In this section, we will apply the Galerkin finite element method [36, 37] to discretize the original problem (TOCP), an infinite-dimensional distributed parameter time optimal control problem, into a semidiscrete

approximation problem governed by a finite-dimensional lumped parameter system.

the function

approximation problem was chosen as an aim of the experiment.

By (2), we know that the

approximation problem of f by polynomials on a domain [OMEGA] is reduced to the well-known

approximation problem of its smooth extension F by polynomials on the square T [4, 10].

This method involves approximating the control function by a piecewise-constant function with possible discontinuities at a set of preassigned switching points, which produces an

approximation problem such that the solution of this approximation is a suboptimal solution to problem ([OCP.sup.h]).

where [phi](x; [a.sub.0],..., [a.sub.n]) is usually a polynomial [P.sub.n](x) of degree at most n, and the

approximation problem can be represented to minimize the error (E):

This problem is so-called the optimal

approximation problem with respect to matrix equation (2) (see e.g., [35,8,11-17,24]).

The function

approximation problem can be stated formally as follows [5].

Special topics include simple C*-algebras, approximation properties for groups, the weak expectation property and local lifting property, weakly exact von Neumann algebras, and such applications as Herrero's

approximation problem and classification of von Neumann algebras.

The weighted

approximation problem is to find a matrix B [element of] [R.sup.m x n] that solves

[34] investigated a rational

approximation problem in connection with the convergence analysis of the ADI iterative method applied to the Stein matrix equation.

Zhang, "Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal

approximation problem," Applied Mathematics and Computation, vol.