Introduction

Abstract

AbstractIn this introductory section we introduce the reader to the main topic of the book, dealing with the so-called Min-Max control problem, where the dynamics of the considered system is described by a first-order vector ordinary differential equation with the right-hand side depending on a parameter that may run over a given parametric set (finite or compact). The performance index is given in the joint (Bolza) form containing the terminal term as well as an integral one defined both on a finite or an infinite horizon. The considered problem consists of the developing of an admissible control law providing the minimum value of the performance index for the worst parameter selection on the plant dynamics. In fact, this Min-Max problem is an optimization problem in a Banach (infinite-dimensional) space. In this section we consider first a Min-Max problem in a finite-dimensional Euclidean space, in order to understand which specific features of a Min-Max solutions arise, what we may expect from their expansion to infinite-dimensional Min-Max problems and to verify whether these properties remain valid or not. We show that two main properties of the solution hold. The joint Hamiltonian (the negative Lagrangian) of the initial optimization problem is equal to the sum (integral) of the individual Hamiltonians calculated over the given parametric set. In the optimal point all loss functions, corresponding to “active indices” (for which the Lagrange multipliers are strictly positive), turn out to be equal. The main question arising here is: “Do these two principal properties, formulated for finite-dimensional Min-Max problems, remain valid for the infinite-dimensional case, formulated in a Banach space for a Min-Max optimal control problem?” The answer is: YES, they do! A detailed justification of this positive answer forms the main contribution of this manuscript.KeywordsFinite-dimensional Euclidean SpaceInitial Optimization ProblemTerminal TermDetailed JustificationSolution HoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.