arXiv2026
Despite the extensive body of work on backstepping for one-dimensional PDEs, results in higher dimensions remain comparatively limited. Most available methods either exploit particular symmetries of the PDE or address problems posed on parallelepiped domains. To the best of our knowledge, the only approach that enables the design of backstepping controllers on non-parallelepiped regions without symmetry assumptions is the domain extension technique. This method, however, presents several drawbacks. In particular, the control input at each time instant is obtained by simulating a PDE on an extended domain, from which the actual input on the original domain is approximated. By contrast, in the one-dimensional setting, once the time-independent backstepping gain kernel is known, the control input can be computed in closed form as a feedback depending solely on the state at that same instant. Moreover, problems such as output-feedback design or adaptive and robust control do not appear straightforward to address with the domain extension method, at least to the best of our knowledge. These considerations motivate the search, whenever possible, for alternatives that preserve the main advantages of one-dimensional backstepping. A motivating example for the domain extension method is the control of the heat equation on a piano-shaped domain, with actuation applied at the tail of the piano. In this extended abstract, we show through a simple calculation that the domain extension method is not required in this setting. Instead, a strategy akin to that used for parallelepiped domains can be adopted. This result constitutes a first instance of a broader framework for backstepping control of asymmetric PDEs posed on non-parallelepiped regions, which we refer to as domains with graph-monotone boundaries. The general framework is developed in a forthcoming paper.