$L_2(\Sigma)$-regularity of the boundary to boundary operator $B^{*}L$ for hyperbolic and Petrowski PDEs

Abstract

This paper takes up and thoroughly analyzes a technical mathematical issue in PDE theory, while—as a by-pass product—making a larger case. The technical issue is the $L_2\left(\Sigma \right)$-regularity of the boundary $\to$ boundary operator $B^{\ast }L$ for (multidimensional) hyperbolic and Petrowski-type mixed PDEs problems, where $L$ is the boundary input $\to$ interior solution operator and $B$ is the control operator from the boundary. Both positive and negative classes of distinctive PDE illustrations are exhibited and proved. The larger case to be made is that hard analysis PDE energy methods are the tools of the trade—not soft analysis methods. This holds true not only to analyze $B^{\ast }L$, but also to establish three inter-related cardinal results: optimal PDE regularity, exact controllability, and uniform stabilization. Thus, the paper takes a critical view on a spate of “abstract” results in “infinite-dimensional systems theory,” generated by unnecessarily complicated and highly limited “soft” methods, with no apparent awareness of the high degree of restriction of the abstract assumptions made—far from necessary—as well as on how to verify them in the case of multidimensional dynamical systems such as PDEs.