Minimum-time strong optimality of a singular arc: the multi-input non involutive case

We consider the minimum-time problem for a multi-input control-affine system, where we assume that the controlled vector fields generate a non-involutive distribution of constant dimension, and where we do not assume a-priori bounds for the controls. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated to a Pontryagin singular arc is sufficient to prove its strong-local optimality. We provide an application of the result to a generalization of Dubins problem

Sufficient optimality conditions for a bang--singular extremal in the minimum time problem

The paper gives second order sufficient conditions for the strong local optimality of a bang-singular extremal in a minimum time problem. The conditions are given in terms of regularity assumptions on the extremal and of the coercivity of the extended second variation associated to the minimum time problem with fixed end-points on the singular arc. The conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where necessary conditions have mild inequalities

Constrained bang-bang-singular extremals

International audienceBy means of Hamiltonian methods we give sufficient conditions for the strong local optimality of a Pontryagin extremal for a Mayer problem where both the end points of admissible trajectories are constrained to smooth manifolds of the state space. The extremal is given by the concatenation of two bang arcs and a partially singular one. Our sufficient conditions amount to regularity conditions on the extremal and the coercivity of a suitable quadratic form

Strong local optimality for a bang-bang-singular extremal: the fixed-free case

International audienceIn this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a partially or totally singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on R n and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of a LQ accessory problem. An example is proposed. All the results are coordinate free so they also hold on a manifold

Minima in control problems with constraints

This paper is devoted to describing second order conditions in the framework of extremal problems, that is, conditions obtained by reducing the optimal control problem to an abstract one in a suitable Banach (or Hilbert) space. The studied problem includes equality constraints both on the end-points and on the state-control trajectory. The second goal is to give a complete description of necessary and sufficient second order conditions for weak local optimality by describing first the associated linear-quadratic problem and then by giving a conjugate point theory for this linear quadratic problem with constraints

Lectures given at the C.I.M.E. Summer School

The lectures gathered in this volume present some of the different aspects of Mathematical Control Theory. Adopting the point of view of Geometric Control Theory and of Nonlinear Control Theory, the lectures focus on some aspects of the Optimization and Control of nonlinear, not necessarily smooth, dynamical systems. Specifically, three of the five lectures discuss respectively: logic-based switching control, sliding mode control and the input to the state stability paradigm for the control and stability of nonlinear systems. The remaining two lectures are devoted to Optimal Control: one investigates the connections between Optimal Control Theory, Dynamical Systems and Differential Geometry, while the second presents a very general version, in a non-smooth context, of the Pontryagin Maximum Principle. The arguments of the whole volume are self-contained and are directed to everyone working in Control Theory. They offer a sound presentation of the methods employed in the control and optimization of nonlinear dynamical systems

Bang bang trajectories with a double switching time in the minimum time problem

Parallel sessionInternational audienceWe consider the minimum time problem with xed end-points on a finite dimensional manifold M in the case when the dynamics is a ne with respect to the control and the control set is a box in R^m. We are interested in giving sufficient conditions for strong local optimality. Our sufficient conditions are given using Hamiltonian methods, but here the presence of a double switch makes the proof of the local invertibility of the maximized Hamiltonian flow much more tricky