By Alexander Y. Khapalov

The aim of this monograph is to deal with the problem of the worldwide controllability of partial differential equations within the context of multiplicative (or bilinear) controls, which input the version equations as coefficients. The mathematical types we study comprise the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and paired hybrid nonlinear dispensed parameter platforms modeling the swimming phenomenon. The booklet deals a brand new, top quality and intrinsically nonlinear method to technique the aforementioned hugely nonlinear controllability problems.

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4 there). In this chapter we focus primarily on the global exact null-controllability problem, which requires a completely different method. 1) is globally exactly null-controllable in H if it can be steered in H from any initial state to the zero-state exactly. Y. 2 Main Results Exact null-controllability. Our main results here are as follows. 1. e. 2) for some positive constant ν0 > 0. 1) vanishes at time T : u(·, T ) = 0. a. (x,t) ∈ Q∞ . e. in Q∞ . 6)). 4 below. , [97]). 1. 6). 1. 1). Our next result deals with the case when n = 1 and α ∈ L2 (QT ) vanishing outside of the given strict subdomain of Ω .

49], h(x,t1 ) > 0 in the interior of Ω , h(x,t1 ) |∂ Ω = 0. 3: Nonnegative Controllability 45 Step 2. Consider any t2 > t1 . On the interval (t1 ,t2 ) we apply a positive constant control v(x) = v (its value will be chosen later). 38) Ω where λk (λk → −∞ as k → ∞) and ωk (x) ( ωk L2 (Ω ) = 1), k = 1, . . are respectively the eigenvalues and eigenfunctions associated with the spectral problem Δ ω = λ ω , ω |∂ Ω = 0 in H01 (Ω ). Consider any number γ > 1 (its value will be chosen more precisely a little bit later) and select a constant (in t and x) control v > 0 such that ev(t2 −t1 ) = γ , namely, v = ln γ .

19)) will be ud (x) = ω1 (x). 4)), in place of the target state. 40) for some t ∗ > t∗ (where λk ’s are the eigenvalues associated with α∗ , λ1 = 0). 5) with α = α∗ . 42) where ρ > 0 is some (fixed) constant. Since λ1 = 0, a < 0 and α (x) = α∗ (x) + a < 0, x ∈ [0, 1]. 24) applies on the interval (t∗ ,t ∗ ): u(·,t ∗ ) − s1+ξ ud L2 (0,1) u(·,t ∗ ) − y(·,t ∗ ) ≤ L2 (0,1) r1 5 5 + y(·,t ∗ ) − s1+ξ ud ≤ C(t ∗ − t∗ )max{ 6 (1− 5 ), 6 (1− + Csξ λ2 /a s1+ξ ud 3r2 5 )} L2 (0,1) smin{r1 ,r2 } L2 (0,1) = o(s1+ξ ) as s → 0+ (we remind the reader that C denotes a generic positive constant).