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Nonlinear conjugate gradient methods for the optimal control of laser surface hardening. (English) Zbl 1062.49034

Summary: Laser surface hardening of steel is formulated in terms of an optimal control problem with bilateral control constraints, where the state equations are composed of a semi-linear heat equation and an ordinary differential equation, which describes the evolution of the high temperature phase. To avoid the melting of the steel we have to impose state constraints for the temperature. These constraints are realized numerically by adding a penalty term to the cost functional. Variants of the nonlinear conjugate gradient method are applied to solve the optimal control problem numerically. A practical line search, which guarantees the strong Wolfe-Powell conditions, is utilized. The behavior of the algorithm is compared to the steepest descent method.

MSC:

49N90 Applications of optimal control and differential games
49M37 Numerical methods based on nonlinear programming
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
78A60 Lasers, masers, optical bistability, nonlinear optics
80A20 Heat and mass transfer, heat flow (MSC2010)
80M50 Optimization problems in thermodynamics and heat transfer

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References:

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