Value function for regional control problems via dynamic programming and Pontryagin maximum principle

In this paper we focus on regional deterministic optimal control problems, i.e., problems where the dynamics and the cost functional may be different in several regions of the state space and present discontinuities at their interface. Under the assumption that optimal trajectories have a locally finite number of switchings (no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem. In turn, we extend to regional optimal control problems the classical sensitivity relations and we prove that the regularity of this value function is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem

L1L^1-optimality conditions for circular restricted three-body problems

In this paper, the L1-minimization for the translational motion of a spacecraft in a circular restricted three-body problem (CRTBP) is considered. Necessary con- ditions are derived by using the Pontryagin Maximum Principle, revealing the existence of bang-bang and singular controls. Singular extremals are detailed, re- calling the existence of the Fuller phenomena according to the theories developed by Marchal in Ref. [14] and Zelikin et al. in Refs. [12, 13]. The sufficient opti- mality conditions for the L1-minimization problem with fixed endpoints have been solved in Ref. [22]. In this paper, through constructing a parameterised family of extremals, some second-order sufficient conditions are established not only for the case that the final point is fixed but also for the case that the final point lies on a smooth submanifold. In addition, the numerical implementation for the optimality conditions is presented. Finally, approximating the Earth-Moon-Spacecraft system as a CRTBP, an L1-minimization trajectory for the translational motion of a spacecraft is computed by employing a combination of a shooting method with a continuation method of Caillau et al. in Refs. [4, 5], and the local optimality of the computed trajectory is tested thanks to the second-order optimality conditions established in this paper

Regularization of chattering phenomena via bounded variation controls

Parallel sessionInternational audience1 Motivation what is chattering? how often does it occur? 2 Convergence results convergence for the perturbed problem rate of convergence and switching times 3 Open problems rate of convergence for the perturbed problem generic rate of convergenc

Regularization of chattering phenomena via bounded variation controls

International audienceIn the control theory, the term chattering is used to refer to fast oscillations of controls, such as an infinite number of switchings over a finite time interval. In this paper, we focus on three typical instances of chattering: the Fuller phenomenon, referring to situations where an optimal control features an accumulation of switchings in finite time; the Robbins phenomenon, concerning optimal control problems with state constraints, where the optimal trajectory touches the boundary of the constraint set an infinite number of times over a finite time interval; and the Zeno phenomenon, for hybrid systems, referring to a trajectory that depicts an infinite number of location switchings in finite time. From the practical point of view, when trying to compute an optimal trajectory, for instance, by means of a shooting method, chattering may be a serious obstacle to convergence. In this paper, we propose a general regularization procedure, by adding an appropriate penalization of the total variation. This produces a family of quasi-optimal controls whose associated cost converge to the optimal cost of the initial problem as the penalization tends to zero. Under additional assumptions, we also quantify quasi-optimality by determining a speed of convergence of the costs