A Tutorial: Introducing Backward-in-Time Uniqueness via Carleman Estimates for Mechanics Students

Abstract This article serves as a tutorial on backward-in-time uniqueness for evolutionary systems and presents an effective approach to establish such uniqueness. The article begins by defining backward-in-time uniqueness and contrasting it with related concepts, such as forward and backward problem uniqueness. The Carleman estimate approach—a powerful and systematic method for proving backward-in-time uniqueness in time-dependent problems—is then introduced, along with a step-by-step framework to guide its application. To reinforce the theoretical foundations, this tutorial-style article presents canonical examples involving ordinary differential equations to illustrate both the concept and the method, and includes a counterexample demonstrating the failure of backward-in-time uniqueness. These concepts are crucial for understanding and controlling systems in real-world and technological applications, such as inverse problems and predictive modeling, yet they are seldom treated in depth in graduate-level mathematical methods curricula. The article addresses this gap, and, as such, its material can be adapted into coursework that introduces these advanced topics and equips students with both theoretical insight and practical research tools. Although the underlying techniques are established, the tutorial’s synthesis, algorithmic structure, and pedagogical presentation constitute a novel contribution to the literature.