Risk minimization under functional constraints on the dynamic disturbance

In this review the application of the Niehans–Savage criterion to control problems under dynamic disturbances is discussed: motivation and formulation of the risk minimizing problem are given; direct relations for the results in different classes of disturbance constraints and solving strategies are provided; the examples of solving process for various problems with this control criteria are given; the results obtained by using the Niehans–Savage criterion are compared with the results based on the classic minimax criterion; the conditions of unimprovability of the strategies with full memory are studied; the optimal risk function as a limit of iterative program construct for the functional of regret is presented; the regularity condition for this functional is given; some additional conditions on the control system to ensure the possibility of numerical implementation of the risk-optimal strategy are considered.