Second-order optimality conditions and other problems in geometric optimal control

This thesis is primarily devoted to second-order optimality conditions in geometric optimal control. After introducing the necessary background, we present a unified framework for second-order necessary and sufficient conditions, based on the second variation, the associated Jacobi curve, conjugate times, and the Maslov index. This preliminary part reorganizes and adapts results available in the literature, with the aim of providing a coherent set of tools for the subsequent applications. We then apply this framework to several classes of extremals and derive the corresponding optimality conditions. First, we study singular extremals for control-affine systems with an L 1 cost, establishing second-order necessary conditions and sufficient conditions for strong local optimality. Next, we consider piecewise-regular extremals obtained by concatenating regular arcs. This framework includes, as particular cases, regular extremals, that is extremals satisfying the strong Legendre condition, and bang-bang extremals. For this class, we characterize the second variation and the associated Jacobi curve, and formulate optimality criteria in terms of the Maslov index of the Jacobi curve. Finally, we partially extend the analysis to nice extremals, obtained by concatenating regular and singular arcs, and derive the corresponding optimality conditions. The last two chapters contain two further contributions in geometric optimal control, independent of the study of second-order optimality. The first concerns the precise computation of the asymptotic motion complexity of a nonadmissible curve for a control-affine system. The second provides an explicit formula for the Lyapunov exponents of a linear switched system alternating between two matrices that generate the Lie algebra sl2(R).