Minimax terminal linear control of a linear dynamic system with discrete time when there is incomplete information on the perturbing processes (Q1921527)

The authors study the linear optimal control problem when the dynamic of the system is described by a linear equation of the form \[ X(n+1) = LX(n) + BU(n) + FV(n) + \xi (n+1), \quad X (n_0) = X_0, \] where \(\xi(n)\) is the standard discrete white noise, \((n= n_0, \dots, n_k)\), \(X_0\) is the initial random vector, \(V(n)\) is a scalar controlling random process, \(L,B,F\) are given matrices and \(U(n)\) is a vector perturbing process. The vector \(U(n)\) is not given and there is partial information available on it such as constraints on the variances and conditions for mutual uncorrelatedness. The process \(U(n)\) appears also in the observation. The cost criterion is a terminal one and of unit rank. The above problem is considered as one in game theory. Necessary and sufficient conditions for optimality for a saddle point of the related game are found and an iterative game process for obtaining a solution to the problem is proposed.