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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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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.

Wydawca

Czasopismo

Rocznik

Tom

1

Numer

2

Strony

141-156

Opis fizyczny

Daty

wydano
2003-06-01
online
2003-06-01

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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  • [13] L. Pandolfi: “On feedback stabilization of functional differential equations”, Boll. Unione Mat. Ital., IV Ser. 11, Suppl. Fasc., Vol. 3, (1975), pp. 626–635.
  • [14] I.G. Petrowsky: “On the Cauchy problem for systems of linear partial differential equations in a domain of nonanalitic functions”, Bull. Mosk. Univ., Ser. A, No. 7, (1938), pp. 1–72.
  • [15] L.S. Pontriagin: “On zeroz on some elementary transcendence functions”, Izv. AN SSSR, Ser. Mat., Vol. 6, (1942), pp. 115–134.
  • [16] R. Rebarber and S. Townley: “Robustness with respect to delay for exponential stability of distributed parameter systems”, SIAM J. Contr. Optim., Vol. 37, (1998), pp. 230–244. http://dx.doi.org/10.1137/S0363012996312453
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bwmeta1.element.doi-10_2478_BF02476004
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