2023/09/05 by Varga Κ. Kalantarov, Kalantarov, Varga, Türker Özsarı +3 · 1 citation
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Numerical methods for differential equations #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2309.02196
We introduce a finite dimensional version of backstepping controller design for stabilizing solutions of PDEs from boundary. Our controller uses only a finite number of Fourier modes of the state of solution, as opposed to the classical backstepping controller which uses all (infinitely many) modes. We apply our method to the reaction-diffusion equation, which serves only as a canonical example but the method is applicable also to other PDEs whose solutions can be decomposed into a slow finite-dimensional part and a fast tail, where the former dominates the evolution in large time. One of the main goals is to estimate the sufficient number of modes needed to stabilize the plant at a prescribed rate. In addition, we find the minimal number of modes that guarantee the stabilization at a certain (unprescribed) decay rate. Theoretical findings are supported with numerical solutions.