On the lack of null-controllability of the heat equation on the half-line

By Sorin Daniel Micu, Enrique Zuazua. 2000

105 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/02124dec1377d655b89d2a57123117283b410c98

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Citations in the corpus

L3,∞-solutions of the Navier-Stokes equations and backward uniqueness (2003, background): “Results of this kind are of interest in control theory (see, for example, [27]).”
On the null-controllability of the heat equation in unbounded domains (2004, background): “On the other hand, its failure can be much more drastic than in the bounded case (when M is the half space andΓ = ∂M , it is proved in [11,12] that initial data with Fourier coefficien that grow less than any exponential are not null-controllable in any time, whereas the initial data with exponentia”
CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, background): “We refer to [164] for a further discussion of this issue.”
Some results on controllability for linear and nonlinear heat equations in unbounded domains (2007, unknown): “On the other hand, the first result on null controllability of parabolic problems was negative: in [19] the authors considered the case Ω = (0,∞) ⊂ R, a ≡ 0 and B ≡ 0, and proved that there is no initial data in any negative Sobolev space that may be driven to zero in finite time by a control functi”
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations (2012, background): “In fact, global Carleman estimates yield a precise knowledge of the “cost” of the control function in the linear 1The problem of null-controllability of the heat equation in the case where Ω is unbounded is entirely different [43,44,46].”
Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends (2011, methodology): “For more bibliographic information on the backward uniqueness for parabolic equations, we recommend the readers refer to [13], [14], [15, 16], [30, 31], [32], [33], [34], [35], [40], [41], [42] and [43].”
Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations (2015, background): “h K>2, so that we ﬁnd that Usolves a nondegenerate heat equation of the form Ut−Uxx+b(X)U= ˜hχ ω˜, where ˜ωis a bounded domain compactly contained both in (−∞,0) and in (0,∞). Adapting a result of [45], the new equation is not controllable, see [16, Remark 4.6]. In particular, proceeding as in [1”
Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form (2008, result): “Using a result of Escauriaza, Seregin and Sverak (see [12] or [13]), that generalizes previous work by Micu and Zuazua (see [26] or [27]), we deduce, as in [7], that (4.”
Carleman estimates for one-dimensional degenerate heat equations (2006, background): “On the other hand, a result of Escauriaza, Seregin and Šverák [12, 13], that generalizes a result of Micu and Zuazua [27], says that the initial conditions that can be controlled to zero in finite time have to be supported in [0, X0).”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, background): “In [145] it was proved that, even if approximate controllability holds, null controllability does not hold for the heat equation in the whole line when the control acts in a bounded subdomain.”
An Introduction to the Controllability of Partial Differential Equations (2004, background): “Although, as shown in [50], using the similarity variables, one can find a spectral decomposition of solutions of the heat equation on the whole or half line, the spectrum is too dense and biorthogonal families do not exist.”
Persistent regional null contrillability for a class of degenerate parabolic equations (2004, background): “Similar results hold when Ω is the half space R+ and when the control is distributed in a bounded control region ω (see Micu and Zuazua [31]).”
Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn (2017, background): “This was showed in [35] for the heat equation in the physical domain R.”