On Bi-metric Regularity of Variational Inclusion Appearing in Affine Optimal Control Problems

This paper deals with different types of stability properties related to the variational inclusion $$0\in \hat{\sigma }(y)+N_{U}(u(y))$$ 0 ∈ σ ^ ( y ) + N U ( u ( y ) ) when the left hand side is perturbed around zero. Here $$U\subset \mathbb {R}^{d}$$ U ⊂ R d is a compact convex polyhedron, $$\hat{\sigma }$$ σ ^ is a function defined on $$\mathbb {R}^{m}$$ R m with values in $$\mathbb {R}^{d}$$ R d and u is the unknown function defined on $$\mathbb {R}^{m}$$ R m with values in U. Variational inclusion problems described previously appear for instance in the formulation of the first order optimality conditions (Pontryagin principle) corresponding to an affine optimal control problem. In this work we get new metric regularity results corresponding to the aforementioned variational inclusion, in particular a bi-metric property is deduced when the corresponding spaces where the functions u and $$\hat{\sigma }$$ σ ^ belong are equipped with appropriated norms. A by-product of this work is the unification of the results concerning the metric regularity properties already obtained separately for $$m=1$$ m = 1 and $$m>1$$ m > 1 .