Optimal Impulsive Control by Aram Arutyunov & Dmitry Karamzin & Fernando Lobo Pereira

Author:Aram Arutyunov & Dmitry Karamzin & Fernando Lobo Pereira
Language: eng
Format: epub
ISBN: 9783030022600
Publisher: Springer International Publishing

4.1 Introduction

The investigation of unbounded nonlinear control systems encounters certain difficulties, which stem from the fact that the passage to the weak-* limit in the right hand of the dynamic system is not feasible. This gives rise to the following phenomenon: The space of Borel vector-valued measures does not include the set of all feasible, or achievable, impulsive controls (that is, those controls which are implemented in practice when approximating a given Borel control vector-valued measure by a sequence of absolutely continuous vector-valued measures). Thus, in order to obtain a proper or well-posed statement of the problem, we are forced to extend the notion of a feasible impulsive control as well as the notion of a feasible trajectory. This is the basic idea for the forthcoming considerations.

There is still a requirement for the control system to be affine w.r.t. the unbounded control parameter v in this chapter, despite the use of the term “nonlinear.” At the same time, the control system is essentially nonlinear w.r.t. state position x. Thus, the objective is now the extension of System (0.5) presented in the Introduction. In spite of increasing complexity, there exist various applications for this particular case, for example, in economics. The model of optimizing the costs of advertisement for two goods has the following form (Vidale-Wolfe model [10]):

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