Classification of generic singularities for the planar time-optimal synthesis.

*(English)*Zbl 0865.49022This paper deals with time-optimal control of single-input, affine control systems on the plane:
\[
dx/dt=f(x)+g(x)u,\quad x\in\mathbb{R}^2,\quad|u|\leq 1,\;f,g\in C^\infty.\tag{\(*\)}
\]
The time-optimal synthesis for the system \((*)\) is constructed algorithmically, leading to a partition of the reachable set of the system into regular submanifolds where the control amounts to \(\pm1\) or becomes singular, whose boundaries consist of so-called frame curves and frame points. At the frame curves and points the minimum time function associated with the time optimal problem loses its smoothness.

The paper presents a complete, local topological classification of the frame curves and frame points for generic reachable systems \((*)\). In particular, the following frame curves have been characterized:

\(\bullet\) a trajectory of \((*)\) corresponding to \(u=-1\),

\(\bullet\) a trajectory of \((*)\) corresponding to \(u=+1\),

\(\bullet\) the topological frontier of the reachable set,

\(\bullet\) a switching curve,

\(\bullet\) a turnpike,

\(\bullet\) an overlap curve.

Simple models of these frame curves are provided and an equivalence theorem is stated. Next, structurally stable frame points (like an intersection of a pair of frame curves) are studied, 22 types of these points have been found and the equivalence to simple models is established.

The paper presents a complete, local topological classification of the frame curves and frame points for generic reachable systems \((*)\). In particular, the following frame curves have been characterized:

\(\bullet\) a trajectory of \((*)\) corresponding to \(u=-1\),

\(\bullet\) a trajectory of \((*)\) corresponding to \(u=+1\),

\(\bullet\) the topological frontier of the reachable set,

\(\bullet\) a switching curve,

\(\bullet\) a turnpike,

\(\bullet\) an overlap curve.

Simple models of these frame curves are provided and an equivalence theorem is stated. Next, structurally stable frame points (like an intersection of a pair of frame curves) are studied, 22 types of these points have been found and the equivalence to simple models is established.

Reviewer: K.Tchoń (Wrocław)

##### MSC:

49K30 | Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |

49K15 | Optimality conditions for problems involving ordinary differential equations |

93B50 | Synthesis problems |

93C10 | Nonlinear systems in control theory |