Note on the cost of the approximate controllability for the heat equation with potential

Phung, Kim Dang

doi:10.1016/j.jmaa.2004.03.059

Cited by 23 publications

(18 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem of obtaining explicit bounds on C(T, u 0 ) received much consideration in the literature (see, for example, [18,12,37,44,33,32,34,10,31]), especially the case of small time, i.e. when T goes to zero.…”

Section: Application To Control Theorymentioning

confidence: 99%

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Nakić¹,

Täufer²,

Tautenhahn³

et al. 2018

Analysis & PDE

View full text Add to dashboard Cite

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ (−∞,E] (H L ) of a Schrödinger operator H L on a cube of side L ∈ N, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameterdependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain. Results Scale-free unique continuation and eigenvalue liftingLet d ∈ N. For L > 0 we denote by Λ L = (−L/2, L/2) d ⊂ R d the cube with side length L, and by ∆ L the Laplace operator on L 2 (Λ L ) with Dirichlet, Neumann or periodic boundary conditions. Moreover, for a measurable and bounded V : R d → R we denote by V L : Λ L → R its restriction to Λ L given by V L (x) = V (x) for x ∈ Λ L , and bythe corresponding Schrödinger operator. Note that H L has purely discrete spectrum. For x ∈ R d and r > 0 we denote by B(x, r) the ball with center x and radius r with respect to Euclidean norm. If the ball is centered at zero we write B(r) = B(0, r).Definition 2.1. Let G > 0 and δ > 0. We say that a sequenceCorresponding to a (G, δ)-equidistributed sequence we define for L ∈ GN the set

show abstract

Section: Application To Control Theorymentioning

confidence: 99%

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Nakić¹,

Täufer²,

Tautenhahn³

et al. 2018

Analysis & PDE

View full text Add to dashboard Cite

show abstract

“…However, it turns out that the problem is closely related to the notion of cost of controllability in the control theory literature. Thus we will use ideas from [Ro95], [Ph04] in order to pass from quantitative uniqueness to the required form of quantitative approximation. Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

The fractional Calderón problem: Low regularity and stability

2020

View full text Add to dashboard Cite

The Calderón problem for the fractional Schrödinger equation was introduced in the work [GSU16], which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant L p or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli-Silvestre extension and a duality argument.

show abstract

“…(In the next section of this paper, we will show such controllability by building up a new type of observability inequality.) The cost of the null controllability and the approximate controllability for heat equations have been extensively studied (see [10,24,9]). It is acknowledged that the order e C T for the upper bound of the cost for controls is sharp in some sense (see [9]).…”

Section: Resultsmentioning

confidence: 99%

Output feedback stabilization for heat equations with sampled-data controls

Lin

Liu

Wang

2020

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we build up an output feedback law to stabilize a sampled-data controlled heat equation (with a potential) in a bounded domain Ω. The feedback law abides the following rules: First, we divide equally the time interval [0, +∞) into infinitely many disjoint time periods, and divide each time period into three disjoint subintervals. Second, for each time period, we observe a solution over an open subset of Ω in the first subinterval; take sample from outputs at one time point of the first subinterval; add a time-invariant output feedback control over another open subset of Ω in the second subinterval; let the equation evolve free in the last subinterval. Thus, the corresponding feedback control is of sampleddata. Our feedback law has the following advantages: the sampling period (which is the length of the above time period) can be arbitrarily taken; the feedback law has an explicit expression in terms of the sampling period; the behaviors of the norm of the feedback law, when the sampling period goes to zero or infinity, are clear. The construction of the feedback law is based on two kinds of approximate null-controllability for heat equations. One has time-invariant controls, while another has impulse controls. The studies of the aforementioned controllability with time-invariant controls need a new observability inequality for heat equations built up in the current work.

show abstract

Note on the cost of the approximate controllability for the heat equation with potential

Cited by 23 publications

References 6 publications

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

The fractional Calderón problem: Low regularity and stability

Output feedback stabilization for heat equations with sampled-data controls

Contact Info

Product

Resources

About