The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control

A classification of phase portraits of optimal synthesis in the vicinity of universal singular manifolds is considered for systems of constant rank that are affine with respect to the control. Both the phase state and the control are assumed to be multidimensional. The classification is based on the order of singular extremals and on the involutiveness (or noninvolutiveness) of the velocity indicatrix. It is shown that the synthesis of optimal trajectories is a fibered space over the base $W$ formed by singular optimal trajectories; the fibers consist of nonsingular optimal trajectories. For a multidimensional control, the singular manifold $W$ is stratified. In the involutive case, the fibers are one-dimensional. In the noninvolutive case, the fibers are multidimensional and contain trajectories with switching at increasing frequency (chattering trajectories); the dimension of the fibers and the structure of the field of trajectories inside the fibers depend on the order of the singular extremals. Application of the theory developed to classical problems of the mechanics of controlled systems and to the evaluation of exact constants in Kolmogorov-type inequalities for derivatives is described.