Functional equations in dynamic programming

where p and q represent the state and decision vectors, respectively, T represents the transformation of the process, and f(p) represents the optimal return function with initial state p. This functional equation can be studied in several ways, either with respect to the type of processes giving rise to (1), or with respect to the precise form of (1), or with respect to the computational aspects of (1). In this survey article, this function will be treated according to the different types of processes. In addition to the optimization problems in dynamic programming as shown in (1), the dynamic programming concept can also be used to solve various types of boundary value problems arising in engineering and physical sciences. The dynamic programming concept without optimization is known as invariant imbedding. The resulting functional equation of invariant imbedding is very similar to (1) except for the absence of the maximization operation. Dynamic programming involves a completely different approach to formulating the problem: Instead of only considering a single problem with a fixed duration, the dynamic programming approach is to colasider a family of problems, with duration of the process ranging from zero to the duration of the original problem. In order to consider these different duration processes, the corresponding initial conditions for these processes must also be calculated and interpolated