On the Problems of Optimal and Minmax Type Control Under Vector Criteria

The topic of this paper is to solve two types of problems in controlled system dynamics formulated in terms of vector-valued criteria whose application depends on the type of ordering for the scalar participants of each such criterion. The first problem is that of optimal control under vector criterion with ordering of the Pareto type. The problem is to indicate the dynamics of the Pareto front. The second is that of finding vector-valued controls of the minmax type. Here the internal problem of dynamic maximization is due to a vector criterion with given type of ordering while the external problem is that of vector-valued dynamic minimization under another type of ordering. A similar situation arises for controls of the maxmin type. The paper indicates a variety of solution formulas that describe vector-valued dynamic interrelations for the problems of minmax and maxmin. The solutions are reached by using the Hamiltonian formalism. The suggested vector type of control problem settings are motivated by structure of system dynamics for physical motions, economics, finance, environmental models and related issues. Examples of applications are indicated. Copyright (C) 2020 The Authors.