Fundamental control principle

The fundamental control principle explains what mutually, i.e., independent r control variables U i , i = 1, 2 , . . . , r , $ U_{i} ,{\text{ i = 1,}}2, \, . \, . \, . \, ,r, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/inline-math7_1.tif"/> i.e., what the control vector U (1.46), U ∈ ℜ K , $ {\mathbf{U}} \in {\Re }^{K} , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/inline-math7_2.tif"/> can achieve at most and what is necessary for them to satisfy in order to govern directly K mutually independent variables Z ₁, Z ₂..., Z _K ,  i.e., to govern directly their vector Z ∈ ℜ K $ {\mathbf{Z}} \in {\Re }^{K} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/inline-math7_3.tif"/> , over some time interval ( t 1 , t 2 ) ⊆ T , t 2 > t 1 , $ (t_{1} , t_{2} ) \subseteq {\mathfrak{T}},t_{2}> t_{1} , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/inline-math7_4.tif"/> i.e., at every moment t ∈ ( t 1 , t 2 ) . $ t \in (t_{1} , t_{2} ). $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/inline-math7_5.tif"/>