Null and approximate controllability for weakly blowing up semilinear heat equations

By Enrique Fernández-Cara, Enrique Zuazua. 2000

406 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/41bc9787e2640df9c8b213e542e38e3b8ac3d01b

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Citations in the corpus

Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients (2002, methodology): “We just mention that we treat the case in which f is only locally Lipschitz continuous as for example in [9] and [13] using approximations of f by C(1) functions.”
Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential (2008, methodology): “Let us briefly mention the related work [12], which presents a study of the control properties of weakly blowing-up semi-linear heat equations, which deals with a similar question as the one asked here.”
A unified controllability/observability theory for some stochastic and deterministic partial differential equations (2010, background): “By the contrary, it is proved in [2, 12] that for some nonlinearities f satisfying lim |s|→∞ | f(s) | s ln(1+ | s |) = 0, (3.”
Null controllability of Grushin-type operators in dimension two (2011, background): “The null and approximate controllability of the heat equation are essentially well understood subjects for both linear and semilinear equations, and for bounded or unbounded domains (see, for instance, [15], [19], [21], [22], [23], [27], [31], [32], [36], [39], [40], [45], [46]).”
Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type (2013, methodology): “In view of the results in [6, 9, 10, 11], we expect that one can extend the results to nonlinearities with slightly superlinear growth and containing gradient terms, as in [12], using more involved regularity theory for (1.”
Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form (2008, background): “Such a generalization allows to obtain, arguing as in [19], [1], the null controllability for the semilinear problem   ut − a(x)uxx + h(t, x, u) = f(t, x)χω(x), (t, x) ∈ (0, T)× (0, 1), u(t, 0) = u(t, 1) = 0, t ∈ (0, T), u(0, x) = u0(x), x ∈ (0, 1).”
Carleman estimates for degenerate parabolic operators with applications to null controllability (2006, methodology): “Several results have also been obtained for semilinear nondegenerate equations, see, in particular, [13], [ 17 ], [18], [19], [22].”
CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, background): “In [83, 84] the problem of null and approximate controllabilty was treated for semi-linear heat equations and, in particular, it was proved that null controllability may hold for some nonlinearities for which, in the absence of control, blow-up phenomena arise.”
Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces (2011, background): “In control theory, global Carleman estimates for parabolic operators yield the null controllability of classes of semi-linear parabolic equations [1, 8 , 9]. This last application was the motivation for the proof of a global Carleman estimate in [7] in the case of a non smooth diffusion coefficient”
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations (2012, background): “By linearization and with a fix point argument, one may consider the controllability of semi-linear parabolic equations for certain forms of non linearities [4,14,18].”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, background): “As we mentioned above, in [69] it was proved that semilinear heat equations are null controllable in an arbitrarily small time and from any open subset ω of the domain Ω, for nonlinearities that, at infinity, grow slower than s log(s).”
Regional controllability of semilinear degenerate parabolic equations in bounded domains (2006, methodology): “Using the fixed point method developed in [ 14 ] for nondegenerate problems we obtain null controllability results for (1.1) when f satisfies generalized Lipschitz conditions.”
Approximate controllability of a class of semilinear systems with boundary degeneracy (2010, background): “Particularly, it has been shown that the approximate controllability is a consequenceofthenullcontrollabilityforthecontrolsystemsgovernedbynondegenerate linear parabolic equations [11, 12 ].”
On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials (2008, background): “2) In [12] the following result was proved: Theorem C.”
Null-controllability of some reaction–diffusion systems with one control force (2006, background): “We refer to [10] where the above formula forτ0 is obtained.”
On the existence of time optimal controls for linear evolution equations (2007, background): “We refer [6, 7, 8, 13, 19] and the rich references cited therein for recent progress on the observability estimates (of L(2)-norm in time) for PDEs.”
An Introduction to the Controllability of Partial Differential Equations (2004, background): “The null controllability property holds in fact in a much more general setting of semilinear heat equations ([23] and [24]).”