Controllability of linear and semilinear non-diagonalizable parabolic systems

Fernández-Cara, Enrique; González-Burgos, Manuel; Teresa, Luz de

doi:10.1051/cocv/2014063

Cited by 13 publications

(21 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…to simplify), the coupling matrix A disturbs the diagonal diffusion matrix D and creates a new "cross" diffusion matrix: D = D − A. When D is not diagonalizable, there are few results (see [26] with a technical assumption on the dimension of the Jordan Blocks of D and the recent preprint [41, Section 3] when C does not depend on time and space).…”

Section: Linear Parabolic Systemsmentioning

confidence: 99%

“…2.3. More restrictive conditions on the initial condition when the target (u * i ) 1≤i≤4 vanishes In the previous section, we have seen that there are invariant quantities in the dynamics of (4) which impose necessary conditions on the initial condition: (23), (26). Moreover, when some coefficients of diffusion d i are equal, we have more invariant quantities in (4) which impose stronger necessary conditions on the initial condition: (24), (27).…”

Section: Case Of 1 Controlmentioning

confidence: 99%

See 1 more Smart Citation

Controllability of a 4 × 4 quadratic reaction–diffusion system

Balc’h

2019

Journal of Differential Equations

View full text Add to dashboard Cite

We consider a 4 × 4 nonlinear reaction-diffusion system posed on a smooth domain Ω of R N (N ≥ 1) with controls localized in some arbitrary nonempty open subset ω of the domain Ω. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction-diffusion system for every N ≥ 1 in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates cross diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for 1 ≤ N ≤ 2 thanks to our local controllabillity result together with a known theorem on the asymptotics of the free nonlinear reaction-diffusion system.

show abstract

Section: Linear Parabolic Systemsmentioning

confidence: 99%

Section: Case Of 1 Controlmentioning

confidence: 99%

Controllability of a 4 × 4 quadratic reaction–diffusion system

Balc’h

2019

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…Then, there exists a (unique) solution y of (1). Moreover, y satisfies the comparison principle (18) ∀t ∈ [0, T ], a.e. x ∈ Ω, y(t, x) ≤ y(t, x) ≤ y(t, x).…”

Section: Maximum Principlementioning

confidence: 99%

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Balc’h

2020

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

We consider the semilinear heat equation posed on a smooth bounded domain Ω of R N with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset ω of Ω. The goal of this paper is to prove the uniform large time global nullcontrollability for semilinearities f (s) = ±|s| log α (2 + |s|) where α ∈ [3/2, 2) which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global nonnegative-controllability (respectively nonpositive-controllability) of the system, i.e., one can steer any initial data to a nonnegative (respectively nonpositive) state in arbitrary time. In particular, one can act locally thanks to the control term in order to prevent the blow-up from happening. The proof relies on precise observability estimates for the linear heat equation with a bounded potential a(t, x). More precisely, we show that observability holds with a sharp constant of the order exp C a 1/2 ∞ for nonnegative initial data. This inequality comes from a new L 1 Carleman estimate. A Kakutani-Leray-Schauder's fixed point argument enables to go back to the semilinear heat equation. Secondly, the uniform large time null-controllability result comes from three ingredients: the global nonnegative-controllability, a comparison principle between the free solution and the solution to the underlying ordinary differential equation which provides the convergence of the free solution toward 0 in L ∞ (Ω)-norm, and the local null-controllability of the semilinear heat equation. ContentsThe proof of Theorem 1.3 is a consequence of the (global) null-controllability of the linear heat equation with a bounded potential (due to Andrei Fursikov and Oleg Imanuvilov, see [23] or [21, Theorem 1.5]) and the small L ∞ perturbations method (see [3, Lemma 6] and [1], [5], [30], [33], [40] for other results in this direction).The following global null-controllability (positive) result has been proved independently by Enrique Fernandez-Cara, Enrique Zuazua (see [22, Theorem 1.2]) and Viorel Barbu under a sign condition (see [4, Theorem 2] or [6, Theorem 3.6]) for Dirichlet boundary conditions. It has been extended to semilinearities which can depend on the gradient of the state and to Robin boundary conditions (then to Neumann boundary conditions) by

show abstract

“…Indeed, because of the new couplings of order 2 that appear, there is a technical restriction on number of equations of system (5.1) to use this method, namely, that n has to be less than or equal to 4. We refer to [15], especially Theorem 1.1, for more details. In the present paper, we will show that, as expected, this condition on the number of equations was only the consequence of the technique used and that it can actually be removed.…”

Section: Controllability Of a Non Diagonalizable Parabolic Systemmentioning

confidence: 99%

Compact perturbations of controlled systems

Duprez¹,

Olive²

2018

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

Using a compactness-uniqueness approach, we show that the Fattorini criterion implies the exact controllability of general compactly perturbed controlled linear systems. We then apply this perturbation result to obtain new controllability results for systems governed by partial differential equations. Notably, we combine it with the fictitious control method to establish the exact controllability of a cascade system of coupled wave equations by a reduced number of controls and with the same control time as the one for a single wave equation. We also combine this perturbation result with transmutation techniques to prove the null controllability in arbitrarily small time of a one-dimensional non diagonalizable system of coupled heat equations with as many controls as equations.

show abstract

Controllability of linear and semilinear non-diagonalizable parabolic systems

Cited by 13 publications

References 29 publications

Controllability of a 4 × 4 quadratic reaction–diffusion system

Controllability of a 4 × 4 quadratic reaction–diffusion system

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Compact perturbations of controlled systems

Contact Info

Product

Resources

About

Controllability of linear and semilinear non-diagonalizable parabolic systems

Cited by 13 publications

References 29 publications

Controllability of a 4 × 4 quadratic reaction–diffusion system

Controllability of a 4 × 4 quadratic reaction–diffusion system

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Compact perturbations of controlled systems

Contact Info

Product

Resources

About

Controllability of a 4 × 4 quadratic reaction–diffusion system

Controllability of a 4 × 4 quadratic reaction–diffusion system