Second-order necessary optimality conditions for semilinear elliptic optimal control problems

Abstract

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.

Keywords:

• First-order necessary optimality condition
• second-order necessary optimality condition
• optimal control
• semilinear elliptic equation
• mixed pointwise constraint
• pure pointwise state constraint

AMS Subject Classifications:

• 49K20
• 35J25

Acknowledgements

The authors wish to express their sincere thanks to the anonymous referees for their helpful suggestions and useful comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

In this research, J. C. Yao was supported by the Grant MOST 102-2221-E-039-017-MY3.