On stabilizability of evolution systems of partial differential equations on ℝn×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

• Autorzy: L. Fardigola
• Języki publikacji: EN

Abstrakty

EN

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system $$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $$ where D x=(-i∂/∂x 1,...-i∂/∂x n), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ℝn, w is an unknown function, u(·,t)=P(D x)w(·,t−h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|2)γ for some Γ, γ∈ℝ depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski-Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e -hλ.

Słowa kluczowe

EN

stabilizability; feedback control; delay; partial differential equation; Fourier transform; 93D15; 35B37; 35A22

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2003
• Tom: 1
• Numer: 2
• Strony: 141-156

Daty

wydano

2003-06-01

online

2003-06-01

Twórcy

autor

L. Fardigola

Bibliografia

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Identyfikatory

DOI

10.2478/BF02476004