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Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

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Abstract

We investigate Hölder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Hölder continuous, while they have the two sided lower Hölder continuity property for less regular constraints. Finally, we provide sufficient conditions for Hölder type regularity of optimal controls.

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Authors and Affiliations

  1. SISSA/ISAS, via Beirut 2–4, 34014, Trieste, Italy

    Piernicola Bettiol

  2. CNRS, CREA, Ecole Polytechnique, 1, Rue Descartes, 75005, Paris, France

    Hélène Frankowska

Authors
  1. Piernicola Bettiol
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  2. Hélène Frankowska
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Correspondence to Piernicola Bettiol.

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Work supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations.

P. Bettiol acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations.

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Bettiol, P., Frankowska, H. Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems. Appl Math Optim 57, 125–147 (2008). https://doi.org/10.1007/s00245-007-9015-8

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