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

Affiliation(s)

Department of Applied Mathematics, Graduate School of System Informatics, Kobe University, Kobe, Japan.

Department of Applied Mathematics, Graduate School of System Informatics, Kobe University, Kobe, Japan.

ABSTRACT

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.

Cite this paper

T. Nambu, "On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems,"*Advances in Pure Mathematics*, Vol. 3 No. 9, 2013, pp. 26-37. doi: 10.4236/apm.2013.39A1005.

T. Nambu, "On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems,"

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.

[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.