The cost of approximate controllability for heat equations: the linear case

By Enrique Fernández-Cara, Enrique Zuazua. Advances in Differential Equations, 2000

335 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/d9abe3e72cf69ae65202699af3c110bd060087ec

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Citations in the corpus

Null controllability of the N-dimensional Stokes system with N−1 scalar controls (2009, background): “Remark 2 From the Carleman inequality (17), one can follow the same steps as in [6] in order to prove the observability inequality (2) for N = 2 and i = 2 and so once Proposition 1 is established, the proof of Theorem 1 is finished.”
DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION (2015, background): “1 Null controllability of the heat equation The null and approximate controllabilities of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains [3, 27, 30, 32, 33, 34, 37, 44, 46, 48, 51, 52, 62, 63] and also with discon”
RECENT RESULTS ON THE CONTROLLABILITY OF LINEAR COUPLED PARABOLIC PROBLEMS: A SURVEY (2011, background): “For a proof, see for instance [37] and [20] (Theorem 2.”
Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient (2004, background): “Inspired in [6], denote by Uad the nonempty set Uad = {v ∈ L(2)(ω × (0, T )) : (y, q) satisfies (5)–(7) with y0 = 0}.”
CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, methodology): “In this section we present some of the results of [41] on the cost of approximate controllability of the system (2.”
Sparse initial data identification for parabolic PDE and its finite element approximations (2015, background): “The interested reader is referred to [14] where this issue is investigated in a more general frame in order to obtain sharp bounds on the cost of approximate control.”
Persistent regional null contrillability for a class of degenerate parabolic equations (2004, result): “See, in particular, Fattorini and Russell [14, 15] for results in one space dimension or under special geometric conditions, Lebeau and Robbiano [25] for a general result without geometric conditions, Fursikov and Imanuvilov [19] for similar results in a more general context, including heat equation”
Null controllability of degenerate heat equations (2005, background): “Introduction Controllability properties of nondegenerate parabolic equations have been widely studied in recent years; concerning the heat equation, see in particular [2, 6, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 34, 35, 37, 38]; concerning the Stokes and Navier-Stokes equations, see in”
Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods (2005, background): “28This type of result has been also proved in the context of the heat equation in [31].”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, methodology): “1 ([74], [68]) There exists a constant C > 0 that only depends on Ω and ω such that the following inequality holds”
Controllability and observability of a heat equation with hyperbolic memory kernel (2009, methodology): “In the context of hyperbolic equations, one may use multipliers [15] or microlocal analysis [2]; while, in the context of parabolic equations, one uses Carleman inequalities [6].”
Controllability results for cascade systems of m coupled parabolic PDEs by one control force (2010, neutral): “This explicit dependence can be obtained arguing as in [9] (also see [6]).”
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]).”