Solution of the Hamilton jacobi bellman uncertainties by the interval version of adomian decomposition method

Optimal control is the policy of getting the optimized control value for minimizing a predefined cost function. Recently, several optimization methods have been introduced for achieving this purpose.1–4 Among these methods, Pontryagins maximum principle5 and the Hamilton Jacobi Bellman equation6 are the most popular. In Pontryagins maximum principle, the optimal control problem will be converted to an ODE problem while the Hamilton Jacobi Bellman method converts the optimal control into a nonlinear partial differential equation. Hamilton Jacobi Bellman method can be utilized in different linear, non-linear and even distributed optimal control problems. Be-cause of difficulty in solving the HJB and the Pontryagins maximum principal method, it is usually necessary to employ the numerical methods to achieve the optimal solutions for the nonlinear practical models. One of the popular semi-numerical methods which is frequently used in the recent years is the Adomian decomposition method.7 Adomian decomposition method and its modifications have been efficiently employed to solve the ordinary and partial differential equations.8–10 Because the method uses no linearization or smallness assumptions in solving the differential equations, it has been an effective method among the other techniques. Generally, in designing the optimal control problems for engineering and practical applications the parameters considered as deterministic, but there is a great deal of uncertain parameters which can greatly affect the system performance. Such an uncertainty can be made by model simplification, manufacture error, design tolerance etc. If the number of the system uncertainties becomes very small, deterministic methods can be employed to solve these problems with little errors. If the number of uncertain parameters has been increased or the ranges of these parameters have become large, the deterministic methods might give the wrong answer. Three different methods have been introduced for solving these uncertain problems: Probabilistic methods, fuzzy methods and interval methods.11 The main purpose of this paper is to introduce an interval version of Adomian decomposition method to solve the Hamilton Jacobi Bellman equation. The proposed method is first applied on a linear optimal control with uncertainties. After that, it is utilized to solve a nonlinear optimal control with uncertain parameters. Finally, a practical distributed system is solved by the proposed method; in this problem, in addition to assuming the presence of uncertainty in parameters, the initial conditions are also considered with uncertainty.