# Infinite-Dimensional Linear Systems,
Optimal Control and Algebraic Riccati Equations

**Author:**
Kalle M. Mikkola

**Email:**
Kalle.Mikkola@hut.fi

**Contact information**
ISBN 951-22-6153-7
ISSN 0784-3143

**Published:**
Helsinki University of Technology Institute of Mathematics Research Reports #A452
Espoo, 2002
**AMS subject classification:**
42A45, 46E40, 46G12, 47A68,
49J27, 49N10, 49N35, 93-02,
93A10, 93B36, 93B52, 93C05,
93C55, 93D15

**Keywords:**
suboptimal H-infinity control,
standard H-infinity problem,
measurement feedback problem;
H-infinity full information control problem,
state feedback problem;
Nehari problem;
LQC, LQR control, H2 problem, minimization;
bounded real lemma, positive real lemma;
dynamic stabilization, controller with internal loop,
strong stabilization, exponential stabilization, optimizability;
canonical factorization, (J,S)-spectral factorization,
(J,S)-inner coprime factorization, generalized factorization,
J-lossless factorization;
compatible operators, weakly regular well-posed linear systems,
distributed parameter systems;
continuous-time, discrete-time;
infinite-horizon;
time-invariant operators, Toeplitz operators, Wiener class,
equally-spaced delays, Popov function;
transfer functions, H-infinity boundary functions,
Fourier multipliers;
Bochner integral, strongly measurable functions,
strong Lp spaces; Laplace transform, Fourier transform

## Abstract

In this monograph, we solve rather general
linear, infinite-dimensional, time-invariant control problems,
including the H-infinity and LQR problems,
in terms of algebraic Riccati equations
and of spectral or coprime factorizations.
We work in the class of (weakly regular)
well-posed linear systems (WPLSs)
in the sense of G. Weiss and D. Salamon.
Moreover, we develop the required theories,
also of independent interest,
on WPLSs, time-invariant operators,
transfer and boundary functions,
factorizations and Riccati equations.
Finally, we present the corresponding theories and results
also for discrete-time systems.

### Dissertation

Dissertation for the degree of Doctor of Science in Technology
to be presented with due permission of
the Department of Engineering Physics and Mathematics,
for public examination and debate
in Council Room at Helsinki University of Technology
(Espoo, Finland) on the 18th of October, 2002, at 12 o'clock noon
## Preface