We consider a Euler-Bernoulli beam, clamped at one extremity and free at the other, to which are attached a piezoelectric actuator and a collocated sensor. We are interested in the stability properties of the system depending on the length/position of the actuator/sensor.

I will discuss some issues arising in the study of the exponential stability of the damped wave equation and its discretizations. I will briefly recall how the exponential decay rate can be measured for the continuous damped wave equation (which have been obtained in articles from Cox-Zuazua in 1d and later by Lebeau in higher dimension). Then, assuming that the damped wave equation is exponentially stable (that is, the energy is exponentially decaying), one can study the stabilization properties of the corresponding space/time/fully-discrete schemes. It has been shown recently that in general, these numerical approximations are NOT uniformly (with respect to the discretization parameters) exponentially stable. However, adding a suitable numerical viscosity operator in the discrete systems, we can correct this phenomenon and guarantee uniform stabilization properties. But, the uniform exponential decay rate of the energy of the discrete systems is not known, since it would require a (asymptotic) spectral theory for the discrete systems on non-selfadjoint operators. I will also present some numerical simulations to give some indications of the expected results.

In a joint paper with Denis Borisov, we introduced a planar waveguide with non-Hermitian parity-time-symmetric Robin boundary conditions. Using perturbation methods, we demonstrated the existence of real weakly-coupled eigenvalues outside the essential spectrum provided that the boundary coupling function is a small perturbation of homogeneous boundary conditions. An open problem is to show the existence of discrete spectra by some qualitatives methods, regardless of the strength of the perturbation.