Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach

Abstract : The exact distributed controllability of the semilinear wave equation ytt − yxx + g(y) = f 1ω, assuming that g satisfies the growth condition |g(s)|/(|s| log 2 (|s|)) → 0 as |s| → ∞ and that g ∈ L ∞ loc (R) has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that g ∈ L ∞ loc (R), that sup a,b∈R,a =b |g (a) − g (b)|/|a − b| r < ∞ for some r ∈ (0, 1] and that g satisfies the growth condition |g (s)|/ log 2 (|s|) → 0 as |s| → ∞, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate 1 + r. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [17 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02975370
Contributor : Arnaud Munch <>
Submitted on : Monday, November 2, 2020 - 7:31:56 PM
Last modification on : Wednesday, November 4, 2020 - 3:29:10 AM

File

LS_WAVE_MUNCH_27_10_2020.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02975370, version 2

Collections

Citation

Arnaud Munch. Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach. 2020. ⟨hal-02975370v2⟩

Share

Metrics

Record views

29

Files downloads

16