Pontryagin’s Principle for Optimal Control Problems Governed by Semilinear Elliptic Equations

  • Eduardo Casas
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)


In this paper we prove an extension of Pontryagin’s principle for optimal control problems governed by semilinear elliptic partial differential equations. The control takes values in a bounded subset, not necessarily convex, of some Euclidean space; the cost functional is of Lagrange type; and some general equality and inequality state constraints are imposed. To derive Pontryagin’s principle we combine a suitable penalization of the state constraints with Ekeland’s principle. In the absence of equality state constraints we establish the optimality conditions in a qualified form for “almost all” problems. In a first stage, the classical spike perturbations of the controls used to derive Pontryagin’s principle are replaced by a new type of variations.

1991 Mathematics Subject Classification

49K20 Secondary 35J65 

Key words and phrases

Pontryagin’s principle semilinear elliptic equations state constraints Ekeland’s variational principle 


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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Eduardo Casas
    • 1
  1. 1.Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Caminos, C. y P.Universidad de CantabriaSantanderSpain

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