A New Type of Distributed Parameter Control Systems: Two-Point Boundary Value Problems for Infinite-Dimensional Dynamical Systems

The mathematical theory of dynamical systems is based on the famous H. Poincaré’s qualitative theory on ordinary differential equations; the works of A. M. Lyapunov and A. A. Andronov also play an essential role in its development. At present, the theory of dynamical systems is an intensively developing branch of modern mathematics, which is closely connected to the theory of ordinary differential equations. Infinite-dimensional dynamical systems generated by nonlinear evolutionary partial differential equations have been rapidly developing in recent years. In this survey note, we will describe a new type of distributed parameter control systems generated by hyperbolic systems of partial differential equations of second order—the two-point boundary value problems. This kind of problems plays an important role in control theory,mathematics,mechanics, physics, and in other areas of sciences and technology. It is well known that there are many deep and beautiful results on the TBVPs for ordinary differential equations of second order. However, according to the authors’ knowledge, only a few of the results on the TBVPs for hyperbolic equations or other nonlinear evolutionary partial differential equations (even for linear or nonlinear wave equations) have been known. In 2010, the first author introduced the TBVPs for linear wave equations (see page 121 of [1]). For simplicity, we consider the following wave equation: