Abstract
We establish conditions for the stabilizability of evolution systems of partial differential equations on \(\mathbb{R}^n \times [0, + \infty )\) by one-dimensional feedback controls. To prove these conditions, we use the Fourier-transform method. We obtain estimates for semialgebraic functions on semialgebraic sets by using the Tarski–Seidenberg theorem and its corollaries. We also give examples of stabilizable and nonstabilizable systems.
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REFERENCES
R. Triggiani, “On the stabilizability problem in Banach space,” J. Math. Anal. Appl., 52, 383–403 (1975).
R. F. Curtain, “Equivalence of input–output stability and exponential stability for infinite dimensional systems,” Math. Syst. Theory, 21, 19–48 (1988).
V. I. Korobov and G. M. Sklyar, “On the problem of strong stabilizability of contracting systems in Hilbert spaces,” Differents. Uravn., 20, No. 11, 1862–1869 (1984).
K. N. Boyadrhiev and N. Levan, “Strong stability of Hilbert space contraction semigroups,” Stud. Sci. Math. Hung., 30, No. 304, 165–182 (1995).
F. A. Khodja and A. Benabdallakh, “Stabilisation de l'équation des ondes par un contrôleur dynamique,” C. R. Acad. Sci. Sér. I. Math., 321, No. 2, 195–198 (1995).
R. Rebarber and S. Townley, “Robustness with respect to delays for exponential stability of distributed parameter systems,” SIAM J. Contr. Optim., 37, No. 1, 230–244 (1998).
I. G. Petrovskii, “On the Cauchy problem for a system of linear partial differential equations in a domain of nonanalytic functions,” Byul. Mosk. Univ., Sek. A, 1, No. 7, 1–72 (1938).
I. M. Gel'fand and G. E. Shilov, Some Problems in the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).
A. Seidenberg, “A new decision method for elementary algebra,” Ann. Math., 60, No. 2, 365–374 (1954).
L. Hörmander, “The analysis of linear differential operators,” in: Differential Operators with Constant Coefficients, Vol. 2, Springer, Berlin (1983).
L. Hörmander, “On the division of distributions by polynomials,” Ark. Mat., 3, No. 6, 555–568 (1958).
L. V. Fardigola, “A criterion for the stabilizability of differential equations with constant coefficients in the entire space,” Differents. Uravn., 36, No. 12, 1699–1706 (2000).
L. V. Fardigola, “On the stabilizability of evolution systems of partial differential equations on ℝn ×; [0, ∞) by feedback control,” Visn. Khark. Univ., Ser. Mat., Prikl. Mat. Mekh., No. 475, 183–194 (2000).
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Fardigola, L.V., Sheveleva, Y.V. On the Possibility of Stabilization of Evolution Systems of Partial Differential Equations on ℝ n × [0, + ∞) Using One-Dimensional Feedback Controls. Ukrainian Mathematical Journal 54, 1556–1565 (2002). https://doi.org/10.1023/A:1023480205549
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DOI: https://doi.org/10.1023/A:1023480205549