Abstract
This chapter is devoted to strongly continuous semigroups of operators. It is well-known that the mentioned semigroups are the basic instrument for investigation of differential equations with constant operators in Banach and Hilbert spaces. Wide classes of autonomous distributed parameter systems are governed by such equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References for Chapter 13
Ahiezer. N. I. and Glazman, I. M. (1981). Theory of Linear Operators in a Hilbert Space. Pitman Advanced Publishing Program, Boston.
Balakrishnan, A.V. (1981). Strong stabilization and the steady state Riccati equation. Applied Mathematics and Optimization. 7, 335–341.
Benchimol, C.D. (1978). A note on weak stabilizability of contraction semigroups, J. Control and Optim., 16. 373–379.
Curtain, R. F. and A.J. Pritchard. (1978). Infinite Dimensional Linear Systems Theory. Lectures Notes in Control and Information Sciences 8, Springer-Verlag, New York.
Curtain, R. F. and H.J. Zwart. (1995). An Introduction to Infinite Dimensional Linear Systems Theory. Springer-Verlag, New York.
Desch, W. and Schappacher. W. (1985). Spectral properties of finitedimensional perturbed linear semigroups, J. Diff. Eqs., 59, 80–102.
Dunford, N and Schwartz, J. T. (1963). Linear Operators, part II, Interscience Publishers, New York. London.
Friedman. A. (1969). Partial Differential Equations. Holt, Rienart and Winston, New York.
Gil’. M. I. (1985). On absolute stability of nonlinear nonstationary distributed systems. Automation and Remote Control, 6, 12–19.
Gil’, M. I. (1995). Norm Estimations for Operator-valued Functions and Applications. Marcel Dekker, Inc. New York.
Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators. Trans. Mathem. Monographs, vol. 18, Amer. Math. Soc, R. I.
Henry, D. (1981). Geometric Theory of Semilmear Parabolic Equations. Lectures Notes in Mathematics, No 840. Springer-Verlag, New York.
Hille, E. and R. S. Phillips. (1957). Functional Analysis and Semigroups. AMS Colloquium Publications, Providence.
Huang, F. (1993). Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Diff. Equations, 104, 307–324.
Kappel, F., Kunish, K. and W. Schappacher (Eds) (1989). Control and Estimation of Distributed Parameter Systems. Birkháuser, Boston.
Krein, S.G. (1971). Linear Differential Equations in Banach space. AMS, Providence, R. I.
Nambu, T. (1994). Robustness of feedback control systems of parabolic type … J. Diff. Eqs., 110, 253–256.
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin.
Rabah, R. and Ionescu, D. (1987). Stabilization problem in Hilbert spaces, Int. J. Control, 46, No 6, 2035–2042.
Russell, D.L. (1978). Controllability and stabilizability theory for linear partial differential equations: recent progress and open problems, SIAM Review, 20, 639–739.
Sakawa, Y. (1984). Feedback control of second order evolution equations with damping, SIAM Control and Optimization, 22, No 3, 343–353.
Slemrod. M. (1989). Feedback stabilization of a linear control system in a Hilbert space with an a priory bounded control. Mathematics of Control Signals and Systems, 2, 265–285.
Tanabe. H. (1979). Equations of Evolutton. Pitman, London— San-Francisco — Melburn.
Tanabe. H. (1997). Functional Analytic Methods for Partial Differential Equations. Marcel Dekker, Inc. New York.
Triggiani, R. (1975). On the stabilization problem in a Banach space, J. Math. Anal. Applications, 52, 308–403.
Wang, P.K.C. (1990). Stabilization and control of distributed systems with time-dependent spatial domains, Journal of Optimization Theory and Appl., 65, No 2, 331–341.
Zabczyk, J. (1975). A note on Co-semigroups, Bull V Acad. Pol. De Sc. Serie Math., 23, 895–898.
Zabczyk, J. (1992). Mathematical Control Theory. An Introduction. Birkhäuser, Boston.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gil’, M.I. (1998). Strongly Continuous Semigroups. In: Stability of Finite and Infinite Dimensional Systems. The Springer International Series in Engineering and Computer Science, vol 455. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-5575-9_13
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5575-9_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-7550-0
Online ISBN: 978-1-4615-5575-9
eBook Packages: Springer Book Archive