Evolutionary Anecdote of Optimal Control and its Application from Brachistochrone Problem

An evocative and nostalgic evolution of optimal control and its application via brachistochrone problem is presented by Euler-Lagrange equation through brachistochrone problem. The solution of brachistochrone problem by variational approach i.e. Euler-Lagrange equation leads to the curve of a cycloid also known as brachistochrone curve. In this paper the evolution of the optimal control law from the rubrics of variational calculus is presented. Eventually the spectacular and novel application of optimal control law is made for the control of unstable inverted pendulum. It is observed from the viewpoint of oscillations and overshoot of transient response that Linear Quadratic Regulator Optimal Controller (LQROC) outperforms the State Space Pole Placement Technique controller (PPTC).