Micro-Macro Backstepping Control of Large-Scale Hyperbolic Systems (Extended Version)

Abstract

We introduce a control design and analysis framework for micro-macro, boundary control of large-scale, $n+m$ hyperbolic PDE systems. Specifically, we develop feedback laws for stabilization of hyperbolic systems at the micro level (i.e., of the large-scale system) that employ a) measurements obtained from the $n+m$ system (i.e., at micro level) and kernels constructed based on an $\infty+\infty$ continuum system counterpart (i.e., at macro level), or b) kernels and measurements both stemming from a continuum counterpart, or c) averaged-continuum kernels/measurements. We also address (d)) stabilization of the continuum (macro) system, employing continuum kernels and measurements. Towards addressing d) we derive in a constructive manner an $\infty+\infty$ continuum approximation of $n+m$ hyperbolic systems and establish that its solutions approximate, for large $n$ and $m$, the solutions of the $n+m$ system. We then construct a feedback law for stabilization of the $\infty+\infty$ system via introduction of a continuum-PDE backstepping transformation. We establish well-posedness of the resulting 4-D kernel equations and prove closed-loop stability via construction of a novel Lyapunov functional. Furthermore, under control configuration a) we establish that the closed-loop system is exponentially stable provided that $n$ and $m$ are large, by proving that the exact, stabilizing $n+m$ control kernels can be accurately approximated by the continuum kernels. While under control configurations b) and c), we establish closed-loop stability capitalizing on the established solutions' and kernels' approximation properties via employment of infinite-dimensional ISS arguments. We provide two numerical simulation examples to illustrate the effectiveness and potential limitations of our design approach.