Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control
By Lucie Baudouin, Otared Kavian, Jean-Pierre Puel. 2005
74 citations recorded by the source corpus.
Abstract We study the Schrodinger equation i ∂ t u + Δ u + V 0 u + V 1 u = 0 on R 3 × ( 0 , T ) , where V 0 ( x , t ) = | x - a ( t ) | - 1 , with a ∈ W 2 , 1 ( 0 , T ; R 3 ) , is a coulombian potential, singular at finite distance, and V 1 is an electric potential, possibly unbounded. The initial condition u 0 ∈ H 2 ( R 3 ) is such that ∫ R 3 ( 1 + | x | 2 ) 2 | u 0 ( x ) | 2 dx ∞ . The potential V 1 is also real valued and may depend on space and time variables. We prove that if V 1 is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/74e6f59713a13533c4bfc325e33f77a1a7a61ddc
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Controllability of the discrete-spectrum Schrödinger equation driven by an external field (2008, methodology): “1The last author was partially supported by a FABER grant by Conseil régional de Bourgogne”
- Local controllability of 1D linear and nonlinear Schr (2010, unknown): “Optimal control techniques have also been investigated for Schrödinger equations with a non-linearity of Hartree type in [11,12] by Baudouin, Kavian, Puel and in [34] by Cances, Le Bris, Pilot.”