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Bellman processes. (English)
Cohen, Guy (ed.) et al., 11th International conference on analysis and optimization of systems: discrete event systems. Sophia-Antipolis, France, June 15-17, 1994. Berlin: Springer-Verlag. Lect. Notes Control Inf. Sci. 199, 302-311 (1994).
The mapping $\lim_{ε\to 0} \log_ε$ defines a morphism between the asymptotics (around 0) of positive real functions of a positive real number and the semifield of real numbers endowed with the min and plus operations. In particular, large deviations send probability calculus into optimization problems. In this paper, optimization and optimal control problems are presented with a formalism analogous to that of probability theory where $+$ and $\times$ laws are replaced by min and $+$ laws. Probability measures correspond to minima of functions that we call cost measures, whereas random variables correspond to constraints on these optimization problems that we call decision variables. Expectation, Laplace transform, Markov chains (resp. processes) and Kolmogorov equations correspond to (optimal) value, Fenchel transform, discrete (resp. continuous) time optimal control problems, and Bellman equations. New results on optimal control equivalent to those of probability such as the law of large numbers and central limit theorems are stated. The Cramer transform is also considered. It leads to a morphism between probability measures endowed with the convolution product and cost measures with convex density endowed with the inf-convolution operation. More generally, it sends independent random variables into independent decision variables.
M.Akian (Le Chesnay)