Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition

Laser surface hardening of steel is formulated in terms of an optimal control problem, where the state equations are a semilinear heat equation and an ordinary differential equation, which describe the evolution of the high temperature phase. The optimal control problem is analyzed and first-order necessary optimality conditions are derived. An error estimate for POD (proper orthogonal decomposition) Galerkin methods for the state system is proved. Finally, a strategy to obtain suboptimal controls using POD is applied to solve some numerical examples.