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In this paper, we are interested in finding the optimal location and shape of the actuators in a stabilization problem. Namely, we consider the one-dimensional wave equation damped by an internal feedback supported on a subdomain $\omega$ of given length. The criterion we want to optimize represents the rate of decay of the total energy of the system. It theoretically involves all the eigenmodes of the operator. From an engineering point of view, it seems more realistic to consider only a finite number of modes, say the N first ones. In that context, we are able to prove existence and uniqueness of an optimal domain $\omega_N^*$: it is the better possible location for the actuators. We characterize this optimal domain and we point out the following strange phenomenon (at least for small lengths): the optimal domain $\omega_N^*$ which is the better one for the N first modes is actually the worse one for the N + 1th mode. This looks like the well-known spillover phenomenon in control theory. At last, we will give some possible extension and open problems in higher dimension.

A Spillover Phenomenon in the Optimal Location of Actuators