Centralized and decentralized solutions of the linear-exponential-Gaussian problem

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Abstract
A particular class of stochastic control problems constrained to different information patterns is considered. This class consists of minimizing the expectation of an exponential cost criterion with quadratic argument subject to a discrete-time Gauss-Markov dynamic system, i.e., the linear-exponential-Gaussian (LEG) control problem. Besides the one-step delayed information pattern previously considered, the classical (includes current observations) and the one-step delayed information-sharing (OSDIS) patterns are assumed, After determining the centralized controller based upon the classical information pattern, the optimal decentralized controller based upon the OSDIS pattern and the solution to a static team problem is found to be affine. A unifying approach to determine controllers based upon these three information patterns is obtained by noting that the value of a quadratic exponent of an exponential function is independent of the information structure. Even though the controllers are determined by a backward recursion of this exponent, the value of the cost criterion is not; rather a coefficient of the exponential delineates the value of the cost criterion with respect to the information patterns. Both necessary and sufficient conditions for the controllers to be minimizing are obtained regardless of the exponential form. The negative exponential form is included which is unimodal but not convex.

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