An improvement to the homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equation

In this paper, the piecewise homotopy perturbation method (PHPM) is employed to solve the Hamilton-Jacobi-Bellman (HJB) equation arising in the optimal control problems. The method is a simple modification of the standard homotopy perturbation method (HPM), in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding HJB equation. Applying the PHPM with He's polynomials reveals that the modified homotopy perturbation is more impressive than the standard HPM. Also, the convergence of the algorithm is discussed in detail. The comparison of the PHPM results with the standard HPM, exact solution, Modal series, multiwavelets spectral method, differential transformations and the measure theory method is made. Simulation examples are employed to test the validity of the PHPM.