An alpha derivative formulation of the Hamilton-Jacobi-Bellman equation Of Dynamic Programming

The time scales calculus, which includes the study of the alpha derivative, is an emerging key area in mathematics. We extend this calculus to approximate dynamic programming. In particular, we investigate application of the alpha derivative, one of the fundamental dynamic derivatives of time scales. We present a alpha-derivative based derivation and proof of the Hamilton-Jacobi-Bellman equation, the solution of which is the fundamental problem in the field of dynamic programming. By drawing together the calculus of time scales and the applied area of stochastic control via approximate dynamic programming, we connect two major fields of research.