Controllability and stabilization of liquid vibration in a container during transportation

Deals with the modeling and the mathematical analysis of problems involving a rectangular container. The container is controlled via a longitudinal acceleration in order to move it from one location to another, and the key problem is the suppression of sloshing during transportation. Practical control problems involving this system have been studied, from a numerical and experimental point of view. For these aspects we refer to Terashima et al. (1993), where the mathematical analysis is not deep enough for the study of controllability or stabilization problems. Here we develop a suitable theoretical framework which is similar to the one used by Mottelet (2000), since the physical system is the same, but with different input and output operators. This framework allows us to show that approximate controllability in finite time does not hold. We also study the stability of the system when the elevation of the surface is measured at the right end of the container, and a static negative acceleration feedback is used. We show that strong stability holds (but with a non-uniform decay), although the perturbation caused by the feedback on the system operator is not dissipative in the natural topology.