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A generic classification of time-optimal planar stabilizing feedbacks. (English) Zbl 0910.93044
One considers the control system \[ \dot{x}=F(x)+G(x)u, \qquad | u| \leq 1. \tag{\(*\)} \] It is obvious by generic assumptions on \(F, C\) that for any time \(\tau\), there exists a set \(A_{\tau}\) of the states such that each state \(x_{0} \in \text{int }A_{\tau}\) may be brought to the origin in a time \(\tau\) that is minimal. One can define a feedback control \(u=\varphi (x)\) having the following property: Each trajectory induced by the closed-loop system \[ \dot{x}=F(x)+G(x)\varphi (x), \quad x_{0} \in \text{int } A_{\tau} \] reaches the origin in a minimum time. The minimum time problem has been solved in detail for the planar controlled object \((*)\). In the paper, the authors outline also an idea of how to generalise all the received results to the case of a general two-dimensional manifold.
Reviewer: W.Hejmo (Kraków)

MSC:
93C10 Nonlinear systems in control theory
49J15 Existence theories for optimal control problems involving ordinary differential equations
93B52 Feedback control
93B50 Synthesis problems
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