An approximation theory for the identification of nonlinear distributed parameter systems. (English) Zbl 0722.47058

Banks, H. T.; Reich, Simeon; Rosen, I. G.

SIAM J. Control Optimization 28, No. 3, 552-569 (1990).

From the authors’ text: “We have developed a general abstract approximation framework for the identification of nonlinear distributed parameter evolution systems. The class of systems to which our theory applies are those whose dynamics can be described by a nonlinear operator that satisfies conditions that are the natural nonlinear extensions, or analogues, of the properties of regularly dissipative, or abstract parabolic, linear operators. The approach we have taken is based on the defining of a sequence of approximating finite-dimensional identification problems in which the systems to be identified are Galerkin approximations to the original, underlying, infinite-dimensional nonlinear dynamics. Under a weak continuity assumption with respect to the known parameters to be identified, equiboundedness and equimonotonicity conditions, and an approximation assumption on the Galerkin subspaces (all of which are readily verified for wide classes of nonlinear distributed systems and finite-element subspaces), we are able to demonstrate that solutions to the approximating problems exist, and, in some sense, approximate (i.e., subsequential convergence) solutions to the original infinite-dimensional identification problem. We have shown that the linear theory presented in [H. T. Banks and K. Itô, Appl. Math. Lett. 1, 13-17 (1988); Control-Theory Adv. Tech. 4, 73-90 (1988)] is a special case of our nonlinear framework and that our results are applicable to a reasonably wide class of nonlinear elliptic operators and corresponding nonlinear parabolic partial differential equations.”
Examples describing some classes of systems to which the general framework developed in this well-written paper applies are given.

Reviewer: S.L.Singh (Rishikesh)

MSC:

47H20	Semigroups of nonlinear operators
93C20	Control/observation systems governed by partial differential equations
93C10	Nonlinear systems in control theory
65J15	Numerical solutions to equations with nonlinear operators

Keywords:

nonlinear heat conduction; abstract approximation framework; identification of nonlinear distributed parameter evolution systems; regularly dissipative; abstract parabolic; approximating finite- dimensional identification problems; Galerkin approximations; equiboundedness; equimonotonicity conditions; Galerkin subspaces; finite- element subspaces; nonlinear elliptic operators; nonlinear parabolic partial differential equations