Interval structure of Runge-Kutta methods for solving optimal control problems with uncertainties

In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique for interval arithmetic is introduced to evaluate the bounds of interval functions. The proposed approach is based on the new forward representation of Hukuhara interval differencing and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies to demonstrate the effectiveness of the proposed method.

Keywords:

Optimal control , Interval analysis , Lagrange multiplier method , Runge-Kutta methods , Hukuhara difference