| Abstract: |
| We consider an optimal control problem where the state has to
approach asymptotically a closed target, while paying an integral cost with a nonnegative Lagrangian. We generalize the partial diferential inequality that usually defines a Control Lyapunov Function by introducing a new, weaker differential inequality, which involves both the Lagrangian and higher order dynamics` directions expressed in form of iterated Lie brackets up to a certain degree k, possibly greater than 1. In particular, we show that the existence of a solution U of the resulting extended relation allows us to construct explicitly a Lie-bracket-based feedback which sample stabilizes the system to the target and, at the same time, provides a bound for the cost, given by a U-dependent function. This means that, with a sample and hold technique, we construct trajectories which take into consideration not only the dynamics` vector fields but also their iterated Lie brackets. This is achieved by prescribing, for every small interval of a time partition, a state-dependent control strategy that makes the corresponding trajectory approximate the direction of an iterated Lie bracket. Furthermore, while the state approaches the target, the feedback regulates the cost as well, i.e. it guarantees a bound on the cost. We call such a solution U a degree-k Minimum Restraint Function. |
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