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 .
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
 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.