Exact controllability theorems for linear parabolic equations in one space dimension

Fattorini, H. O.; Russell, David L.

doi:10.1007/bf00250466

Cited by 369 publications

(432 citation statements)

References 6 publications

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“…Also, the present authors have shown in an earlier paper [6] that if the X" are real and satisfy the asymptotic relationship Xn = K(n + a)e + o(nff-1) (n -> ») (1.6) where K > 0, f) > 1 and a is real, then there exist constants K > 0, Kp > 0 such that ||g»IU*«..-, < K exp [(K, + o(l))X"1/?J in > 1) (1.7)…”

Section: If Rn G E (Nsupporting

confidence: 73%

See 1 more Smart Citation

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

Fattorini¹,

Russell²

1974

Quart. Appl. Math.

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153

169

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Section: If Rn G E (Nsupporting

confidence: 73%

“…As in [6], we begin by studying the problem formulated above in the special case where T = + ». The first result, proved in Sec.…”

Section: If Rn G E (Nmentioning

confidence: 99%

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

Fattorini¹,

Russell²

1974

Quart. Appl. Math.

Self Cite

153

169

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“…Fattorini and D.L. Russell (see [19,20]) through the moment method. The controllability of the N -dimensional case, still for the scalar equation (n = 1), has been established later by G. Lebeau and L. Robbiano in [31] and by A. Fursikov and O. Yu.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Unlike the scalar case, from the results in [10] we infer that the approximate and null boundary controllability properties for non-scalar parabolic systems are, in general, not equivalent. In [21], [6] and [10], the authors use the moment method (see [19,20]) to prove the positive null controllability result at time T . They carry out a study on bounds of biorthogonal families to exponentials associated to complex sequences.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

Khodja

Benabdallah

González-Burgos

et al. 2016

Journal of Mathematical Analysis and Applications

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We consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllability. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0.

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“…This allows to make the well known reduction to some properties of entire functions and exponential sums (cf. [FR71] and references in [Mil04]). Indeed, we point out in the appendix that some abstract systems with finite dimensional inputs satisfying a condition based on Müntz theorem on the completeness of sets of exponentials have a large set of non controllable initial states.…”

Section: Application To the Fractional Diffusionmentioning

confidence: 99%

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

Miller

2006

Math. Control Signals Syst.

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Abstract. This paper introduces a "spectral observability condition" for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one dimensional output systems from Müntz-Szász theorem on the closed span of sets of power functions.In this paper, an observability condition on the spectral subspaces of a negative self-adjoint operator is introduced which ensures fast controllability, i.e. the semigroup generated by this operator is null-controllable in arbitrarily small time. In this asymptotic, it also ensures an upper bound for the controllability cost, i.e. the supremum, over every initial state with norm one, of the norm of the optimal input function which steers it to zero (cf. section 1). This spectral observability condition is the abstract version of a property proved for the Dirichlet Laplacian ∆ on a compact manifold observed on any non empty region in [LZ98, JL99] (cf.[Mil05] for non-compact manifolds).It applies to the semigroup generated by the fractional Laplacian on manifolds −(−∆) α as long as α > 1/2. This semigroup is widely used to describe physical systems exhibiting anomalous diffusions (cf. references in sect.2.1). Thus new interior null-controllability results for such fractional diffusions with non-constant coefficients in any dimension are deduced (the one dimension problem with constant coefficients and one dimensional input was recently considered in [MZ04]). In particular, as the control time T tends to 0, the controllability cost grows at most like C β exp(c β /T β ) where C β and c β are positive constants and β > 1/(2α − 1) (n.b. a lower bound of the same form with equality β = 1/(2α − 1) holds in the case α = 1 which corresponds to the heat equation). It is proved in the appendix that a similar one dimensional problem is not controllable from the boundary forThis last result is deduced from a more general remark of independent interest on the lack of controllability of any finite linear combination of eigenfunctions of systems with one dimensional input, based on the generalized Müntz theorem on the completeness of sets of exponentials.1. The main result in the abstract setting.After recalling the duality between controllability and observability for parabolic semigroups, this section states the main definition and theorem.1.1. The abstract setting. Let the generator A be a positive self-adjoint operator with domain D(A) on the Hilbert space H of states which we identify with its dual.2000 Mathematics Subject Classification. 93B05, 34K35, 58J35.

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Exact controllability theorems for linear parabolic equations in one space dimension

Cited by 369 publications

References 6 publications

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

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