A source identification problem related to mathematical model of laser surface heating

A mathematical model of two-dimensional laser surface heating for the hardening of metallic materials is proposed. The model is governed by the heat equation u t − Δ u = m ( t ) δ γ ( x − ω ( t ) ) , ( x , t ) ∈ Ω , with the pointwise source term δ γ ( y ) , satisfying the initial u ( x , 0 ) = g ( x ) and boundary u ( x , t ) = 0 , x ∈ ∂ Ω , conditions. The pair of source terms 〈 m ( t ) , ω ( t ) 〉 is assumed to be unknown. The two-valued ( m ( t ) = 0 or m ( t ) = m 0 > 0 ) function m ( t ) is treated as the intensity of the laser beam, and the function ω ( t ) describes the laser beam trajectory. The identification problem consists of determining the pair of source terms 〈 m ( t ) , ω ( t ) 〉 such that the corresponding heat function u ( x , t ) satisfies the condition ‖ u − v ‖ L 2 ( Ω ) ≤ ε , where the smooth function v ( x , t ) is assumed to be known (experimentally), and ε > 0 is a given-in-advance parameter. Besides the existence result, the structure of the optimal trajectory is also described.