From the Publisher:
This book presents the newly developed theory of non-harmonic Fourier series and its applications to the control of distributed parameter systems The book begins with the modern theory of exponentials, using an operator theory approach, then extends this to the method of moments--one of the most powerful tools in control theory Applications to various types of control problems are discussed Researchers in control theory, functional analysis, and partial differential equations will find much to interest them in this treatise

Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems

This paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods. The main focus is on the property of observability, corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary. The mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion, as is often desired in engineering applications. Observability/controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper. However, when the wave equation is discretized by finite difference methods, the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem. In other words, the classical convergence (consistency + stability) property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high-frequency spurious solutions for which the group velocity vanishes. This phenomenon is analyzed and several remedies are suggested, including filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. 
We also briefly discuss these issues for the heat, beam, and Schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability/controllability properties at the discrete level. We conclude with a list of open problems and future subjects for research.

http://benasque.org/benasque/2005pde/2005pde-talks/023sirev.pdf

Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods

In this chapter we present some of the recent progresses done on the problem of controllability of partial differential equations (PDE). Control problems for PDE arise in many different contexts and ways. A prototypical problem is that of controllability. Roughly speaking it consists in analyzing whether the solution of the PDE can be driven to a given final target by means of a control applied on the boundary or on a subdomain of the domain in which the equation evolves. In an appropriate functional setting this problem is equivalent to that of observability which concerns the possibility of recovering full estimates on the solutions of the uncontrolled adjoint system in terms of partial measurements done on the control region. Observability/controllability properties depend in a very sensitive way on the class of PDE under consideration. In particular, heat and wave equations behave in a significantly different way, because of their different behavior with respect to time reversal. In this paper we first recall the known basic controllability properties of the wave and heat equations emphasizing how their different nature affects their main controllability properties. We also recall the main tools to analyze these problems: the so-called Hilbert uniqueness method (HUM), multipliers, microlocal analysis and Carleman inequalities. We then discuss some more recent developments concerning equations with low regularity coefficients, equations with potentials, bang-bang controls, etc. We also analyze the way control and observability properties depend on the norm and regularity of these coefficients, a problem which is also relevant when addressing nonlinear models. We then present some recent results on coupled models of wave–heat equations arising in fluid–structure interaction. We also present some open problems and future directions of research.

Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

In this review, concerning parabolic equations, we give self-contained descriptions on derivations of Carleman estimates; methods for applications of the Carleman estimates to estimates of solutions and to inverse problems.Moreover limited to parabolic equations, we survey the previous and recent results in view of the applicability of the Carleman estimate. We do not intend to pursue any general treatments of the Carleman estimate itself but by showing it in a direct manner, we mainly aim to demonstrate the applicability of the Carleman estimate to the estimation of solutions and inverse problems.

https://iopscience.iop.org/article/10.1088/0266-5611/25/12/123013/pdf

Carleman estimates for parabolic equations and applications

Linear operators part ii (spectral theory)

: In the report the authors study harmonic properties of sequences (e - (lambda sub k)t) of real exponential functions. Linear independence results, including estimates on the norms of biorthogonal functions, are obtained in the space (L sup 2) over the interval = or > 0, but 0. The results are uniform in that they depend only upon certain separation requirements on the (lambda sub k) rather than upon the individual sequence (lambda sub k). The results are used to study the boundary value controllability of the heat equation in the unit ball of (R sup n). (Author)

/pdf/uniform-bounds-on-biorthogonal-functions-for-real-1jssvyf10m.pdf

D. L. Russell

Papers

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations