Some Topics Related to the Controllability Operator

Abstract

What can we say about the controllability operator?

In this chapter, we collect facts and properties of the controllability operator. Firstly, in Section 8.1 we show that the ε _n (x ₀)–balls which are removed as part of the satisfaction of viability are continuous functions. This leads to establishing continuity of the controllability operator. Secondly, in Section 8.2 we consider the lattice properties of the control laws. Two orderings of the control law classes are defined, one weak and one strong ordering. Having this, it is established that the set of control law classes with the order relation and over set intersection and union form a lattice. Next conditions for satisfying the order relations are derived. Thirdly, in Section 8.3 homotopies are defined to consider the variation in the value of the controllability operator relative to the base admissible control law class PWC^Δ which corresponds to the collection of piecewise continuous functions over the sampling interval Δ. A conclusion is made in Section 8.4.