Vietnam Journal of Mathematics

, Volume 44, Issue 1, pp 181–202 | Cite as

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

  • Eduardo Casas
  • Fredi TröltzschEmail author


Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order conditions cannot be expected to hold. For this case, new second-order conditions are established that are based on different types of critical cones. Depending on the choice of this cones, the second-order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.


Optimal control Parabolic equation Semilinear equation Second-order optimality conditions Weak local minimum Strong local minimum 

Mathematics Subject Classification (2010)

49J20 49K20 



The first author was partially supported by Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P. The second is supported by DFG in the framework of the Collaborative Research Center SFB 910, project B6.


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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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