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

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.