The modulating function method for state estimation and feedback of infinite-dimensional systems

TL;DR

This paper extends the modulating function method to infinite-dimensional systems, enabling effective state reconstruction and feedback for PDEs with boundary control, including unbounded operators and distributional solutions.

Contribution

It introduces a novel extension of the modulating function approach to infinite-dimensional systems, facilitating state estimation and feedback in PDEs with boundary and point evaluations.

Findings

Effective state reconstruction via convolution with null controls

Handling unbounded feedback operators with distributional solutions

Enabling pointwise feedback in PDE boundary control

Abstract

We investigate state feedback and observation for infinite-dimensional linear systems, including a variety of partial differential equations with boundary control and observation. We extend the modulating function approach to infinite-dimensional systems. This approach, simply put, involves reconstructing part of the state by convolving with null controls of the adjoint system. We show how this method aids in state reconstruction, and we also examine distributional solutions of the adjoint system, showing their ability to handle unbounded feedback operators. This enables us to use feedback from spatial point evaluations in partial differential equations.