Second-Order Necessary Conditions for a Strong Local Minimum in a Control Problem with General Control Constraints

  • Hélène Frankowska
  • Nikolai P. Osmolovskii


We establish some second-order necessary optimality conditions for strong local minima in the Mayer type optimal control problem with a general control constraint \(U \subset \mathrm{I\! R}^m\) and final state constraint described by a finite number of inequalities. In the difference with the main approaches of the existing literature, the second order tangents to U and the second order linearization of the control system are used to derive the second-order necessary conditions. This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of set-valued maps lead to fairly general statements. We illustrate the results by considering the case of control constraints defined by inequalities involving functions with positively or linearly independent gradients of active constraints, but also in the cases where the known approaches do not apply.


Optimal control End point constraints General control constraints Second-order necessary conditions Second order tangents Differential inclusion 

Mathematics Subject Classification

Primary 49K15 Secondary 34H05 49K24 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Hélène Frankowska
    • 1
  • Nikolai P. Osmolovskii
    • 2
    • 3
    • 4
  1. 1.CNRS, IMJ-PRG, UMR 7586, Sorbonne UniversitésParisFrance
  2. 2.University of Technology and HumanitiesRadomPoland
  3. 3.Systems Research InstitutePolish Academy of SciencesWarszawaPoland
  4. 4.Moscow State University of Civil EngineeringMoscowRussia

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