Fractal functions with continuous Weil-type derivatives of variable order in control of distributed systems

Abstract

Properties of fractal functions which are not differentiable in the classical sense but have continuous Weil-type derivatives of variable order at each point are studied. It is shown that the Weierstrass, Takagi, and Besicovitch classical fractal functions have such derivatives. An example of an oscillatory system controlling which requires constructing a fractal control function having a Weil-type derivative of variable order at each point is considered.