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The article explores the structure of optimal control solutions to the boundary control problem for the heat equation y_{t}=y_{xx} in finite (t,x) domains, when the set of admissible control values is bounded. The states y(t,.), admissible control trajectories u(.), and functions entering the formulation of initial and boundary conditions, are members of appropriate L² spaces, which together with the linearity of the problem imply the (least-squares) existence of the optimal solution from the beginning. Therefore the work concentrates in assessing when the numerical methods at hand can identify the optimal control trajectory and the eventual location of its noticeable points, specially the points of discontinuity of u (switching) or u (saturation). In physically existing systems the boundary conditions (the temperature y(t,0) at one end of a bar, for instance) can not be manipulated as an ordinary control variable u(t), so the setup adopted in the article, allowing for infinitely countable switching points, is perhaps too demanding for industrial control purposes. Still, the article illustrates the richness in the qualitative behavior of solutions generated by the imposing of restrictions into the set of control values. This diversity, notably including the appearance of bang-bang solutions, also appears when applying the Pontryagin Maximum Principle to some systems governed by ODEs; so the authors´ approach may become attractive not only to the mathematically oriented audience but also to those readers interested in the subtleties of optimal control problems for finite-dimensional systems. The optimal boundary control problem, as posed here, asks for the control to generate the `best feasible approximation´ to a target spatial profile. But its study turns also relevant to elucidate the `exact´ (in L²) reachability question, namely: is it possible to modify the boundary condition(s), as time t proceeds and through some control u(t), to accomplish that at final time T the state profile y(T,.) of the solution become equal to some desired function y_{d}(.)? A basic theorem based on the uniqueness of solutions asserts that the `final´ condition can not be arbitrarily imposed to the heat equation when the initial and boundary conditions have already been chosen and remain fixed. Some theoretical answers to the reachability question are reviewed by the authors, and partial advances are worked out when numerical perturbations are present. Results for the (simpler) Neumann and Robin boundary conditions are used in the paper to study the Dirichlet case in an asymptotic fashion. A remarkable contribution consists in exploiting the regularity of the adjoint state to show that the number and location of switching points in optimal bang-bang controls can be decided numerically. The exponential moment problem is also used as a powerful tool in obtaining necessary conditions for reachability of a desired final state; another interesting aspect for the numerical analyst. Although only partial answers are proved by the authors in some cases, the extent of the effort illustrates the difficulty of translating theorems into numerical certainties in the presence of bang-bang and similar control trajectories. In particular, the `countable bang-bang principle´ (theorem 2.3) for Neumann and Robin conditions turns ambiguous under perturbations, and therefore meticulous investigations and estimations were needed to classify the effect of numerical discretization. In fact, this level of detail in treating so many cases sort of precludes grasping the whole picture: the paper lacks a `Conclusions´ section where the results were recapitulated/contrasted against the announces of the `Introduction´. But, in short, the final opinion of the reviewer is much in favor of the quality of the work done by the authors, and its reading is recommended.