Controllability of partial differential equations governed by Alexander Y. Khapalov PDF

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By Alexander Y. Khapalov

ISBN-10: 3642124127

ISBN-13: 9783642124129

The objective 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 versions we learn contain the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and matched hybrid nonlinear disbursed parameter platforms modeling the swimming phenomenon. The ebook bargains a brand new, top quality and intrinsically nonlinear technique to technique the aforementioned hugely nonlinear controllability problems.

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3. 1) Step 1. 1 in place of u0 and ud . Denote hd (x) . 27) v∗ (x) = ln h0 (x) Then hd (x) = ev∗ (x) h0 (x). 1. 29) for any number s > 0. Step 2. 13). 6) on (0, T ). Consider any v ∈ L∞ (Ω ) and denote g = h − y. 13), we obtain: gt = vg + Δ h in QT , g |t=0 = 0. Thus t ev(x)(t−τ ) Δ h(x, τ )d τ , t ∈ [0, T ]. 29), namely, v(x) = sv∗ (x), where we now treat the positive number s as a parameter (its value will be selected later in Step 4). 30) yields: g(·,t) 2 L2 ( Ω ) ≤ e2st ln c2 − 1 s ln c2 Δh 2 , L2 (Qt ) Qt = Ω × (0,t), t ∈ [0, T ].

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). 4) and ξ ∈ (0, min{r1 , r2 } − 1). 30 2 Global Nonnegative Controllability of the 1-D Semilinear Parabolic Equation Thus, we showed that u(·,t ∗ ) = s1+ξ ud + o(s1+ξ ) as s → 0 + .

13). Consider any pair of initial and target states h0 , hd ∈ L2 (Ω ), which are nonnegative (almost everywhere) in Ω and h0 = 0. Since we study the issue of approximate controllability and because the set of infinitely differentiable functions with compact support (denote it by C0∞ (Ω )) is dense in L2 (Ω ), without loss of generality we can further assume that hd ∈ C0∞ (Ω ), hd = 0, hd (x) ≥ 0 ∀x ∈ Ω . 35) We plan to approximate hd by using three static bilinear controls, applied subsequently in time: (a) Firstly, we will use v = 0 on some time-interval (0,t1 ) to steer our system to a state h(·,t1 ) which is strictly positive in the interior of Ω .

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