## Section: New Results

### Trajectory optimization - PMP approach

#### Interior-point approach for solving optimal control problems

Participants : F. Alvarez [ U. Chile ] , J. Bolte [ U. Paris 6 ] , F. Bonnans, F. Silva.

In this INRIA Report [30] We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If the control of the initial problem satisfies strict complementarity conditions for its Hamiltonian except for a finite number of times, the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem.

Our results provide several types of efficiency measures of the penalization
technique: error estimations of the control for L^{s} norms (s in [1, + ] ), error estimations
of the state and the adjoint state in Sobolev spaces W^{1, s} (s in [1, + ) ) and also error estimates for the value function.
For the L^{1} norm and the logarithmic penalty,
the optimal results are given. In this
case we indeed establish that the penalized control and the value function
errors are of order O(|log|) .

#### Interior-point approach for solving elliptic type optimal control problems

Participants : F. Bonnans, F. Silva.

In this INRIA Report [33] , we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and it is constrained to be nonnegative. The approach is to consider an associated family of penalized problems, parametrized by >0 , whose solutions define a central path converging to the solution of the original problem. Our aim is to characterize the solutions of the penalized problems by developing an asymptotic expansion around the solution of the original problem. This approach allows us to obtain some specific error bounds in various norms and for a general class of barrier functions. In this manner, we generalize previous results that were obtained in the ODE framework.

#### Application to hydropower models

Participants : S. Aronna, F. Bonnans, P. Lotito [ U. Tandil ] , A. Dmitruk [ Moscow State U. ] .

Nous avons obtenu des conditions nécessaires et suffisantes du second ordre pour une problème avec un arc singulier et une seule commande. Un article est en préparation.

#### Study of optimal trajectories with singular arcs for space launcher problems

Participants : P. Martinon, F. Bonnans, E. Trélat, J. Laurent-Varin [ Direction des lanceurs, CNES Evry ] .

In the frame of research contracts with the CNES, we have studied since 2006 trajectory optimization for the atmospheric climbing phase of space launchers. One major axis was to investigate the existence of optimal trajectories with non-maximal thrust arcs (i.e. singular arcs), both from the theoretical and numerical point of view. The physical reason behind this phenomenon is that aerodynamic forces may make high speed ineffective (namely the drag term, proportional to the speed squared). Our main axis is an indirect method (Pontryagin's Minimum Principle and shooting method) combined with a continuation approach. We studied in [25] the theoretical aspects on the generalized Goddard problem, and conducted the numerical experiments for a typical Ariane 5 mission to the geostationary transfer orbit ([25] ). We then moved on to the study of a prototype reusable launcher with wings, for which we considered a more complex aerodynamic model (lift force) as well as a mixed state-control constraint limiting the angle of attack. While our work seem to indicate that optimal trajectories involve a full thrust, the homotopic approach was able to deal with the constraint quite smoothly.

#### Analysis of Optimal Control Problems with State Constraint

Participants : F. Bonnans, A. Hermant.

We have in [20] improved the results and given shorter proofs for the analysis of state constrained optimal control problems presented by the authors in [21] , concerning second order optimality conditions and the well-posedness of the shooting algorithm. The hypothesis for the second order necessary conditions is weaker, and the main results are obtained without reduction to the normal form used in that reference, and without analysis of high order regularity results for the control. In addition, we provide some numerical illustration. The essential tool is the use of the “alternative optimality system”.

#### Application to technological changes

In the report [36] we study a two-stage control problem, in which model parameters ("technology") might be changed at some time. An optimal solution to utility maximisation for t his class of problems needs to thus contain information on the time, at which the change will take place (0, finite or never) as well as the optimal control strategies before and after the change. For the change, or switch, to occur the "new technology"d value function needs to dominate the "old technology" value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and study when the graphs can intersect and hence whether the switch can occur. Using this characterisation we analyse a technology switching problem.

#### Formulations as problems with occupational measures

We obtain in [26] a linear programming characterization for the minimum cost associated to finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizon. The reflection is given by the normal cone to a proximal retract set.

#### Reflected differential games

In the paper [27] we analyze the existence of a -possible discontinuous- value for a zero sum two-player reflected differential game under Isaacs' condition. We characterize the value function as the unique solution -in a suitable sense- to a variational inequality, namely the Hamilton Jacobi Isaacs differential inclusion.