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The structure of time-optimal trajectories for single-input systems in the plane: The general real analytic case. (English) Zbl 0664.93034
The paper studies time-optimal trajectories for systems of the form: $$x'=f(x)+u\cdot g(x)$$, $$u\in [-1,1]$$, in the case f($$\cdot)$$ and g($$\cdot)$$ are real-analytic vector fields on a two-dimensional real- analytic manifold, M.
Using the theory of subanalytic sets and some control-theoretic arguments, the behaviour of local time-optimal trajectories is very thoroughly analysed to prove, under fairly general hypotheses, that, locally, the time-optimal trajectories are finite concatenations of “bang-bang” and singular arcs with bounds on the number of switchings. It is also proved that for any time-optimal trajectory x($$\cdot): [a,b]\to M$$ there exists a time-optimal trajectory y($$\cdot): [a,b]\to M$$ such that: $$y(a)=x(a)$$, $$y(b)=x(b)$$ and y($$\cdot)$$ is a finite concatenation of X-, Y- and Z-trajectories where $$X=f-g$$, $$Y=f+g$$ and Z is the corresponding singular vector field.
Reviewer: S.Mirica

MSC:
 93C10 Nonlinear systems in control theory 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 57R27 Controllability of vector fields on $$C^\infty$$ and real-analytic manifolds 93C15 Control/observation systems governed by ordinary differential equations
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