Verification of some functional inequalities via polynomial optimization
Fantuzzi G (2022)
Publication Type: Conference contribution
Publication year: 2022
Journal
Publisher: Elsevier B.V.
Book Volume: 55
Pages Range: 166-171
Conference Proceedings Title: IFAC-PapersOnLine
Event location: Gif sur Yvette
DOI: 10.1016/j.ifacol.2022.09.018
Abstract
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form f(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.
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APA:
Fantuzzi, G. (2022). Verification of some functional inequalities via polynomial optimization. In Filippova Tatiana (Eds.), IFAC-PapersOnLine (pp. 166-171). Gif sur Yvette, FR: Elsevier B.V..
MLA:
Fantuzzi, Giovanni. "Verification of some functional inequalities via polynomial optimization." Proceedings of the 18th IFAC Workshop on Control Applications of Optimization, CAO 2022, Gif sur Yvette Ed. Filippova Tatiana, Elsevier B.V., 2022. 166-171.
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