Narrow Results

In this paper, we make research on the approximation of functions with fractal dimension in continuous functions space. We first investigate fractal dimension of the linear combination of continuous functions with different fractal dimensions. Then, fractal winding of continuous functions has been given. Furthermore, based on Weierstrass theorem and Weierstrass function, we establish a theorem that continuous functions with the non-integer fractal dimension can be approximated by the linear combination of trigonometric polynomials with the same fractal dimension. Condition about continuous functions with fractal dimension one and two has also been discussed elementary.

A distributed control problem for cooperative parabolic systems governed by Schrödinger operator is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii-Milyutin Theorem given by Walczak (1984, On some control problems Acta Univ. Lod. Folia Math., 1, 187-196), the optimality conditions are derived for the Neumann problem. Finally, several mathematical examples for derived optimality conditions are presented.