On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems

Takao Nambu^{*}

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Unlike regular stabilizations, we construct in the paper a specific feedback control system such that *u*(*t*) decays exponentially with the designated decay rate, and that some non-trivial linear functionals of *u* decay exactly faster than . The system contains a dynamic compensator with another state *v* in the feedback loop, and consists of two states *u* and *v*. This problem entirely differs from the one with static feedback scheme in which the system consists only of a single state *u*. To show the essential difference, some specific property of the spectral subspaces associated with our control system is studied.

References

[1] R. F. Curtain, “Finite Dimensional Compensators for Parabolic Distributed Systems with Unbounded Control and Observation,” SIAM Journal on Control and Optimization, Vol. 22, 1984, pp. 255-276.

http://dx.doi.org/10.1137/0322018

[2] T. Nambu, “On Stabilization of Partial Differential Equations of Parabolic Type: Boundary Observation and Feedback,” Funkcialaj Ekvacioj, Vol. 28, No. 3, 1985, pp. 267-298.

[3] T. Nambu, “An L2(Ω)-Based Algebraic Approach to Boundary Stabilization for Linear Parabolic Systems,” Quarterly of Applied Mathematics, Vol. 62, No. 4, 2004, pp. 711-748.

[4] T. Nambu, “A New Algebraic Approach to Stabilization for Boundary Control Systems of Parabolic Type,” Journal of Differential Equations, Vol. 218, No. 1, 2005, pp. 136-158. http://dx.doi.org/10.1016/j.jde.2005.03.013

[5] T. Nambu, “Alternative Algebraic Approach to Stabilization for Linear Parabolic Boundary Control Systems,” Mathematics of Control, Signals, and Systems, 2013, in Press. http://dx.doi.org/10.1007/s00498-013-0108-4

[6] Y. Sakawa, “Feedback Stabilization of Linear Diffusion Systems,” SIAM Journal on Control and Optimization, Vol. 21, No. 5, 1983, pp. 667-676.

http://dx.doi.org/10.1137/0321040

[7] S. Ito, “Diffusion Equations,” American Mathematical Society, Providence, 1992.

[8] S. Agmon, “Lectures on Elliptic Boundary Value Problems,” Van Nostrand, Princeton, 1965.

[9] A. E. Taylor, “Introduction to Functional Analysis,” John Wiley & Sons, New York, 1958.

[10] T. Nambu, “Stability Enhancement of Output for a Class of Linear Parabolic Systems,” Proceedings of Royal Society Edinburgh, Section A, Vol. 133A, No. 1, 2003, pp. 157-175.

[11] T. Nambu, “Stabilization and Decay of Functionals for Linear Parabolic Control Systems,” Proceedings of the Japan Academy, Series A, Vol. 84, No. 2, 2008, pp. 19-24. http://dx.doi.org/10.3792/pjaa.84.19

[12] T. Nambu, “On State and Output Stabilization of Linear Parabolic Systems,” Funkcialaj Ekvacioj, Vol. 52, No. 3, 2009, pp. 321-341. http://dx.doi.org/10.1619/fesi.52.321

[13] T. Nambu, “Stabilization and a Class of Functionals for Linear Parabolic Control Systems,” Proceedings of Royal Society Edinburgh, Section A, Vol. 140, No. 1, 2010, pp. 153-174. http://dx.doi.org/10.1017/S0308210508000978

[14] G. N. Watson, “A Treatise on the Theory of Bessel Functions,” Cambridge University Press, Cambridge, 1922.

[15] N. Levinson, “Gap and Density Theorems,” American Mathematical Society Colloquium Publications, New York, 1940.

[16] E. C. Titchmarsh, “The Theory of Functions,” The Clarendon Press, Oxford, 1939.

[17] Y. Sakawa and T. Matsushita, “Feedback Stabilization of a Class of Distributed Systems and Construction of a State Estimator,” IEEE Transactions on Automatic Control, Vol. 20, No. 6, 1975, pp. 748-753.

http://dx.doi.org/10.1109/TAC.1975.1101095

[18] R. Courant and D. Hilbert, “Methods of Mathematical Physics, I,” Wiley Interscience, New York, 1953.