Elements of analytical solutions constructor in a class of time-optimal control problems with the break of curvature of a target set

P. D. Lebedev ; A. A. Uspenskii

Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 132, pp. 370-386

Résumé

A planar velocity control problem with a disc indicatrix and a target set with a smooth boundary having finite discontinuities of second-order derivatives of coordinate functions is considered. We have studied pseudo-vertices-special points of the goal boundary that generate a singularity for the optimal control function. For non-stationary pseudo-vertices with discontinuous curvature, one-way markers are found, the values of which are necessary for analytical and numerical construction of branches of a singular set. It is proved that the markers lie on the border of the spectrum-the region of possible values. One of them is equal to zero, the other takes an invalid value $-\infty.$ In their calculation, asymptotic expansions of a nonlinear equation expressing the transversality condition are applied. Exact formulas for the extreme points of branches of a singular set are also obtained based on markers. An example of a control problem is presented, in which the constructive elements are obtained using the developed methods (pseudo-vertex, its markers, and the extreme point of a singular set), are sufficient to construct a singular set and an optimal result function in an explicit analytical form over the entire area of consideration.

Détail

Keywords: velocity, optimal result function, singular set, transversality, Hamilton-Jacobi equation, bisector, minimax solution, diffeomorphism.