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Classification of generic singularities for the planar time-optimal synthesis. (English) Zbl 0865.49022
This 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.

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
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