Séance Séminaire

Séminaire ACSIOM

Formules séquentielles pour le cône normal à des ensembles de sous-niveau

Let $X$ be a reflexive Banach space and let $\Phi$ be an extended real-valued lower semicontinuous convex function on $X$. Given a real $\lambda$ and the sublevel set $S=[\Phi\leq \lambda]$, we establish at $\overline{x}\in S$ the following formula for the normal cone to $S$ $$N_S(\overline{x})=\limsup_{x\to \overline{x}}\, \mathbb{R}_+\, \partial \Phi(x) \quad \mbox{if} \quad \Phi(\overline{x})=\lambda, \quad \quad(\star)$$ without any qualification condition. The case $\Phi(\overline{x})<\lambda$ is also studied. Here $\partial \Phi$ stands for the subdifferential of $\Phi$ in the sense of convex analysis. The proof is based on the sequential convex subdifferential calculus. The formula $(\star)$ is extended to nonreflexive Banach spaces via the use of nets. The normal cone to the intersection of finitely many sublevel sets is also examined, thus leading to new formulae without qualification condition. Our study goes beyond the convex framework: when $\dim X<+\infty$, we show that the inclusion of the left member of $(\star)$ into the right one still holds true for a locally Lipschitz continuous function. Finally, an application of the formula $(\star)$ is given to the study of the asymptotic behavior of some gradient dynamical system.