Well-posed linear systems – a survey with emphasis on conservative systems. (English) Zbl 0990.93046

Weiss, George; Staffans, Olof J.; Tucsnak, Marius

Int. J. Appl. Math. Comput. Sci. 11, No. 1, 7-33 (2001).

The paper under review is, as it is seen from its title, a survey of the literature on well-posed linear systems. The notion of such systems was introduced simultaneously by Yu. L. Shmulyan [Invariant subspaces of semigroups and Lax–Phillips scheme, Dep. VINITI, No. 8009-1386, Odessa, 1986, 49 pp. (Russian)] and D. Salamon [Trans. Am. Math. Soc. 300, 383-431 (1987; Zbl 0623.93040)]. Unfortunately, Shmulyan’s paper was not translated from the Russian and until now is not known to general Western readers. Note also that some significant parts of this theory are found already in the paper of J. W. Helton [Proc. IEEE 64, 145-160 (1976)].
Briefly, a well-posed linear system is a time-invariant linear system \(\Sigma\) such that on any finite time interval \([0,\tau ]\) the operator \(\Sigma_\tau\) mapping the pair consisting of the initial state \(x(0)\) and the input function \(u\) to the pair consisting of the final state \(x(\tau)\) and the output function \(y\) is bounded. The input space \(U\), the state space \(X\), and the output space \(Y\) are Hilbert spaces, and the input and output functions are of class \(L^2_{\text{loc}}\). For any \(u\in L^2_{\text{loc}}([0,\infty);U)\) and any \(\tau\geq 0\) denote by \(P_{\tau}u\) its truncation to the interval \([0,\tau ]\). Then the well-posed system \(\Sigma\) consists of the family of bounded operators \(\Sigma =(\Sigma_\tau)_{\tau\geq 0}\) such that \(\operatorname {col}(x(\tau),P_{\tau}y)=\Sigma_{\tau}\operatorname {col}(x(0),P_{\tau}u)\).
Some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators are described. Connections with scattering theory of P. Lax and R. Phillips are considered. In this relation, let us mention again the contribution of Yu. L. Shmulyan [op. cit.], and the paper of D. Arov and M. Nudelman [Integral Equ. Oper. Theory 24, 1-45 (1996; Zbl 0838.47004)] based on it, that studies passive systems, i.e., such well-posed systems for which the operators \(\Sigma_\tau :X\times L^2([0,\tau ];U)\rightarrow X\times L^2([0,\tau ];Y)\) are contractive for all \(\tau\geq 0\).
A special case of passive systems are conservative systems, i.e., well-posed systems with the operators \(\Sigma_\tau\) being unitary. This particular subclass of well-posed systems is examined in the paper under review. The authors describe a simple way to generate conservative systems via a second-order differential equation with damping in a Hilbert space. Also, they give an example of a well-posed linear system modeling a Rayleigh beam with piezoelectric actuator, and obtain results on stability, controllability and observability of the conservative system, which they get from the original system after adding a damping term to its corresponding second-order differential equation.

Reviewer: Dmitriy Kalyuzhniy-Verbovetzky (Odessa)

MSC:

93C05	Linear systems in control theory
93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
93D20	Asymptotic stability in control theory
93B07	Observability
93C25	Control/observation systems in abstract spaces
93B05	Controllability

Keywords:

well-posed linear system; operator semigroup; conservative system; scattering theory; time-flow inversion; second-order damped differential equations in Hilbert space; beam equation; passive systems; stability; controllability; observability

Citations:

Zbl 0623.93040; Zbl 0838.47004