Null Controllability for Degenerate Parabolic Operators
Null Controllability for Degenerate Parabolic Operators
Null Controllability for Degenerate Parabolic Operators
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More general 1-d problems• divergence <strong>for</strong>m• Martinez & Vancostenoble (2006) u t − ( a(x)u x)x = χ ωf• Alabau & C Fragnelli (2006) u t − ( a(x)u x)x + g(u) = χ ωf• non-divergence C & Fragnelli & Rocchetti (2007, 2008)u t − a(x)u xx − b(x)u x = χ ω f• using (invariant) measure ρ(x)dx = 1Ra(x) e x b(s)1/2 a(s) ds dx[ ]operator trans<strong>for</strong>ms a(x)u xx + b(x)u x = 1ρ(x) ρ(x)a(x)ux( {self-adjoint L 2 ρ 0, 1) := u ∈ L 2 (0, 1) ∣ ∫ 10 u2 ρ dx < ∞ }• degenerate/singular problemsVancostenoble & Zuazua (2008), Vancostenoble (2009)u t − ( x α u x)x −λx β u = χ ωfx