WIAS Preprint No. 315, (1997)

Optimal control of laser hardening


  • Hömberg, Dietmar
    ORCID: 0000-0001-9460-5729
  • Sokolowski, Jan

2010 Mathematics Subject Classification

  • 35K55 49K20 80A22


  • Laser treatment of materials, phase transitions, state constraints, optimality conditions




We present a mathematical model for the laser surface hardening of steel. It consists of a nonlinear heat equation coupled with a system of five ordinary differential equations to describe the volume fractions of the occuring phases.

Existence, regularity and stability results are discussed.

Since the resulting hardness can be estimated by the volume fraction of martensite, we formulate the problem of surface hardening in terms of an optimal control problem. To avoid surface melting, which would decrease the workpiece's quality, state constraints for the temperature are included.

We prove differentiability of the solution operator and derive necessary conditions for optimality.

Appeared in

  • Adv. Math. Sci., 8 (1998), pp.911-928

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