Consider the problem of finding a control parameter $p(t)$ in the following \begin{equation} \frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}+p(t)u(x,t)+f(x,t), \ ~~\ 0\leq x \leq1 ,\ ~~\ 0< t \leq1 \end{equation} with the initial conditions \begin{equation} \ u(x,0)=h(x), \ ~~\ 0\leq x \leq1 \end{equation} and boundary conditions \begin{equation} \ u(0,t)=g_1(x), \ ~~\ 0< t \leq1 \\ u(1,t)=g_2(x), \ ~~\ 0< t \leq1 \end{equation} with an additional condition which describes the overspecification over a portion of spatial domain \begin{equation} \int_{0}^{s(t)} u(x,t) dx = E(t), \ ~~\ 0< t \leq1, \ ~~\ 0< s(t) \leq1 \end{equation}
I know the existence and uniqueness of the solution but I don't have any idea whether it is stable or not. Please help me out with the stability of the solution.