Beyond Bellman's principle of optimality; the principle of "real-time optimality" (RTO)

Bellman's principle of optimality and his dynamic programming technique for computing optimal sequential-decisions may not apply to problems involving uncertain, non-noisy exogenous-variables. In this paper, we show that if the uncertain behavior of non-noisy exogenous-variables can be modeled by a class of spline-expressions, with known basis-functions and unknown, "stepwise-constant" weighting-coefficients, one can introduce a pseudo state-vector and a generalization of the principle of optimality, called real-time optimality, which enables rational, real-time sequential-decisions that, are "optimal" in a certain practical, causal sense. The results have wide applications in business, engineering, defense, competitive sports, etc.