The $C$-regularized semigroup method for partial differential equations with delays

Xin  Yu; Guojie  Zheng; Chao Xu

doi:10.3934/dcds.2016024

Article Contents

2016, Volume 36, Issue 9: 5163-5181. Doi: 10.3934/dcds.2016024

This issue Previous Article Periodic solutions of the planar N-center problem with topological constraints Next Article Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity

The $C$-regularized semigroup method for partial differential equations with delays

1.
Laboratory of Information & Control Technology, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100
2.
College of Mathematics & Information Science, Henan Normal University, Xinxiang 453007
3.
State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou, Zhejiang 310027

Received: January 2015

Revised: January 2016

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.

Keywords:

Mathematics Subject Classification: Primary: 47D60, 47D06; Secondary: 35R10.

Citation:

Related Papers

Cited by

References

[1]	W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.doi: 10.1007/BF02774144.
[2]	A. Bátkai and S. Piazzera, Semigroups and linear differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20.doi: 10.1006/jmaa.2001.6705.
[3]	A. Bátkai and S. Piazzera, Semigroups for Delay Equations, A. K. Peters, Wellesley, 2005.
[4]	P. N. Chen and H. S. Qin, Controllability of linear systems in Banach spaces, Syst. Control Lett., 45 (2002), 155-161.doi: 10.1016/S0167-6911(01)00177-3.
[5]	E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc., 55 (1987), 181-208.doi: 10.1112/plms/s3-55.1.181.
[6]	R. deLaubenfels and E. Families, Functional Calculi and Evolution Equations, Springer-Verlag, 1994.
[7]	R. deLaubenfels, Matrices of operators and regularized semigroups, Math. Z., 212 (1993), 619-629.doi: 10.1007/BF02571680.
[8]	R. deLaubenfels, $C$-semigroups and the Cauchy problem, J. Funct. Anal., 111 (1993), 44-61.doi: 10.1006/jfan.1993.1003.
[9]	K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
[10]	B. Z. Guo, J. M. Wang and S. P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Syst. Control Lett., 54 (2005), 557-574.doi: 10.1016/j.sysconle.2004.10.006.
[11]	J. K. Hale, Functional Differential Equations, Appl. Math. Sci., Vol. 3, Springer-Verlag, 1971.
[12]	M. Hieber, Laplace transforms and $\alpha$-times integrated semigroups, Forum Math., 3 (1991), 595-612.doi: 10.1515/form.1991.3.595.
[13]	M. Hieber, Integrated semigroups and differential operators on $L^p$ spaces, Math. Ann., 291 (1991), 1-16.doi: 10.1007/BF01445187.
[14]	L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-140.doi: 10.1007/BF02547187.
[15]	F. T. Iha and C. F. Schubert, The spectrum of partial differential operators on $L^p(R^n)$, Trans. Amer. Math. Soc., 152 (1970), 215-226.
[16]	C. Kaiser, Integrated semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 292 (2004), 328-339.doi: 10.1016/j.jmaa.2003.10.031.
[17]	H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.doi: 10.1016/0022-1236(89)90116-X.
[18]	C. C. Kuo, On perturbation of $\alpha$-times integrated $C$-semigroups, Taiwanese J. Math., 14 (2010), 1979-1992.
[19]	Y. S. Lei and Q. Zheng, The application of $C$-semigroups to differential operators in $L^p(R^n)$, J. Math. Anal. Appl., 188 (1994), 809-818.doi: 10.1006/jmaa.1994.1464.
[20]	Y. S. Lei, W. H. Yi and Q. Zheng, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc., 69 (1994), 144-170.doi: 10.1112/plms/s3-69.1.144.
[21]	Y. S. Lei and Q. Zheng, Exponentially bounded $C$-semigroups and integrated semigroups with nondensely defined generators II: Perturbation (in Chinese), Acta Math. Sci., 13 (1993), 428-434.
[22]	K. S. Liu and Z. Y. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.doi: 10.1137/S0363012996310703.
[23]	I. V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman & Hall, London, 2001.doi: 10.1201/9781420035490.
[24]	I. V. Mel'nikova and A. Filinkov, Integrated semigroups and $C$-semigroups, well-posedness and regularization of differential-operator problems, Russian Math. Surveys, 49 (1994), 115-155.doi: 10.1070/RM1994v049n06ABEH002449.
[25]	F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pac. J. Math., 135 (1988), 111-155.doi: 10.2140/pjm.1988.135.111.
[26]	M. Schechter, Spectra of Partial Differential Operators, $2^{nd}$, North Holland, Elsevier, 1986.
[27]	X. L. Song and J. G. Peng, Lipschitzian semigroups and abstract functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 2346-2355.doi: 10.1016/j.na.2009.10.035.
[28]	N. Tanaka, On perturbation theory for exponentially bounded $C$-semigroups, Semigroup Forum, 41 (1990), 215-236.doi: 10.1007/BF02573392.
[29]	N. Tanaka and I. Miyadera, Exponential bounded $C$-semigroups and intgrated semigroups, Tokyo, J. Math., 12 (1989), 99-115.doi: 10.3836/tjm/1270133551.
[30]	G. S. Wang and L. J. Wang, The Bang-Bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett., 56 (2007), 709-713.doi: 10.1016/j.sysconle.2007.06.001.
[31]	G. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Diff. Eq., 20 (1976), 71-89.doi: 10.1016/0022-0396(76)90097-8.
[32]	G. Weiss, Optimal control of systems with a unitary semigroup and with colocated control and observation, Syst. Control Lett., 48 (2003), 329-340.doi: 10.1016/S0167-6911(02)00276-1.
[33]	J. Wu, Theory and Application of Partial Functional Differential Equations, Appl. Math. Sci., Vol. 119, Springer-Verlag, New York, 1996.doi: 10.1007/978-1-4612-4050-1.
[34]	Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators, Sciense Press, Beijing, 2014.