Featured Review:; Numerical Analysis for Elliptic Optimal Control Problems

Abstract

SIAM Review, Volume 68, Issue 3, Page 687-689, September 2026.
Two fundamental issues arise in the study of an optimal control problem. First, we aim to establish the well-posedness of the problem, meaning that it has at least one solution. While the uniqueness of the solution is interesting, it is not crucial. The analysis of the state equation—a partial differential equation in the book at hand—is the starting point in proving the existence of an optimal control. Essential to this analysis is proving the existence and uniqueness of a solution to the state equation for every control and establishing the continuity of the control-to-state relation. The differentiability of the control-to-state relation is also essential for subsequent analyses. Some multistate control problems exist where the existence or uniqueness of a state corresponding to certain controls may fail, and this requires a different approach. However, such problems are not addressed in the present book.