## Section: New Results

### Optimal control with singular arcs

Participants : Pierre Martinon, Andrei Dmitruk [Moscow State University] , Pablo Lotito [U. Tandil, Argentina] , Soledad Aronna, Frédéric Bonnans.

These studies enter in the framework of the PhD thesis of S. Aronna, supervised by J.F. Bonnans and P. Lotito, that ended in December 2011.

In the paper [21] we deal with optimal control problems for systems affine in the control variable. We have nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we get a second order sufficient condition for the scalar control case. The results use in an essential way the Goh transformation.

In the report [22] , we design a shooting algorithm applied to optimal control problems for which all control variables enter linearly in the Hamiltonian. This shooting algorithm is non standard, in particular since there are more equations than unknowns, and extends some previous algorithms designed for specific structures. We start investigating the case having only initial-final state constraints and free control variable, and afterwards we deal with control bounds. The shooting algorithm is locally well-posed and quadratically convergent if the derivative of its associated shooting function is injective at the optimal solution. The main result of this paper is to provide a sufficient condition for this injectivity, that is very close to the second order necessary condition. We prove that this sufficient condition guarantees the stability of the optimal solution under small perturbations and the well-posedness of the shooting algorithm for the perturbed problem. We present numerical tests that validate our method.

In the report [20] we deal with optimal control problems for systems that are affine in one part of the control variables and nonlinear in the rest of the control variables. We have finitely many equality and inequality constraints on the initial and final states. First we obtain second order necessary and sufficient conditions for weak optimality. Afterwards, we propose a shooting algorithm, and we show that the sufficient condition above-mentioned is also sufficient for the injectivity of the shooting function at the solution.