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On singular extremals in the time minimal control problem in $${\mathbb{R}}^ 3$$. (English) Zbl 0602.49027
The author studies the time minimal control problem of a single-input generic system in $$E^ 3$$ given by $$dx/dt=X(x)+uY(x)$$ where X and Y are analytic and the singular controls are defined by a feedback of the form $$u(x)=\Delta '(x)/\Delta (x).$$ The author classifies the local behaviour of singular trajectories near points x such that $$\Delta (x)=0$$ when X is quadratic and Y a constant. The analysis is applicable to the Euler equation of a rigid body control problem.
Reviewer: E.Chukwu

##### MSC:
 93B99 Controllability, observability, and system structure 49K15 Optimality conditions for problems involving ordinary differential equations 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 70E15 Free motion of a rigid body 70Q05 Control of mechanical systems 93C15 Control/observation systems governed by ordinary differential equations
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