Other Extension Constructions in the Space of Solutions

Abstract

In the previous chapters of this book, the developed “picture” of applications of FAM-theory for the extension of integral constraints has been given. The characteristic singularity of the above-mentioned investigation is the immersion of the space of “ordinary” controls in the space of weakly absolutely η-continuous FAM, where η is a given nonnegative FAM. However, on the basis of a highly general construction of proposition 5.2.1, the scheme has many other applications. In particular, these applications can be realized in the class of FAM. In this chapter, we consider some problems of such a kind. One of these is connected with the investigation of different realizations of “pure impulse” control. Here we keep in mind the realization of “ordinary” controls in the form of combinations of Dirac measures. In this case, we have a process of control “with pushs” [17, 23]. Consider, on a profound level, a simple model of such a kind. Let

$$ \dot x(t) = A(t)x(t),{\text{ x(}}{{\text{t}}_0}{\text{) = }}{{\text{x}}_0} $$

(7.1.1)

be a system functioning in the n-dimensional phase space R ⁿ on the finite time interval [t ₀, θ ₀], t ₀ < θ ₀. In (7.1.1), x ₀ ∈ R ⁿ is an initial state and A(·) is a matriciant on [t ₀, θ ₀] with continuous components A _i,j (·), $i \in \overline {1,n} {\text{ j}} \in \overline {1,n}$. Consider a situation when, in the process of movement, a finite collection of “saltusses” of the phase state is possible. These saltusses are defined by “pushs” a _i b(t _i ), where a _i ∈ R and t _i ∈ [t ₀, θ ₀[; b(·) here is supposed to be a bounded vector function on [t ₀, θ ₀[ with values in R ⁿ.