Zajtsev Y. Minimax prediction of solutions to initial boundary value problems of transmission for parabolic equations under incomplete data.

Methods of determination of minimax mean square prediction estimates for linear functionals from solutions to initial boundary value problems of transmission for linear parabolic partial differential equations of the second order under incomplete data are elaborated in the manuscript. We consider the cases when the restrictions on unknown deterministic functions (right-hand sides of equations, boundary, initial conditions, conditions on the interfaces) are given completely or partly. In all the cases the theorems on a general form of minimax mean square prediction estimates of functionals have been proved and the errors of prediction have been established. We prove the statements on a reduction of minimax prediction problems to certain problems of optimal control of parabolic equations with discontinuous coefficients and with the quadratic performance criterions. Analogous results have been also obtained for a problem of minimax mean square estimation of functionals from right-hand sides of parabolic e quations with discontinuous coefficients.