2019/07/09 by Arnaud Münch, Munch, Arnaud, Carlos Castro +1
Engineering · Mathematics · #58K55 #93B05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1907.04118
openalex publication_date 2019/07/09 · openalex created_date 2023/04/19 · openalex updated_date 2026/07/28
The Petrowsky type equation ytt^\eps+\eps yxxxx^\eps - yxx^\eps=0, \eps>0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √(\eps) occurring at the extremities, these boundary controls get singular as \eps goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y^\eps then derive a rigorous second order asymptotic expansion of the control of minimal L2-norm, with respect to the parameter \eps. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.