An approach for solving of a moving boundary problem (Q1428989)

The authors introduce a ``new'' approach for solving 1D Stefan problems. They indeed consider the parabolic problem : \(V_{t}=V_{xx}\) posed in \((0,s(t) ) \times (0,+\infty ) \), where \(s\) is the moving boundary which starts at \(a\) at time 0. The boundary conditions : \(V_{x}(0,t) =g(t) \) and \(V(s(t) ,t) =0\)\ are imposed together with the Stefan condition : \( -dV/dx(s(t) ,t) =s^{\prime }(t) \). The authors consider the objective function : \(H=\int_{0}^{T}\int_{0}^{s(t) }(V_{xx}-V_{t}) ^{2}\,dx\,dt\) which involves the unknown moving boundary \(s\). Introducing an appropriate change of scale and new unknown functions, the authors deal with an objective function which integrates a much more complicated expression but on the fixed domain \( (0,T) \times (0,1) \). Five control functions \(u\), \( v_{1}\), \(v_{2}\), \(v_{3}\), \(v_{4}\) and \(v_{5}\) indeed appear in the new integrand, which are linked to the time derivative of \(s\) or to partial derivatives of the transformed solution \(Z\). The authors then define the notion of admissible triple \((Z,u,v) \) for this control problem. They finally prove that the control problem is equivalent to the minimization of a linear functional on a set of Radon measures. This last problem is proved to have a solution \(\mu ^{\ast }\). Using classical results [see J. E. Rubio's book, Control and Optimization. The linear treatment of nonlinear problems. Manchester University Press (1986)], they approximate the optimal measure \(\mu ^{\ast }\) by a finite combination of atomic measures. This leads to the minimization of a linear expression under linear constraints. The last part of the paper presents a numerical example which illustrates this procedure. This approach however seems to be restricted to 1D-Stefan problems.