Shape optimization of piezoelectric sensors dealing with spill-over instability

This paper deals with the recurrent problem of spill-over that induces instability when the control strategy is based on uncollocated transducers. We first introduce the concept of modal control and the need to limit controllability and observability to a finite set of modes in order to avoid residual coupling effects which is the source of spill-over. In the second part of the paper, we establish the typical form of controllability and observability gramians when the distributed system is discretized on its modal components. An adequate norm is also introduced to optimize these operators. The last part aims at showing the obvious advantage of optimizing the piezoelectric density of sensors in order to achieve efficient control. An example is treated and shows some typical results obtained by our method.