This article analyzes infinitesimal characterizations of commutativity of locally Lipschitz continuous vector fields that depend on a time-varying control parameter. Such a family of vector fields is said to commute if for every choice of control functions the flows of the corresponding time-varying vector fields commute. The article presents and proves a characterization of such commutativity in terms of the vanishing of a set-valued bracket that extends the classical Lie brackets to this nonsmooth case. A second characterization is provided in terms of an invariance condition for lifts of multi-time paths. The results are applied and discussed in terms of commutativity of families of optimal control problems that involve concatenations of different dynamical constraints and different cost functionals.
Commutativity of Control Vector Fields and inf-commutativity
RAMPAZZO, FRANCO
2010
Abstract
This article analyzes infinitesimal characterizations of commutativity of locally Lipschitz continuous vector fields that depend on a time-varying control parameter. Such a family of vector fields is said to commute if for every choice of control functions the flows of the corresponding time-varying vector fields commute. The article presents and proves a characterization of such commutativity in terms of the vanishing of a set-valued bracket that extends the classical Lie brackets to this nonsmooth case. A second characterization is provided in terms of an invariance condition for lifts of multi-time paths. The results are applied and discussed in terms of commutativity of families of optimal control problems that involve concatenations of different dynamical constraints and different cost functionals.Pubblicazioni consigliate
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