On non-improvability of full-memory strategies in problems of optimization of the guaranteed result

TL;DR

This paper proves that in certain control systems with disturbances, full-memory strategies cannot be improved upon and constructs an approximately optimal strategy, supported by an example.

Contribution

It demonstrates the non-improvability of full-memory strategies in guaranteed result optimization under specific disturbance conditions.

Findings

Optimal guaranteed result matches that of quasi-strategies.

Constructed an ε-optimal full-memory strategy.

Provided an illustrative nonlinear example.

Abstract

The paper addresses the problem of optimization of a guaranteed (worst case) result for a control system driven by a controlling side in presence of a dynamical disturbance. The disturbances as functions of time are subject to functional constraints belonging to a given family of constraints. The latter family is known to the controlling side that does not observe the disturbance and uses full-memory strategies to form the control actions. The study is focused on the case where disturbance varies in open-loop disturbances chosen in advance and the case where the disturbances are restricted to a $L_2$--compact set fixed in advance but unknown to the controlling side. In these cases it is shown that the optimal guaranteed result is non-improvable in the sense that it coincides with that obtained in the class of quasi-strategies -- nonantisipatory transformations of disturbances into controls. An $\varepsilon$--optimal full-memory strategy is constructed. An illustrative nonlinear example is given.