ABSTRACT
Time delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. More than fifty years ago, N. Minorski in his book Nonlinear Oscillations considered the problem of stabilizing the rolling of a ship by the activated tanks method in which ballast water is pumped from one position to another. Many interesting applied models which described by second order delay differential equations have been considered recently.
Two basic proportional (adaptive) control models are widely used: a standard feedback controller u(t) = K(x(t) − x*) with the controlling force proportional to the deviation of the system from the attractor, where x* is an equilibrium of the equation, and the delayed feedback controller u(t) = K(x(t − *) − x(t)). A proportional control fails if there exist rapid changes to the system that come from an external source, and to keep the system steady under an abrupt change, a derivative control was used. The purpose of the present chapter is to design a universal controller for second order nonlinear equations by combining these two types. A wide class of unstable second order equations is stabilized by applying a damping control u(t) = K(Λ 1 x(t)+Λ 2 x2(t)) and an adaptive delayed feedback controller. The results are based on the stability tests obtained in previous chapters of this book.
To illustrate the application of the results obtained in the chapter, stabilization of the sunflower equation was considered at the end of the chapter.
