Mathematical problems of control theory. An introduction (Q2771729)

This volume presents introductory and research aspects of control theory, embedding the development in applied situations. Often books start with applications as motivation for the theory, which constitutes the rest of the text. Here one finds concrete problems throughout the volume. This makes sense as various topics studied in control theory stem from such problems, and the field can be fully appreciated when one keeps in mind its origins in applications.NEWLINENEWLINENEWLINEAnother characteristic is the use of the qualitative theory of differential equations and analytical methods.NEWLINENEWLINENEWLINEThe first chapter presents issues in stability, and the Watt governor is introduced. A study of its transients is performed.NEWLINENEWLINENEWLINEIn chapter two, linear circuits motivate the frequency response description of input output linear plants. The transfer function is obtained using the tool of the Laplace transform.NEWLINENEWLINENEWLINEChapter three shows fundamental properties of linear systems (in the state space description as well as in the frequency domain): controllability, observability and the Nyquist stability criterion. The normal forms of Brunovsky and Kalman are included. Then, one finds a study of nineteen pages on a problem proposed by \textit{R. Brockett} [Open problems in mathematical systems and control theory (1999; Zbl 0945.93005)]. The subject is the existence of periodic gains for the stabilization of linear systems. The gains are supposed to take two constant values on two corresponding subintervals of the time axis. It is supposed that there are associated stable spaces and invariant spaces (the latter potentially unstable). One supposes moreover that between the subintervals, there exists a time-varying gain such that the associated fundamental matrix sends the first potentially unstable space into the second stable space. Under a condition on associated eigenvalues, one obtains the required existence of the stabilization gain. Other more special cases are studied. Let us observe that since the conditions are sufficient, a conservatism is introduced, and that moreover the natural controllability assumption is dissimulated in the conditions. Let us mention that the problem has been treated in a constructive and systematic manner in the reviewer's thesis [Periodic feedback for linear systems and optimal control of bilinear systems (1999; Zbl 0945.93002)].NEWLINENEWLINENEWLINEChapter four focuses on two-dimensional systems that can be easily visualized with phase portraits. The spacecraft, electric machine, synchronization and population dynamics examples are studied.NEWLINENEWLINENEWLINEChapter five is a pendant on discrete systems with its own examples.NEWLINENEWLINENEWLINEChapter six presents a proof of the Popov theorem, which differs from the one found by \textit{M. Vidyasagar} [Nonlinear systems analysis (1978)], where it was based on the Kalman-Yakubovich lemma. Here, the proof uses the fact that the Fourier transformation is a unitary operator.NEWLINENEWLINENEWLINEThe reviewer is sympathetic to the pedagogical standpoint found in this book. Rather than a heavy and precisely drawn course where the construction of the exposition remains opaque (and the intelligent reader will use time in decomposing and reconstructing the theory), one has here a light introduction proposing paths that bring together motivating applications, important known facts and methods as well as open problems. Instead of imposing a point of view, the reader will build a working knowledge, and his creativity will be stimulated so that he can develop his own perspective.