Brief Overview of Systems of Second Order: Design of Optimal Feedback Control

We consider interesting in the control theory problem of design of an optimal feedback control. In other words,
we want to design optimal control as function of the state (phase) coordinates [1,2]. This problem is important
one, but at the same time is difficult and not always may be solved. The nonlinear autonomous system of second
order in general form is considered here. The restrictions imposed on the control input can depend on the state
coordinates of the system. The goal of the control is to maximize or minimize one phase coordinate of our
system while other coordinate takes a prescribed in advance value. Optimal control problems for the systems of
second order considered in the literature most frequently are associated with driving both phase coordinates to a
prescribed in advance state. With our statement of the problem, an optimal control can be designed as function
of the state coordinates (as feedback control) for more general kind of the systems.
As an example, we have explicitly (analytically) solved the problem of maximizing or minimizing the amplitude
of the swing oscillations.