By Henry M. Paynter
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Additional resources for Analysis and design engineering systems
E, d) denotes a nonempty set of transitions on (E, d). e. D( f (y), f (z)) ≤ λ · d(y, z) for any y, z ∈ E. Furthermore assume α := sup α ( f (z)) < ∞. z∈E ◦ Fix an element x0 ∈ E and a curve y(·) : [0, T ] −→ E with y (t) = 0/ for all t ∈ [0, T ]. 3 Feedback Leads to Mutational Equations 39 Then there exists a unique solution x(·) : [0, T ] −→ E to the initial value problem ◦ x (·) f x(·) x(0) = x0 In addition, it satisfies the following inequality for all t ∈ [0, T ] d x(t), y(t) ≤ d(x0 , y(0)) · e(α +λ ) t + t 0 e(α +λ ) (t−s) · inf D f (y(s)), ϑ ◦ ds.
Let Θ (E, d) be a nonempty set of transitions on a metric space (E, d) and, x(·) : [0, T ] −→ E denotes a curve. For t ∈ [0, T [, the set ◦ x (t) := ϑ ∈ Θ (E, d) lim h↓0 1 h · d ϑ (h, x(t)), x(t + h) = 0 is called mutation of x(·) at time t. Remark 11. For every transition ϑ on (E, d) and initial element x0 ∈ E, the curve xx0 (·) := ϑ (·, x0 ) : [0, 1] −→ E has ϑ in its mutation at each time t ∈ [0, 1[: ◦ ϑ ∈ xx0 (t) for every t ∈ [0, 1[. ) in Definition 1. In regard to real-valued functions, the classical concepts of derivative and integral are closely related by the fundamental theorem of calculus.
Even simple examples, however, indicate obstacles in the current mutational framework. Consider just an annulus expanding isotropically at a constant speed 1. After a finite period, the “hole” in the center of the annulus disappears suddenly. Hence, the topological boundary of the expanding annulus does not evolve continuously (in the sense of Painlev´e-Kuratowski). The classical Pompeiu-Hausdorff distance between the boundaries of such an annulus K ⊂ RN and its expanding counterpart Bt (K) ⊂ RN does not have to be continuous with respect to time t and thus, it is unsuitable for comparing topological boundaries in regard to transitions.
Analysis and design engineering systems by Henry M. Paynter