## 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 q3 , 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.