Null Controllability for Degenerate Parabolic Operators

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Outline1 Examples of degenerate parabolic problems2 Null controllability for degenerate parabolic operatorsintroduction to null controllabilityone dimensional case3 Higher dimensional problems
Outline1 Examples of degenerate parabolic problems2 Null controllability for degenerate parabolic operatorsintroduction to null controllabilityone dimensional case3 Higher dimensional problems
Page 1 and 2: Null Controllabilityfor Degenerate
Page 4: degenerate parabolic operatorsO ⊂
Page 9 and 10: 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 and 62:
the simplest exampleω =]a, b[⊂
Page 63 and 64:
why null controllability fails for
Page 65 and 66:
Page 67 and 68:
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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