We discuss stabilization around trajectories of the continuity equation with nonlo cal vector fields, where the control is localized, i.e., it acts on a fixed subset of the configuration space. We first show that the correct definition of stabilization is the following: given an initial error of order epsilon , measured in Wasserstein distance, one can improve the final error to an order epsilon (1+kappa) with kappa > 0. We then prove the main result: assuming that the trajectory crosses the subset of control action, stabilization can be achieved. The key problem lies in regularity issues: the reference trajectory needs to be absolutely continuous, while the initial state to be stabilized needs to be realized by a small Lipschitz perturbation or by being in a very small neighborhood of it.
TRAJECTORY STABILIZATION OF NONLOCAL CONTINUITY EQUATIONS BY LOCALIZED CONTROLS
Pogodaev N.
;
2024
Abstract
We discuss stabilization around trajectories of the continuity equation with nonlo cal vector fields, where the control is localized, i.e., it acts on a fixed subset of the configuration space. We first show that the correct definition of stabilization is the following: given an initial error of order epsilon , measured in Wasserstein distance, one can improve the final error to an order epsilon (1+kappa) with kappa > 0. We then prove the main result: assuming that the trajectory crosses the subset of control action, stabilization can be achieved. The key problem lies in regularity issues: the reference trajectory needs to be absolutely continuous, while the initial state to be stabilized needs to be realized by a small Lipschitz perturbation or by being in a very small neighborhood of it.
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