Optimal observation of the one-dimensional wave equation

In this paper, we consider the homogeneous one-dimensional wave equation on [0, π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0, π]. Let L ∈ (0, 1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0, π] of Lebesgue measure Lπ. 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 6= 1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following [8, 9], 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.