September  2016, 36(9): 5163-5181. doi: 10.3934/dcds.2016024

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, China

2. 

College of Mathematics & Information Science, Henan Normal University, Xinxiang 453007, China

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 2016

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.
Citation: Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
References:
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W. Arendt, Vector valued Laplace transforms and Cauchy problems,, Israel J. Math., 59 (1987), 327.  doi: 10.1007/BF02774144.  Google Scholar

[2]

A. Bátkai and S. Piazzera, Semigroups and linear differential equations with delay,, J. Math. Anal. Appl., 264 (2001), 1.  doi: 10.1006/jmaa.2001.6705.  Google Scholar

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A. Bátkai and S. Piazzera, Semigroups for Delay Equations,, A. K. Peters, (2005).   Google Scholar

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P. N. Chen and H. S. Qin, Controllability of linear systems in Banach spaces,, Syst. Control Lett., 45 (2002), 155.  doi: 10.1016/S0167-6911(01)00177-3.  Google Scholar

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R. deLaubenfels and E. Families, Functional Calculi and Evolution Equations,, Springer-Verlag, (1994).   Google Scholar

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R. deLaubenfels, Matrices of operators and regularized semigroups,, Math. Z., 212 (1993), 619.  doi: 10.1007/BF02571680.  Google Scholar

[8]

R. deLaubenfels, $C$-semigroups and the Cauchy problem,, J. Funct. Anal., 111 (1993), 44.  doi: 10.1006/jfan.1993.1003.  Google Scholar

[9]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar

[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.  doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

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J. K. Hale, Functional Differential Equations,, Appl. Math. Sci., (1971).   Google Scholar

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M. Hieber, Laplace transforms and $\alpha$-times integrated semigroups,, Forum Math., 3 (1991), 595.  doi: 10.1515/form.1991.3.595.  Google Scholar

[13]

M. Hieber, Integrated semigroups and differential operators on $L^p$ spaces,, Math. Ann., 291 (1991), 1.  doi: 10.1007/BF01445187.  Google Scholar

[14]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces,, Acta Math., 104 (1960), 93.  doi: 10.1007/BF02547187.  Google Scholar

[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.   Google Scholar

[16]

C. Kaiser, Integrated semigroups and linear partial differential equations with delay,, J. Math. Anal. Appl., 292 (2004), 328.  doi: 10.1016/j.jmaa.2003.10.031.  Google Scholar

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H. Kellermann and M. Hieber, Integrated semigroups,, J. Funct. Anal., 84 (1989), 160.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

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C. C. Kuo, On perturbation of $\alpha$-times integrated $C$-semigroups,, Taiwanese J. Math., 14 (2010), 1979.   Google Scholar

[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.  doi: 10.1006/jmaa.1994.1464.  Google Scholar

[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.  doi: 10.1112/plms/s3-69.1.144.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0363012996310703.  Google Scholar

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I. V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches,, Chapman & Hall, (2001).  doi: 10.1201/9781420035490.  Google Scholar

[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.  doi: 10.1070/RM1994v049n06ABEH002449.  Google Scholar

[25]

F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem,, Pac. J. Math., 135 (1988), 111.  doi: 10.2140/pjm.1988.135.111.  Google Scholar

[26]

M. Schechter, Spectra of Partial Differential Operators,, $2^{nd}$, (1986).   Google Scholar

[27]

X. L. Song and J. G. Peng, Lipschitzian semigroups and abstract functional differential equations,, Nonlinear Analysis: Theory, 72 (2010), 2346.  doi: 10.1016/j.na.2009.10.035.  Google Scholar

[28]

N. Tanaka, On perturbation theory for exponentially bounded $C$-semigroups,, Semigroup Forum, 41 (1990), 215.  doi: 10.1007/BF02573392.  Google Scholar

[29]

N. Tanaka and I. Miyadera, Exponential bounded $C$-semigroups and intgrated semigroups,, Tokyo, 12 (1989), 99.  doi: 10.3836/tjm/1270133551.  Google Scholar

[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.  doi: 10.1016/j.sysconle.2007.06.001.  Google Scholar

[31]

G. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces,, J. Diff. Eq., 20 (1976), 71.  doi: 10.1016/0022-0396(76)90097-8.  Google Scholar

[32]

G. Weiss, Optimal control of systems with a unitary semigroup and with colocated control and observation,, Syst. Control Lett., 48 (2003), 329.  doi: 10.1016/S0167-6911(02)00276-1.  Google Scholar

[33]

J. Wu, Theory and Application of Partial Functional Differential Equations,, Appl. Math. Sci., (1996).  doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[34]

Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators,, Sciense Press, (2014).   Google Scholar

show all references

References:
[1]

W. Arendt, Vector valued Laplace transforms and Cauchy problems,, Israel J. Math., 59 (1987), 327.  doi: 10.1007/BF02774144.  Google Scholar

[2]

A. Bátkai and S. Piazzera, Semigroups and linear differential equations with delay,, J. Math. Anal. Appl., 264 (2001), 1.  doi: 10.1006/jmaa.2001.6705.  Google Scholar

[3]

A. Bátkai and S. Piazzera, Semigroups for Delay Equations,, A. K. Peters, (2005).   Google Scholar

[4]

P. N. Chen and H. S. Qin, Controllability of linear systems in Banach spaces,, Syst. Control Lett., 45 (2002), 155.  doi: 10.1016/S0167-6911(01)00177-3.  Google Scholar

[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.  doi: 10.1112/plms/s3-55.1.181.  Google Scholar

[6]

R. deLaubenfels and E. Families, Functional Calculi and Evolution Equations,, Springer-Verlag, (1994).   Google Scholar

[7]

R. deLaubenfels, Matrices of operators and regularized semigroups,, Math. Z., 212 (1993), 619.  doi: 10.1007/BF02571680.  Google Scholar

[8]

R. deLaubenfels, $C$-semigroups and the Cauchy problem,, J. Funct. Anal., 111 (1993), 44.  doi: 10.1006/jfan.1993.1003.  Google Scholar

[9]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar

[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.  doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

[11]

J. K. Hale, Functional Differential Equations,, Appl. Math. Sci., (1971).   Google Scholar

[12]

M. Hieber, Laplace transforms and $\alpha$-times integrated semigroups,, Forum Math., 3 (1991), 595.  doi: 10.1515/form.1991.3.595.  Google Scholar

[13]

M. Hieber, Integrated semigroups and differential operators on $L^p$ spaces,, Math. Ann., 291 (1991), 1.  doi: 10.1007/BF01445187.  Google Scholar

[14]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces,, Acta Math., 104 (1960), 93.  doi: 10.1007/BF02547187.  Google Scholar

[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.   Google Scholar

[16]

C. Kaiser, Integrated semigroups and linear partial differential equations with delay,, J. Math. Anal. Appl., 292 (2004), 328.  doi: 10.1016/j.jmaa.2003.10.031.  Google Scholar

[17]

H. Kellermann and M. Hieber, Integrated semigroups,, J. Funct. Anal., 84 (1989), 160.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[18]

C. C. Kuo, On perturbation of $\alpha$-times integrated $C$-semigroups,, Taiwanese J. Math., 14 (2010), 1979.   Google Scholar

[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.  doi: 10.1006/jmaa.1994.1464.  Google Scholar

[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.  doi: 10.1112/plms/s3-69.1.144.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0363012996310703.  Google Scholar

[23]

I. V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches,, Chapman & Hall, (2001).  doi: 10.1201/9781420035490.  Google Scholar

[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.  doi: 10.1070/RM1994v049n06ABEH002449.  Google Scholar

[25]

F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem,, Pac. J. Math., 135 (1988), 111.  doi: 10.2140/pjm.1988.135.111.  Google Scholar

[26]

M. Schechter, Spectra of Partial Differential Operators,, $2^{nd}$, (1986).   Google Scholar

[27]

X. L. Song and J. G. Peng, Lipschitzian semigroups and abstract functional differential equations,, Nonlinear Analysis: Theory, 72 (2010), 2346.  doi: 10.1016/j.na.2009.10.035.  Google Scholar

[28]

N. Tanaka, On perturbation theory for exponentially bounded $C$-semigroups,, Semigroup Forum, 41 (1990), 215.  doi: 10.1007/BF02573392.  Google Scholar

[29]

N. Tanaka and I. Miyadera, Exponential bounded $C$-semigroups and intgrated semigroups,, Tokyo, 12 (1989), 99.  doi: 10.3836/tjm/1270133551.  Google Scholar

[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.  doi: 10.1016/j.sysconle.2007.06.001.  Google Scholar

[31]

G. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces,, J. Diff. Eq., 20 (1976), 71.  doi: 10.1016/0022-0396(76)90097-8.  Google Scholar

[32]

G. Weiss, Optimal control of systems with a unitary semigroup and with colocated control and observation,, Syst. Control Lett., 48 (2003), 329.  doi: 10.1016/S0167-6911(02)00276-1.  Google Scholar

[33]

J. Wu, Theory and Application of Partial Functional Differential Equations,, Appl. Math. Sci., (1996).  doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[34]

Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators,, Sciense Press, (2014).   Google Scholar

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