On the “bang-bang” principle for nonlinear evolution inclusions

Abstract

Summao'. In this paper we establish the existence of extremal solutions for a class of nonlinear evolution inclusions defined on an evolution triple of Hilbert spaces. Then we show that these extremal solutions are in fact dense in the solutions of the original system. Subsequently we use this density result to derive nonlinear and infinite dimensional versions of the "bang-bang" principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail.

Свойства Множества Допустимых Пар “Траектория - Управление” Эволюционных Управляемых Систем Первого Порядка

“…Связь (аналог следствия 4.1) между решениями обыкновенного дифференциаль ного включения с выпуклозначной, компактнозначной правой частью F(t,x) и с правой частью extF (t,x) изучалась в [1]- [3], [5]- [7] и др., а для эволюционных включений-в [4], [10], [12], [30] и др.…”

Baire's category and the bang-bang property for evolution differential inclusions of contractive type

“…This was introduced in 1982 by De Blasi and Pianigiani [9][10][11] (starting from a generic type result proved by Cellina [5]) in order to study the existence of solutions of some classes of non-convex valued differential inclusions in Banach spaces, without hypotheses of compactness. Subsequently, the Baire method has been employed in different contexts by several authors including Bressan and Colombo [2], Papageorgiou [27], Suslov [32] for ordinary differential inclusions, and Bressan and Flores [3], Dacorogna and Marcellini [8], De Blasi and Pianigiani [12,13] for partial differential inclusions. An account of results obtained by means of the Baire method and a view on some recent problems concerning differential inclusions can be found in Pianigiani [30] and Cellina [6].…”

Baire category and boundary value problems for ordinary and partial differential inclusions under Carathéodory assumptions

“…In the following years the Baire category method, and variants of it, were employed by several authors (see Bressan and Colombo [2], Bressan and Piccoli [4], De Blasi and Pianigiani [12,13], Hu and Papageorgiou [17], Papageorgiou [21], Suslov [22], Tolstonogov [23,24]) to treat other problems, as the investigation of extremal solutions, the study of the topological structure of the solution sets, the existence of solutions of evolution differential inclusions, the bang-bang property, etc.…”