Null Controllability for Degenerate Parabolic Operators

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concluding remarks• <strong>for</strong> n = 1 null controllability yields strong Feller property oftransition semigroup associated to SPDEdu(t) = (au x (t)) x dt + χ ω dW (t)needed to study invariant measures• application to inverse problemsu t − ( x α u x)x = r(t, x)f (x) |r(t 0, ·)| > 0f (·)←−Tu(t 0 , ·)u x (·, 1)ref0 1C & Tort & Yamamoto (in progress)
concluding remarks• <strong>for</strong> n = 1 null controllability yields strong Feller property oftransition semigroup associated to SPDEdu(t) = (au x (t)) x dt + χ ω dW (t)needed to study invariant measures• application to inverse problemsu t − ( x α u x)x = r(t, x)f (x) |r(t 0, ·)| > 0f (·)←−Tu(t 0 , ·)u x (·, 1)ref0 1C & Tort & Yamamoto (in progress)
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Null Controllabilityfor Degenerate
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degenerate parabolic operatorsO ⊂
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Outline1 Examples of degenerate par
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examples of degenerate problems•
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{ut − ( (1 − x 2 )u x)(1 − x
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{ut − ( (1 − x 2 )u x)(1 − x
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Fleming-Viot diffusion processevolu
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Fleming-Viot diffusion processevolu
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• X(·, x)invariant sets for stoc
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• X(·, x)invariant sets for stoc
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two invariance problemsProblemgiven
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Page 31 and 32:
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 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: the simplest problem in 2dn = 2⎧
Page 87 and 88: concluding remarks• for n = 1 nul
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Null Controllability for Degenerate Parabolic Operators

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