Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force

González-Burgos, Manuel; Teresa, Luz de

doi:10.4171/pm/1859

Cited by 93 publications

(99 citation statements)

References 12 publications

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“…As a particular case of the result in Section 4 of [18] (see also [5]), system (1.3) is null controllable whenever …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Duprez

2017

ESAIM: COCV

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Abstract. This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval I ⊂⊂ R, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technical tool for obtaining these results is the moment method.Mathematics Subject Classification. 93B05, 93B07.

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“…As a particular case of the result in Section 4 of [18] (see also [5]), system (1.3) is null controllable whenever …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…1.1). In [18], the authors obtain the null controllability of system (1.1) at any time under condition (1.4), so we will not consider this case and we will always suppose that Supp p∩ω = ∅. This implies that there exists x 0 ∈ ω such that p(x 0 ) = 0.…”

Section: Solving the Moment Problemmentioning

confidence: 99%

Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Duprez

2017

ESAIM: COCV

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“…The previous controllability results have been extended in [26] to n ≥ 2 when system (1.4) has a particular structure: cascade systems. To this end, the authors assume a generalization of assumption (1.5) on the coupling matrix A(·, ·) and, again, use Carleman inequalities for the adjoint problem for proving the null controllability result.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Supp q ∩ ω = ∅ : In this case, system (1.2) is a particular case of system (1.4) (C ≡ 0) where the coefficient a 12 = q satisfies condition (1.5) with σ = 1 and ω 0 could be a connected component of the interior of the set Supp q ∩ ω = ∅. From very well-known results (see for instance [35], [25] or [26]), we deduce that system (1.2) is null controllable at time T for any T > 0, that is to say, the minimal time of distributed null controllability is zero: T 0 (q) = 0.…”

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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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“…The second one assumes locally distributed couplings, and locally distributed controls, but then with a non-empty intersection between the coupling and the control regions. We refer to [47,8,9,23,24,32,18,40,36] and to the survey paper [10] for such results and to the references therein. Let us further mention an interesting result by Coron, Guerrero and Rosier [18], which proves local null controllability results for nonlinearly coupled 2-systems of parabolic equations with a nonlinear coupling term.…”

Section: Some Overview On the Literature For Controlled Coupled Systemsmentioning

confidence: 99%

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

Alabau‐Boussouira

2013

Math. Control Signals Syst.

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AMS(MOS) subject classifications: 34G10, 35B35, 35B37, 35L90, 93D15, 93D20.Key words and phrases: Boundary observability. Locally distributed observability. Boundary control. Locally distributed control. HUM. Indirect controllability. Hyperbolic systems. Parabolic systems. Schrödinger equations. Cascade systems. Geometric conditions. Abstract linear evolution equations. Insensitizing controls. Optimal conditions. AbstractWe study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE's and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in [2,4] for symmetric coupled systems, to cascade systems which are examples of non symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos Lebeau and Rauch [11]. By duality, this solves the exact controllability, by a single control, of 2-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger 2-coupled cascade systems under (GCC) and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa [47]. Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by J.-L. Lions [35] and later on by Dáger [19] and Tebou [44].

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Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force

Cited by 93 publications

References 12 publications

Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

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

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

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