An approximation of problems of optimal control on the coefficients of elliptic convection-diffusion equations with an imperfect contact matching condition
We consider nonlinear optimization problems for processes described by non-self-adjoint elliptic equations of convection-diffusion problems with an imperfect contact matching conditions. These are the problems with a jump of the coefficients and of the solution on the interface; the jump of the solution is proportional to the normal component of the flux. Variable coefficients multiplying the highest and the lowest derivatives in the equation and the coefficients by nonlinear terms in the equations of state are used as controls. Finite difference approximations of optimization problems are constructed and investigated. For the approximation of state equations we propose a new “modified difference scheme” in which the variable grid coefficients in the principal part of the difference operator are computed using method other than traditionally applied in the theory of difference schemes. The problem's correctness is investigated. The accuracy estimation of difference approximations with respect to the state are obtained. Convergence rate of approximations with respect to cost functional is estimated, too. Weak convergence with respect to control is proved. The presence of a non-self-adjoint operator causes certain difficulties in constructing and studying approximations of differential equations describing discontinuous states of controlled processes, in particular, in proving the difference approximations well-posedness, and in studying the relationship between the original optimal control problem and the approximate mesh problem. The approximations are regularized. The obtained results will be heavily used later in solving problems associated with the development of effective methods for the numerical solution to the constructed finite-dimensional mesh optimal control problems and their computer implementation.