Stabilization of controllable systems: some new insights and an application to the water tank

Abstract

In this talk, we show how the ideas of backstepping for PDEs can be understood in the framework of system equivalence, which allows us to relate this popular stabilization method to other well-known stabilization methods such as the Gramian method (and Linear Quadratic Regulation) and pole-shifting. The backstepping method has yielded explicit feedback laws to stabilize many different types of PDEs, which can then be used to achieve null controllability with a constructive control, or to stabilize nonlinear systems.We then focus on a variant of backstepping for internal distributed controls, which uses a Fredholm-like transformation instead of the ‘usual’ Volterra transformation of the second kind. We will illustrate it on the stabilization of a 1-D water tank. It can be shown, using a moments method with some sharp estimates, that the linearized systems around non-uniform steady states are controllable in Sobolev spaces (up to conservation of mass). We use this partial controllability result to construct exponentially stabilizing feedbacks for the linearized water-tank system around non-uniform steady states. This shows that the backstepping method can be adapted to more complex hyperbolic systems, despite the additional difficulties due to the coupling terms, and the conservation of mass.