Null Controllability for Degenerate Parabolic Operators

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the simplest exampleω =]a, b[⊂⊂]0, 1[ a(x) = x α (α > 0)u t − ( x α u x)x = χ ωf , u(0, x) = u 0 (x)C & Martinez & Vancostenoble (2008)⎧⎪⎨ false α ≥ 2 (→{regional null controllability)n. c.any b.c. 0 ≤ α < 1 weak⎪⎩ true 0 ≤ α < 2Neumann b.c. 1 ≤ α < 2 strongTregional 0 ω 1Figure: regional null controllability
why null controllability fails for α ≥ 2• classical change of variable (Courant-Hilbert)y(x) =∫ 1xdss α/2 U(y(x), t) = x −α/4 u(x, t)transforms equation into˜ω =]˜b, ã[ boundedU t − U yy + V α (y)U = χ eω F0 < y < ∞V α (y) = α 2( 34 α − 1) 1[2−(2−α)y] 2 bounded for α ≥ 2• Escauriaza, Seregin, Sverak (2003, 2004)U(·, T ) = 0 ⇐⇒ spt ( U(·, 0) ) ⊂ [0, ã[• u t − ( x α u x)x = χ ωf{no [0, 1]yes if spt(u 0 ) ⊂]a, 1]
Page 1 and 2:
Null Controllabilityfor Degenerate
Page 4:
degenerate parabolic operatorsO ⊂
Page 7 and 8:
Outline1 Examples of degenerate par
Page 9 and 10:
examples of degenerate problems•
Page 11 and 12: examples of degenerate problems•
Page 13 and 14: examples of degenerate problems•
Page 15 and 16: {ut − ( (1 − x 2 )u x)(1 − x
Page 17 and 18: {ut − ( (1 − x 2 )u x)(1 − x
Page 19 and 20: Fleming-Viot diffusion processevolu
Page 21 and 22: Fleming-Viot diffusion processevolu
Page 23 and 24: • X(·, x)invariant sets for stoc
Page 25 and 26: • X(·, x)invariant sets for stoc
Page 27 and 28: two invariance problemsProblemgiven
Page 33 and 34: Theoremconditions for invariance(a)
Page 39 and 40: controlled parabolic equationsω
Page 41 and 42: the null controllability problemwan
Page 43 and 44: oadmap to null controllability• s
Page 45 and 46: oadmap to null controllability• s
Page 47 and 48: difficulties in degenerate casefor
Page 49 and 50: difficulties in degenerate casefor
Page 51 and 52: Outline1 Examples of degenerate par
Page 53 and 54: one space dimensiona ∈ C([0, 1])
Page 55 and 56: one space dimensiona ∈ C([0, 1])
Page 57 and 58: well-posednessin both cases{D(A) ={
Page 59 and 60: well-posednessin both cases{D(A) ={
Page 61: the simplest exampleω =]a, b[⊂
Page 65 and 66: why null controllability fails for
Page 67 and 68: why null controllability fails for
Page 69 and 70: Carleman’s estimate 0 < α < 2w t
Page 71 and 72: Carleman’s estimate 0 < α < 2w t
Page 73 and 74: adjoint problem for 0 < α < 2 (⋆
Page 75 and 76: observability v t + ( x α v x)x =
Page 77 and 78: More general 1-d problems• diverg
Page 79 and 80: More general 1-d problems• diverg
Page 81 and 82: the simplest problem in 2dn = 2⎧
Page 83 and 84: the simplest problem in 2dn = 2⎧
Page 85 and 86: concluding remarks• for n = 1 nul
Page 87 and 88: concluding remarks• for n = 1 nul
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Null Controllability for Degenerate Parabolic Operators

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