
