{"ID":5935878,"CreatedAt":"2026-07-07T01:22:02.77346169Z","UpdatedAt":"2026-07-07T02:10:06.972658124Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03031","arxiv_id":"2607.03031","title":"Quantitative rapid boundary stabilization via modal decomposition and its application to the Allen-Cahn equation","abstract":"We investigate quantitative rapid stabilization for the one-dimensional Allen--Cahn equation and develop a quantitative modal decomposition approach that makes explicit the dependence of the feedback laws and stabilization costs on the prescribed decay rate. We construct an explicit feedback law on the finite-dimensional unstable modes via Ackermann's formula. The explicit structure of the feedback allows us to derive quantitative low-frequency estimates, which, combined with the frequency Lyapunov method, yield quantitative stabilization estimates. Together with the stabilization framework of [37], the resulting estimates can be adapted to a broader class of one-dimensional parabolic models. We further construct piecewise feedback laws that yield the null controllability with control costs and finite-time stabilization.","short_abstract":"We investigate quantitative rapid stabilization for the one-dimensional Allen--Cahn equation and develop a quantitative modal decomposition approach that makes explicit the dependence of the feedback laws and stabilization costs on the prescribed decay rate. We construct an explicit feedback law on the finite-dimension...","url_abs":"https://arxiv.org/abs/2607.03031","url_pdf":"https://arxiv.org/pdf/2607.03031v1","authors":"[\"Shengquan Xiang\",\"Yu Xiao\",\"Can Zhang\"]","published":"2026-07-03T07:20:43Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\"]","methods":"[]","has_code":false}