Cooperative solutions in multi-person quadratic decision problems: Performance-measure statistics and cost-cumulant control paradigm

In cost-cumulant control for the class of multi-person single-objective decision problems characterized by quadratic random costs and state-feedback information structures, individual decision makers share state information with their neighbors and then autonomously determine decision strategies to achieve the desired goal of the group which is a minimization of a finite linear combination of the first k cumulants of a finite-horizon integral-quadratic performance-measure associated with a linear stochastic system. Since this problem formulation is parameterized by the number of cost cumulants, the scalar coefficients in the linear combination and the group of decision makers, it may be viewed both as a generalization of linear-quadratic Gaussian control, when the first cumulant is minimized by a single decision maker and of the problem class of linear-quadratic identical-goal stochastic games when the first cumulant is minimized by multiple decision makers. Using a more direct dynamic programming approach to the resultant initial-cost problem, it is shown that the decision laws associated with multiple persons are linear and are found as the unique solutions of the set of coupled differential matrix Riccati equations, whose solvability guarantees the existence of the closed-loop feedback decision laws for the corresponding multi-person single-objective decision problem.