Optimal Observation of the One-dimensional Wave Equation
By Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Journal of Fourier Analysis and Applications, 2013
63 citations recorded by the source corpus.
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≠1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following Hébrard 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.
Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/081568418a6c96ae7579d005dd30a753b2098753
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In Computational Theory and Mathematics
- Emmanuel Trélat
- Enrique Zuazua
- Yannick Privat
- Sylvain Ervedoza
- Xu Zhang
- Piermarco Cannarsa
- Genni Fragnelli
- Nazim Idrisoglu Mahmudov
In Control and Systems Engineering
- Emmanuel Trélat
- Enrique Zuazua
- Yannick Privat
- Sylvain Ervedoza
- Xu Zhang
- Piermarco Cannarsa
- Genni Fragnelli
- Nazim Idrisoglu Mahmudov
In Mathematical Physics
- Enrique Zuazua
- Emmanuel Trélat
- Yannick Privat
- Sylvain Ervedoza
- Xu Zhang
- Genni Fragnelli
- Piermarco Cannarsa
- Judith Vancostenoble
Citations in the corpus
- Sparse stabilization and optimal control of the Cucker-Smale model (2013, unknown): “Also in optimal control with partial differential equation constraints it became rather popular to use L1minimization to enforce sparsity of controls [9, 13, 14, 31, 49, 58, 65], for instance in the optimal placement of actuators or sensors (see also the recent works [53, 52, 51] where the sparsity ”
- Optimal location of controllers for the one-dimensional wave equation (2013, background): “They have inspired our own works such as the recent one [20] in which we investigate similar questions for the optimal observability of the wave equation (with practical issues for the optimal placement of sensors).”