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Sufficient second-order optimality conditions for convex control constraints

Wachsmuth, Daniel

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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 an 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 a L∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier-Stokes equations.