2021/04/10 by Simon Kerschbaum, Kerschbaum, Simon, Joachim Deutscher +1
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2104.04854
openalex publication_date 2021/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper considers the backstepping state feedback control of coupled\nlinear parabolic PDEs with spatially varying coefficients and bilateral\nactuation. By making use of the folding technique, a system representation with\nunilateral actuation is obtained, allowing to apply the standard backstepping\ntransformation. To ensure the regularity of the solution, the folded system is\nsubject to unusual folding boundary conditions, which lead to additional\nboundary couplings between the PDEs. Therefore, the solution of the\ncorresponding kernel equations determining the transformations is a very\nchallenging problem. A systematic approach to derive the corresponding integral\nequations is proposed, allowing to solve them with the method of successive\napproximations. By making use of a Volterra and a Volterra-Fredholm\ntransformation, the closed-loop system is mapped into a cascade of stable\nparabolic systems. This allows a simple proof of exponential stability in the\nL2-norm with the decay rate as design parameter. The bilateral state\nfeedback stabilization of an unstable system of two coupled parabolic PDEs and\nthe comparison to the application of an unilateral controller demonstrates the\nresults of the paper.\n