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Preprint 30-2004

Sufficient second-order optimality conditions for convex control constraints

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Author(s) : Daniel Wachsmuth

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 30-2004

MSC 2000

49K20 Problems involving partial differential equations
26E25 Set-valued functions
49J53 Set-valued and variational analysis
76D05 Navier-Stokes equations

Abstract :
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.

Keywords : Optimal control, sufficient second-order conditions, strongly active sets, convex control constraints, measurable set-valued functions, Navier-Stokes equations

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