In this paper, we are interested in the computation of the unknown initial state for the simulation and prediction of PDE systems where the solution measures are partially known over a time interval. Such a problem is usually solved by an ill-posed optimal control problem. Based on an appropriate collocation approximation, we obtain a discrete inverse problem. To solve this problem, a non-standard discrete approach is used. This allows to obtain a transformation of the original problem into a well-posed ones based on the zero controllability of a discrete system. The desired control is then calculated as well as the discrete approximation of the initial state values.
Mathematics Subject Classification. 35K55; 35J60; 80A22.