Infinite-step backstepping for a heat equation-like PDE with arbitrarily many unstable eigenvalues

We consider feedback transformations of the backstepping/feedback linearization type that have been prevalent in finite dimensional nonlinear stabilization, and, with the objective of ultimately addressing nonlinear PDE, generate the first such transformations for a linear PDE that can have an arbitrary finite number of open-loop unstable eigenvalues. These transformations have the form of recursive relationships and the fundamental difficulty is that the recursion has an infinite number of iterations. Naive versions of backstepping lead to unbounded coefficients in those transformations. We show how to design them such that they are sufficiently regular (not continuous but L² ). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE