Optimal Observation of the One-dimensional Wave Equation

In this paper, we consider the homogeneous one-dimensional wave equation on [0,pi] with Dirichlet boundary conditions, and observe its solutions on a subset omega of [0,pi]. Let La(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets omega of [0,pi] of Lebesgue measure L pi. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L not equal 1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following H,brard and Henrot (Syst. Control Lett., 48:199-209, 2003; SIAM J. Control Optim., 44:349-366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.