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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”.

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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