Abstract
In this paper we discuss necessary and sufficient second order optimality conditions for control problems of semilinear elliptic equations with pointwise control constraints as well as integral state constraints. We consider the case of a boundary control problem, the state equation being described by a Neumann problem. The presence of state constraints along with pointwise control constraints causes some known difficulties in the proof of optimality conditions. Due to the specific integral form of the state-constraints, here we are able to considerably tighten the gap between second order necessary and sufficient condition. The analysis is based on some constraint qualification.
This research was partially supported by European Union, under HCM Project number ER-BCHRXCT940471, and by Deutsche Forschungsgemeinschaft, under Project number Tr 302/1–2. The first author was also supported by Dirección General de Investigación Científica y Técnica (Spain).
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References
F. Bonnans. Second order analysis for control constrained optimal control problems of semilinear elliptic systems. To appear in Appl. Math. Optimization.
E. Casas and F. Tröltzsch. Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. To appear in Appl. Math. Optimization.
E. Casas, F. Tröltzsch, and A. Unger. Second order sufficient optimality conditions for a nonlinear elliptic control problem. J. for Analysis and its Appl. 15 (1996), pp. 687–707.
E. Casas, F. Tröltzsch, and A. Unger. Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. To appear in SIAM J. Control Optimization.
A.L. Dontchev, W.W. Hager, A.B. Poore, and B. Yang. Optimality, stability, and convergence in nonlinear control. Appl. Math. Optimization 31 (1995), No. 3, pp. 297–326.
H. Goldberg, and F. Tröltzsch. Second order sufficient optimality conditions for a class of non-linear parabolic boundary control problems. SIAM J. Control Optimization 31 (1993), pp. 1007–1027.
H. Maurer. First and second order sufficient optimality conditions in mathematical programming and optimal control. Math. Programming Study 14 (1981), pp. 163–177.
H. Maurer, and J. Zowe. First-and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979), pp. 98–110.
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© 1999 Springer Basel AG
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Casas, E., Tröltzsch, F., Unger, A. (1999). Second Order Optimality Conditions for Some Control Problems of Semilinear Elliptic Equations with Integral State Constraints. In: Hoffmann, KH., Leugering, G., Tröltzsch, F., Caesar, S. (eds) Optimal Control of Partial Differential Equations. ISNM International Series of Numerical Mathematics, vol 133. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8691-8_8
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DOI: https://doi.org/10.1007/978-3-0348-8691-8_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9731-0
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