Difference approximations and regularization of problems of optimal control for parabolic equations with controls in the coefficients

Problems of optimal control for parabolic equations with controls both in the coefficients of the equation of state, which depend on x and t, and in the coefficients of boundary conditions of the third kind, and with solutions from W-2(2,1) (QT) are analysed. The question of whether such problems are well-posed in a weak topology is considered. Two difference approximations A(p), p = 1, 2, of extremal problems are constructed. Estimates O (gamma(h tau)((p))), p = 1, 2, of the approximations A(p), p = 1, 2, for the state in a mesh norm V-2(1,0) and for the functional are obtained, from which it follows, in particular, that gamma(h tau)((p)) = /h/(3/4), P = 1, 2, if tau similar to root/h/(3), and gamma(h tau)((p)) = root/h/(p), p = 1, 2, if tau similar to /h/(2). These estimates are obtained with no additional a priori assumptions regarding the smoothness of the generalized solutions of the state of the control process. Tikhonov regularization of the approximations is carried out.