Second order analysis for the optimal control of parabolic equations under control and final state constraints

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• 作者：F. J. Silva
• 关键词：Optimal control ; Parabolic equations ; Box constraints for the control ; Finitely many constraints for the state ; Robinson constraint qualification ; Second order optimality conditions ; Quadratic hamiltonian ; 49J20 ; 49K20 ; 35Q93
• 刊名：Set-Valued and Variational Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：24
• 期：1
• 页码：57-81
• 全文大小：435 KB
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• 作者单位：F. J. Silva (1)
  
  1. XLIM - DMI UMR CNRS 7252 Faculté des Sciences et Techniques, Université de Limoges, Limoges, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L 1 and \(L^{\infty }\) topologies. Keywords Optimal control Parabolic equations Box constraints for the control Finitely many constraints for the state Robinson constraint qualification Second order optimality conditions Quadratic hamiltonian