The structure of optimal synthesis in a neighbourhood of singular manifolds for problems that are affine in control

The question of the classification of the phase portraits of optimal synthesis in a neighbourhood of a singular universal manifold is discussed for systems of constant rank that are affine in control. Both phase state and control are assumed to be many-dimensional. The classification is based on the order of the singular extremals and the property of involutiveness (or otherwise) of the velocity indicator. The synthesis of optimal trajectories is shown to be a space fibred over the base $W$ consisting of singular optimal trajectories; its fibres are non-singular optimal trajectories. If the control is many-dimensional, then $W$ is a stratified manifold. In the involutive case the fibres are one-dimensional. In the non-involutive case the fibres are many-dimensional and contain chattering trajectories; the dimension of the fibres and the structure of the field of trajectories in the fibres depend on the order of the singular extremals.