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 ut−Δu=m(t)δγ(x−ω(t))ut−Δu=m(t)δγ(x−ω(t)), (x,t)∈Ω(x,t)∈Ω, with the pointwise source term δγ(y)δγ(y), satisfying the initial u(x,0)=g(x)u(x,0)=g(x) and boundary u(x,t)=0u(x,t)=0, x∈∂Ωx∈∂Ω, conditions. The pair of source terms 〈m(t),ω(t)〉〈m(t),ω(t)〉 is assumed to be unknown. The two-valued (m(t)=0m(t)=0 or m(t)=m0>0m(t)=m0>0) function m(t)m(t) is treated as the intensity of the laser beam, and the function ω(t)ω(t) describes the laser beam trajectory. The identification problem consists of determining the pair of source terms 〈m(t),ω(t)〉〈m(t),ω(t)〉 such that the corresponding heat function u(x,t)u(x,t) satisfies the condition ‖u−v‖L2(Ω)≤ε‖u−v‖L2(Ω)≤ε, where the smooth function v(x,t)v(x,t) is assumed to be known (experimentally), and ε>0ε>0 is a given-in-advance parameter. Besides the existence result, the structure of the optimal trajectory is also described.