## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Stabilization and decay of functionals for linear parabolic control systems

Takao Nambu

#### Abstract

We construct a specific feedback control scheme for a class of linear parabolic systems such that some nontrivial linear functionals of the state decay faster than the state, while the state is stabilized. In particular, we raise a new question of pole allocation which is subject to constraint, and derive the necessary and sufficient condition: an essential extension of the well known result by W. M. Wonham (1967).

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 2 (2008), 19-24.

Dates
First available in Project Euclid: 4 February 2008

https://projecteuclid.org/euclid.pja/1202136633

Digital Object Identifier
doi:10.3792/pjaa.84.19

Mathematical Reviews number (MathSciNet)
MR2386960

Zentralblatt MATH identifier
1141.93051

Subjects
Primary: 93D15: Stabilization of systems by feedback
Secondary: 35B35: Stability

#### Citation

Nambu, Takao. Stabilization and decay of functionals for linear parabolic control systems. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 2, 19--24. doi:10.3792/pjaa.84.19. https://projecteuclid.org/euclid.pja/1202136633

#### References

• S. Agmon, Lectures on elliptic boundary value problems, D. Van Nostrand Co., Inc., Princeton, N.J., 1965.
• R. F. Curtain, Finite-dimensional compensators for parabolic distributed systems with unbounded control and observation, SIAM J. Control Optim. 22 (1984), no. 2, 255–276.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983.
• D. G. Luenberger, Observers for multivariable systems, IEEE Automatic Control AC-11 (1966), 190–197.
• T. Nambu, On stabilization of partial differential equations of parabolic type: boundary observation and feedback, Funkcial. Ekvac. 28 (1985), no. 3, 267–298.
• T. Nambu, Stability enhancement of output for a class of linear parabolic systems, Proc. Roy. Soc. Edinburgh Sec. A 133A (2003), no. 1, 157–175.
• T. Nambu, An $L^{2}(\Omega)$-based algebraic approach to boundary stabilization for linear parabolic systems, Quart. Appl. Math. 62 (2004), no. 4, 711–748.
• T. Nambu, A new algebraic approach to stabilization for boundary control systems of parabolic type, J. Differential Equations 218 (2005), no. 1, 136–158.
• Y. Sakawa, Feedback stabilization of linear diffusion systems, SIAM J. Control Optim. 21 (1983), no. 5, 667–676.
• W. M. Wonham, On pole assignment in multi-input controllable linear systems, IEEE Trans. Automat. Control AC-12 (1967), 660–665.