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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.

##### 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
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