Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints

Estimates of reachable sets of control systems with bilinear–quadratic nonlinearities

“…The above generalization is based on a combination of the techniques described above and the results of [11,12]. Remark 1.…”

“…Remark 1. [11] To determinate simpler estimate of the reachable set W (t 0 + σ) we introduce small parameter ε > 0 and embed the degenerate ellipsoid W (t 0 , σ, ν) in nondegenerate ellipsoid E ε w(t 0 , σ, ν), O ε (t 0 , σ, ν) :…”

“…and lim ε→+0 h(W (t 0 , σ), W ε (t 0 , σ)) = 0. The passage to the family of nondegenerate ellipsoids enables one to use the algorithms of [11,17] and construct an external estimate E ε (w + (σ), O + (σ)) of the union of ellipsoids W ε (t 0 , σ). Therefore we get ellipsoidal estimates of the reachable set W (t 0 + σ)…”

“…Lemma 1 [11]. The reachable set X(T ) is the projection of W (T + µ) at the subspace of variables z:…”

“…, m). At this step we find the ellipsoidal estimate for the union of a finite family of ellipsoids [11,17].…”