On controllability problem for quasilinear heat conductivity equation (Q2719465)

The authors discuss the local solvability of a converse problem for a quasilinear equation of heat conductivity NEWLINE\[NEWLINE \begin{aligned} &v_t - \Delta v = F(x,t),\tag{1}\\ &v\big|_{t=0} = 0,\tag{2}\\ &v\big|_{S_T} = 0,\tag{3}\\ &v\big|_{t=t_1} = \chi(x),\tag{4} \end{aligned} NEWLINE\]NEWLINE where \( S_T = \partial\Omega\times [0,T]\), \( 0<t_1\leq T\), \( 0<T<\infty\), \( \partial\Omega\in C^2(n\geq 2)\), \( F(x,t) = h(x,t)f(x)\), \( h(x,t)\) and \(\chi(x)\) are given functions, \(v\) and \(f\) are unknown functions. Under some additional assumptions this problem may be interpreted as a controllability problem. In the paper a theorem on local controllability of the system is proved.