While addressing in this book the numerical solution of controllability problems for systems governed by partial differential equations, we had the opportunity to encounter a variety of concepts and methods whose applicability goes much beyond the solution of genuine control problems. Among these concept and methods let us mention convex duality, space–time discretization of partial differential equations, numerical methods for the solution of large linear systems, least-squares formulations, optimization algorithms, and so on. In Chapter 7, we have shown while formulating a given problem as a controllability that one may gain access to powerful solution methods. Such a situation is not unique as shown by the following example inspired from work in progress by R. Azencott, A.M. Ramos and the first author (see Azencott, Glowinski, and Ramos, 2007). The (relatively simple) problem that we consider is part of a large research program on shape identification and pattern recognition (largely motivated by medical applications); it can be described as follows:

Let Γ0 be a rectifiable (or piece of) bounded curve in ℝ2; suppose that one wishes to know how close is Γ0 to another curve ΓR (the curve of reference) which is also rectifiable and bounded. The idea here is to introduce a distance between Γ0 and ΓR (rigid displacement and similarity invariant in general, but these conditions can be relaxed if necessary, or more conditions can be added).