کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
6417092 1338528 2015 33 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
2d Grushin-type equations: Minimal time and null controllable data
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
2d Grushin-type equations: Minimal time and null controllable data
چکیده انگلیسی

We study internal null controllability for degenerate parabolic equations of Grushin-type Gγ=∂xx2+|x|2γ∂yy2(γ>0), in the rectangle (x,y)∈Ω=(−1,1)×(0,1).Previous works proved that null controllability holds for weak degeneracies (γ small), and fails for strong degeneracies (γ large). Moreover, in the transition regime and with strip shaped control domains, a positive minimal time is required.In this paper, we work with controls acting on two strips, symmetric with respect to the degeneracy. We give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time T (when the system is not null controllable): their regularity depends on the control domain and the time T.We also prove that, with a control that acts on one strip, touching the degeneracy line {x=0}, then Grushin-type equations are null controllable in any time T>0 and for any degeneracy γ>0.Our approach is based on a precise study of the observability property for the one-dimensional heat equations satisfied by the Fourier coefficients in variable y. This precise study is done, through a transmutation process, on the resulting one-dimensional wave equations, by lateral propagation of energy method.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Differential Equations - Volume 259, Issue 11, 5 December 2015, Pages 5813-5845
نویسندگان
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