Sufficient second-order optimality conditions for convex control constraints

In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in Rn. The constraint is of the form u(x)∈U(x), where U is a set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in an L∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier–Stokes equations.