Abstract
The optimal pursuit problem and a variant of the final value control problem being set in the Hilbert space are considered in this paper, and these two problems are solved in the same framework using the orthogonal projection. The product space is constructed from, input (control) and output (or state) spaces, the norm of which coincides with the performance index itself. The operator relating input to output (state) is represented by its graph in the product space. Then the optimal control is obtained as the orthogonal projection of the desired output (state) onto the graph. The orthogonal projection method shows that even for the improper systems, such as differentiator, the optimal pursuit problem is worthwhile to be considered and its unique optimal control exists. This method offers the optimal control in the explicit form. The computational procedure for obtaining the optimal control is straightforward and then can be easily performed.
