Carleman estimates for degenerate parabolic operators with applications to null controllability

By Fatiha Alabau-Boussouira, Piermarco Cannarsa, Genni Fragnelli. 2006

258 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/dd35896ed6208f3104f7334dd0b70cad0efedbf4

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Citations in the corpus

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).”
Carleman estimates for degenerate parabolic equations with first order terms and applications (2010, background): “…controllability 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…”
Regional controllability of semilinear degenerate parabolic equations in bounded domains (2006, background): “In particular in [20] P. Martinez and J. Vancostenoble consider the linear equation ut (aux)x = h, while in [ 1 ] the authors consider the semilinear case ut (aux)x + f(t,x,u) = h, where the function f satisfies conditions like those of Hypothesis 3.1.”
Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type (2013, background): “Recently, many results are obtained even in the case where diffusion coefficients degenerate at the boundary, see [1] and the references therein.”
Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form (2008, methodology): “In [1], [7], [8], [9] and [24] observability results for the adjoint problem of (1.”
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, 8, 9, 10, 11, 39] in the one-dimensional case, and [12] for the multi-dimensional one.”
Carleman estimates and observability inequalities for parabolic equations with interior degeneracy (2012, background): “At a first glance, one may think that our results can be obtained just by a “translation” of the ones obtained in [3], but this is not the case.”
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].”
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.”
Null controllability of Grushin-type operators in dimension two (2011, background): “On the other hand, for γ ∈ (0, 1/2) we will follow the strategy of [1, 10, 37], adapting the weight β to the nonsmooth coefficient |x|2γ (see (78)–(79)).”
CONTROLLABILITY OF 1-D COUPLED DEGENERATE PARABOLIC EQUATIONS (2009, methodology): “This inequality was basically proved in [6, 1, 8].”