Carleman estimates for one-dimensional degenerate heat equations

By Patrick S. Martinez, Judith Vancostenoble. 2006

119 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7b6c3c36f57cb5fec25c91d4269e90f7f6e2ac9

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Citations in the corpus

Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form (2008, background): “Observe that we need well-posedness in L2(0, 1) since the results of [24] for (1.”
2d Grushin-type equations: Minimal time and null controllable data (2015, background): “In particular, null controllability still holds for weak degeneracies and fails for too strong ones, see [12, 13, 11, 1, 29, 10, 9, 8, 20, 34].”
Null-controllability of hypoelliptic quadratic differential equations (2016, unknown): “This feature is well understood for parabolic equations that degenerate on the domain boundary, see [1, 9, 10, 11, 12, 42] in the one-dimensional case, and [13] for the multi-dimensional one.”
Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential (2008, methodology): “This corresponds precisely to the weight given in [17] for dealing with singular 1-d heat-type equation and in [19] when dealing with the observability around the singularity.”
DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION (2015, background): “Several extensions of the above results are available in one space dimension, see [2, 49] for equations in divergence form, [20, 19] for operators in nondivergence form, and [18, 35] for cascade systems.”
Carleman estimates for degenerate parabolic equations with first order terms and applications (2010, background): “…properties of linear one-dimensional degenerate parabolic equations with zero order terms are nowadays well known; see for instance [1,6,12,7] In [5] the authors study the property of null controllability of the system (4) with ﬁrst order terms when the coeﬃcient c only depends on…”
Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations (2015, background): “dition, even if a′ exists, considering the well-posedness for the last equation, additional conditions are necessary: for instance, for the prototype a(x) = xK, well-posedness is guaranteed if K ≥ 2 ([44]). However, in [16] the authors prove that if a(x) = xK the global null controllability fails ex”
Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems (2010, background): “In [33, 1], the authors treat the case of a degenerate operator −(a(x)ux)x with a general coefficient a(x) that vanishes at x = 0 and/or x = 1, possibly with a weakly nonlinear term f(u).”
Regional controllability of semilinear degenerate parabolic equations in bounded domains (2006, result): “However, the previous results have been improved, recently, in [1] and in [20] (in the semilinear and in the linear case), where a global null controllability is proved in the weakly and in the strongly degenerate case.”