$L^1$-Minimization for Mechanical Systems

Second-order subdifferential. Extremal problems for operational inclusions

In the work, the property of the second-order subdifferential is studied and second-order optimality conditions are obtained for the minimization problem. We also obtained necessary and sufficient conditions for an extremum for the extremal problem of operator inclusions.

Strict abnormal extremals in nonholonomic and kinematic control systems

In optimal control problems, there exist different kinds of extremals, that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic systems as long as they are equivalent. With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be …

Time minimization problem on the group of motions of a plane with admissible control in a half-disk

We study a time minimization problem on the group of motions of a plane with admissible control in a half-disk. The considered control system describes a model of a car that can move forward on a plane and turn in place. Optimal trajectories of this system are used in image processing for the detection of salient lines. In particular, such trajectories are used in the analysis of medical images when searching for blood vessels in photos of the human retina. The problem is of interest in geometric control theory as a model example in which the set of admissible controls contains zero at the bou…

广义Kantorovich-Rubinstein范数的极值点与稀疏优化

Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms

本文刻画了广义Kantorovich-Rubinstein范数下水平集的极值点结构，推导了一阶最优性条件，提出带线性收敛保证的高效求解算法，数值实验验证了其收敛特性。

Extremal points of total generalized variation balls in 1D: characterization and applications

The total generalized variation (TGV) is a popular regularizer in inverse problems and imaging combining discontinuous solutions and higher order smoothing. In particular, empirical observations suggest that its order two version strongly favors piecewise affine functions. In the present manuscript, we formalize this statement for the one-dimensional TGV-functional by characterizing the extremal points of its sublevel sets with respect to a suitable quotient space topology. These results imply that 1D TGV-regularized linear inverse problems with finite dimensional observations admit piecewise …

$L^1$-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 pap…