Parameter Estimation in the Stefan Problem

We consider the problem of estimating an unknown time-dependent diffusion coefficient in a Stefan problem. We shall treat the one-dimensional case here, for which the partial differential equation model is given by $$ \begin{gathered} u_t = \left( {a\left( t \right)u_x } \right)_x {\text{ 0 < t}} \leqslant {\text{T,0 < x < s(t),}} \hfill \\ {\text{a}}\left( t \right)u_x \left( {0,t} \right) = g\left( t \right){\text{ 0 < t < T,}} \hfill \\ u = \left( {s\left( t \right),t} \right) = 0{\text{ 0}} \leqslant {\text{t}} \leqslant {\text{T,}} \hfill \\ {\text{u}}\left( {x,0} \right) = \phi \left( x \right){\text{ 0}} \leqslant {\text{x}} \leqslant {\text{b,}} \hfill \\ \end{gathered} $$ (1.1) $$\begin{gathered} \dot s\left( t \right) = - \gamma a\left( t \right)u_x \left( {s\left( t \right),t} \right){\text{ 0 < t}} \leqslant {\text{T,}} \hfill \\ {\text{s}}\left( 0 \right) = b. \hfill \\ \end{gathered} $$ (1.2)