Null Controllability for Degenerate Parabolic Operators

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Theoremconditions <strong>for</strong> invariance(a) ⇐⇒ (b) ⇐⇒ (c)(a) O invariant{(i) L0 d Γ (x) ≥ 0(b) ∀x ∈ Γ(ii) 〈A(x)∇d Γ (x), ∇d Γ (x)〉 = 0(c) O invariantref Friedman & Pinsky (1975)Da Prato & Frankowska (2004)C & Da Prato & Frankowska (2009)stochastic viability . . .invariance =⇒ L 0 degenerate on Γ
Theoremconditions <strong>for</strong> invariance(a) ⇐⇒ (b) ⇐⇒ (c)(a) O invariant{(i) L0 d Γ (x) ≥ 0(b) ∀x ∈ Γ(ii) 〈A(x)∇d Γ (x), ∇d Γ (x)〉 = 0(c) O invariantref Friedman & Pinsky (1975)Da Prato & Frankowska (2004)C & Da Prato & Frankowska (2009)stochastic viability . . .invariance =⇒ L 0 degenerate on Γ
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 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 29 and 30: two invariance problemsProblemgiven
Page 31: two invariance problemsProblemgiven
Page 35 and 36: Theoremconditions for invariance(a)
Page 37 and 38: 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 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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