Augmented Lagrangian Method For Discretized Optimal Control Problems

Abstract

In this paper, we are concerned with one-dimensional time invariant optimal control problem, whose objective function is quadratic and the dynamical system is a differential equation with initial condition .Since most real life problems are nonlinear and their analytical solutions are not readily available, we resolve to approximate solutions. Our aim is to develop a numerical scheme to solve one dimensional optimal control problem. A discretization of the performance index using trapezoidal rule and the state equation using crank-Nicholson is adopted. By parameter optimization, this results into a sparse non-linear programming problem. With the aid of Augmented Lagrangian method, a quadratic function with a control operator (penalized matrix) amenable to conjugate gradient method is generated. Numerical experiments verify the efficiency of the proposed technique which compares much more favourably to the existing scheme.

Keywords: Trapezoidal rule, Euler’s method, Augmented Lagrangian method, and Conjugate gradient method.