By Alexander Y. Khapalov

The aim of this monograph is to handle the difficulty 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 study contain the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and paired hybrid nonlinear allotted parameter platforms modeling the swimming phenomenon. The ebook deals a brand new, top of the range and intrinsically nonlinear method to process the aforementioned hugely nonlinear controllability problems.

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Example text

Then, we “stretch” the state u∗ to approximate the desirable target state ud . 2. 15). Step 1. Select any t∗ > 0. On the interval (0,t∗ ) we intend to apply a negative constant control α (x) = λ (its value will be chosen later). 35) 0 √ where in this case λk = −(π k)2 , ωk (x) = 2 sin π kx. Fix any s ∈ (0, 1) and select now (a constant on (0,t∗ ) bilinear control) λ = λ (t∗ , s) < 0 such that eλ t∗ = s. 35) that y(·,t∗ ) → sy0 (·) = su0 as t∗ → 0 + . 1) (uniform over all the possible multiple solutions).

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).

K x j−1 Note that gk (x) are strictly separated from zero in (0, 1). Thirdly, each of such piecewise constant functions can in turn be approximated in L2 (0, 1) by continuous piecewise linear functions that vanish at x = 0, 1 and everywhere else are strictly positive, whose graphs, accordingly, do not have vertical pieces. , by using pieces of circles of “sufficiently small” radia with centers located on the bisectors of the angles generated by the corresponding adjacent straight lines of the graphs (so that these lines are tangent to the associated circles).

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