Dynamic programming for noncausal problems

The theory of dynamic programming as used in most control applications relies heavily on the causal structure of the underlying dynamics. In this survey paper, we will show noncausal problems, such as those arising in partial differential equations, multidimensional smoothing and filtering, and digital image processing can often be converted into causal problems by imposing a causal vector structure on the process. The resulting problem can then be handled as a vector multistage decision process using dynamic programming. We will present this approach for some simple two-dimensional deterministic problems. Applications to the numerical solution of partial differential equations and image restoration will be shown. Although this paper covers a body of theory and applications which have been developed over the past decade, the approach presented here is considerably more unified than appears elsewhere.