On data assimilation for quasilinear parabolic problems (Q2730694)

The data assimilation problems for identifying the initial condition have been studied by optimal control methods for various classes of evolution equations by many researchers. In this paper the authors consider the following data assimilation problem: find \(u\) and \(T\) such that NEWLINE\[NEWLINEC(T){\partial T\over\partial t}- \text{div}(L\text{ grad }T)= f(t,\overline x),\quad t\in (0,{\mathcal T}),\quad \overline x\in\Omega,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINET|_{t=0}= u,\quad T|_{\Gamma_1}= T_2(t),\quad {\partial T\over\partial n}\Biggr|_{\Gamma_2}= 0,\quad S(u)= \text{inf}_{\widetilde u}S(\widetilde u),NEWLINE\]NEWLINE where NEWLINE\[NEWLINES(\widetilde u)= {\alpha\over 2} \|\widetilde u-\widehat T\|^2_{L^2(\Omega)}+{1\over 2} \int^T_0\|\widetilde T- \widehat T\|_{L^2(\Omega)} dt.\tag{2}NEWLINE\]NEWLINE Here \(\Omega\in \mathbb{R}^m\), \(1\leq m\leq 3\), is a bounded domain with piecewise boundary \(\partial\Omega= \Gamma_1\cup \Gamma_2\), \(T= T(l,\overline x)\), \(u= u(l,\overline x)\) are unknown functions, and \(\widetilde T\) is the solution to the initial boundary value problem (1) with \(\widetilde T|_{t=0}= \widetilde u\). We assume that the function \(L= L(\overline x)\), \(C= C(T)\), \(f(t,\overline x)\), \(\widehat T_0(\overline x)\), \(T_1(t)\), \(\widehat T(t,\overline x)\) are given and \(\widehat T_0\), \(\widehat T\) are obtained from observational data, \(\overline x= (x_1,\dots, x_m)^T\in \Omega\), \(n\) is the outward normal unit vector to \(\partial\Omega\), \({\mathcal T}<\infty\).NEWLINENEWLINENEWLINEThe authors study the properties of the nonlinear operator of the data assimilation problem (1)--(2) and prove the solvability of this problem in a specific class of functional spaces. The Newton algorithm for the optimality system is given.