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## Open Mathematics

2003 | 1 | 2 | 141-156
Tytuł artykułu

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

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EN
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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.
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EN
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Tom
Numer
Strony
141-156
Opis fizyczny
Daty
wydano
2003-06-01
online
2003-06-01
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autor
Bibliografia
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• [4] L.V. Fardigola: “On a nonlocal two-point boundary-value problem in a layer for an equation with variable coefficients” (in Russian), Sibirsk. Mat. Zh., Vol. 38, (1997), pp. 424–438, English translation in: Siberian Math. J., Vol. 38, (1997), pp. 367–379.
• [5] L.V. Fardigola: “On stabilizability of evolution systems of partial differential equations on ℝn ×[0,+∞) by feedback control”, Visnyk Kharkivs'kogo Universytetu. Ser. Matematyka, Prykladna Matematyka i Mekhanika, Vol. 475, (2000), pp. 183–194.
• [6] L.V. Fardigola: “A criterion for stabilizability of differential equations with constant coefficients on the whole space”, (in Russian), Differ. uravn., Vol. 36, (2000), pp. 1699–1706, English translation in: Differ. Equ., Vol. 36, (2000), pp. 1863–1871.
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• [18] J.M. Sloss, I.S. Sadek, J.C. Bruch, S. Aldali: “Stabilization of structurally damped systems by time-delayed feedback control”, Dyn. Stab. Syst., Vol. 7, (1992), pp. 173–178.
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