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Controllability along a trajectory: A variational approach. (English) Zbl 0797.49015
“The aim of this paper is to give high-order conditions for a point of a reference trajectory to be interior to the reachable set. This property is linked to the minimum-time problem by the fact that if $$\widehat x$$ is an optimal trajectory on $$[0,T]$$, then, for each $$t\in [0,T]$$, $$\widehat x(t)$$ belongs to the boundary of $$R(\xi_ 0,t)$$, i.e. the reachable set at time $$t$$ from the initial point $$\xi_ 0= \widehat x(0)$$ of the optimal trajectory. Therefore, each sufficient condition for $$\widehat x(t)$$ to belong to the interior of $$R(\xi_ 0,t)$$ yields a necessary condition for $$\widehat x$$ to be time optimal. The high-order conditions are of particular interest in the case of nonlinear systems for which the Pontryagin maximum principle may not be sufficient to single out a unique candidate and singular trajectories may appear.”
“For controllability along a possibly nonstationary trajectory, the idea is to use some known conditions based on the relations at a point in the Lie algebra associated with the system in order to construct high-order variations of the trajectory.”
“Our techniques can be used for both bounded and unbounded controls. Many examples are given”.

##### MSC:
 49K15 Optimality conditions for problems involving ordinary differential equations 93B05 Controllability 93C10 Nonlinear systems in control theory 93C99 Model systems in control theory
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