Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators

Kunisch, Karl; Rodrigues, Sérgio S.

doi:10.1051/cocv/2018054

Cited by 26 publications

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References 36 publications

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“…We refer to the works [6,[8][9][10]12,16,24,40,43,49] and references therein. See also the comments in [31,Sect. 6.5].…”

Section: )mentioning

confidence: 99%

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Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Rodrigues¹

2020

Evolution Equations &Amp; Control Theory

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It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depend on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.2010 Mathematics Subject Classification. 93D15, 93C10, 93B52, 93C20.is globally exponentially stable, with the feedback control operatorprovided the conditionholds true. In (1.3) and (1.4), 1 is the identity operator, λ > 0 is an arbitrary constant, and P E ⊥ M U M stands for the oblique projection in H onto the closed subspace U M along the closed subspace E ⊥ M . Where U M := span{Ψ i | i ∈ {1, 2, . . . , M }} is the linear span of our M linearly independent actuators and E M := span{e i | i ∈ M}, with M = {1, 2, . . . , M }, is the linear span of "the" first M linearly independent eigenfunctions of the diffusion operator A : D(A) → H, with domain D(A) d,c −→ H. Further, α M +1 is the (M + 1)st eigenvalue of A. The eigenvalues of A, denoted by α i , are supposed to satisfyIt is not difficult to see that we can follow the arguments in [31, Thms. 3.5, 3.6, and Rem. 3.8] to conclude that system (1.2) is still stable if we replace (1.4) by F(y) = Ay + λ1y. Observe that (1.5) concerns a single M ∈ N and a single pair (U M , E M ). The following result, which follows straightforwardly from the sufficiency of (1.5), concerns a sequence of pairs (U M , E M ) M ∈N . Theorem 1.2. Assume that we can construct a sequence (U M , E M ) M ∈N such that P E ⊥ M U M L(H) ≤ C P remains bounded, with C P > 0 independent of M . Then system (1.2) is globally exponentially stable for large enough M , with F(y) ∈ {λ1y, Ay + λ1y}. P E M (ςA+λ1)P E M ,M,N U M , because P E M commutes with both A and 1 and because P E ⊥ M U M P E M = P E ⊥ M U M . Notice also that when F is linear and N = 0, then y → K F ,M U M (t, y) is linear, while y → K F ,M,N U M (t, y) is nonlinear.1.2. Motivation and short comparison to previous works. We find systems in form (1.1) when, for example, we want to stabilize a system to a...

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“…We refer to the works [6,[8][9][10]12,16,24,40,43,49] and references therein. See also the comments in [31,Sect. 6.5].…”

Section: )mentioning

confidence: 99%

“…The general properties asked for A, A rc , and N will be precised later on. In the linear case, N = 0, is has been proven in [31] that the closed-loop systeṁ y + Ay + A rc (t)y − K F ,M U M (t, y) = 0, y(0) = y 0 ∈ H, (…”

Section: Introductionmentioning

confidence: 99%

Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Rodrigues¹

2020

Evolution Equations &Amp; Control Theory

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“…In the recent work [13] a new explicit feedback control for nonautonomous parabolic equations such as is exponentially stabilising, provided we can find a sequence of actuator domains ω M 1 , . .…”

Section: Introductionmentioning

confidence: 99%

“…We underline that verifying that the operator norm (1.3b) remains bounded is challenging since a nonorthogonal projection has an operator norm strictly bigger than 1 and, depending on the choice of U M , a finite constant C P satisfying (1.3b) may not exist. However, in [13,Sections 4.6 and 4.7], it was observed numerically that the operator norm will likely remain bounded for suitable placements of the actuators. The main contribution of this paper is to give a rigorous proof of this numerical observation.…”

Section: Introductionmentioning

confidence: 99%

“…However, the inclusion of a general convention term in system (1.1) under Neumann boundary conditions is not covered by the results in [13]. See in particular [13,Assumption 2.3]. Here the main goal is the investigation of the properties of the oblique projection P E ⊥ M UM , appearing in the feedback control (1.2), rather than investigating the regularity properties of the system, that is why here we consider only the case of a reaction term, for which the results in [13] still hold for Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections

Rodrigues

Sturm

2018

IMA Journal of Mathematical Control and Information

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In recently proposed stabilisation techniques for parabolic equations, a crucial role is played by a suitable sequence of oblique projections in Hilbert spaces, onto the linear span of a suitable set of M actuators, and along the subspace orthogonal to the space spanned by "the" first M eigenfunctions of the Laplacian operator. This new approach uses an explicit feedback law, which is stabilising provided that the sequence of operator norms of such oblique projections remains bounded.The main result of the paper is the proof that, for suitable explicitly given sequences of sets of actuators, the operator norm of the corresponding oblique projections remains bounded.In the final part of the paper we provide numerical results, showing the performance of the explicit feedback control for both Dirichlet and Neumann homogeneous boundary conditions. 2010 Mathematics Subject Classification. 93D15,47A75,47N70.. supplemented with either homogeneous Dirichlet or Neumann boundary conditions, has been proposed. The reaction coefficient a = a(x, t) ∈ R is given and allowed to be time and space dependent. The scalar functions u 1 (t), u 2 (t), . . . , u M (t) are time-dependent controls at our disposal. The indicator functions 1 ωi = 1 ωi (x) associated with domains ω i ⊂ (0, L), are our actuators. Note that in this way, our control on the right hand side of (1.1), is finite dimensional, that is, for all given time t ≥ 0 our control is a linear combination of our (finite number of) actuators. Further, it is supported (localised) in the union M j=1 ω j . In [13, Sect. 3 and 5] it has been proven that for all positive λ > 0, the explicit feedback control M j=1 u j (t)1 ωj defined by y → Ky := P E ⊥ M UM (−∆y + ay − λy) (1.2)

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Learning an Optimal Feedback Operator Semiglobally Stabilizing Semilinear Parabolic Equations

2021

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Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators

Cited by 26 publications

References 36 publications

Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections

Learning an Optimal Feedback Operator Semiglobally Stabilizing Semilinear Parabolic Equations

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