文章基本信息

标题：Verification of some functional inequalities via polynomial optimization
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作者：Giovanni Fantuzzi
期刊名称：IFAC PapersOnLine
印刷版ISSN：2405-8963
出版年度：2022
卷号：55
期号：16
页码：166-171
DOI：10.1016/j.ifacol.2022.09.018
语种：English
出版社：Elsevier
摘要：AbstractMotivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the formf(I1(u), . . ., Ik(u)) ≥ 0 where f is a polynomial, u is any function satisfying prescribed constraints, and I1(u), . . .,Ik(u) are integral functionals whose integrands are polynomial in u, its derivatives, and the integration variable. We show that such functional inequalities can be strengthened into sufficient polynomial inequalities, which in principle can be checked via semidefinite programming using standard techniques for polynomial optimization. These sufficient conditions can be used also to optimize functionals with affine dependence on tunable parameters whilst ensuring their nonnegativity. Our approach relies on a measure-theoretic lifting of the original functional inequality, which extends both a recent moment relaxation strategy for PDE analysis and a dual approach to inequalities for integral functionals.
关键词：KeywordsFunctional inequalitiesmoment relaxationspolynomial optimizationLyapunov methodspartial differential equationsdistributed parameter systems