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September  2021, 10(3): 491-509. doi: 10.3934/eect.2020077

New results on controllability of fractional evolution systems with order $ \alpha\in (1,2) $

1. 

Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

2. 

Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China

* Corresponding author: Yong Zhou

Received  April 2020 Revised  April 2020 Published  September 2021 Early access  June 2020

This paper addresses some interesting results of mild solutions to fractional evolution systems with order $ \alpha\in (1,2) $ in Banach spaces as well as the controllability problem. Firstly, we deduce a new representation of solution operators and give a new concept of mild solutions for the objective equations by the Laplace transform and Mainardi's Wright-type function, and then we proceed to establish a new compact result of the solution operators when the sine family is compact. Secondly, the controllability results of mild solutions are obtained. Finally, an example is presented to illustrate the main results.

Citation: Yong Zhou, Jia Wei He. New results on controllability of fractional evolution systems with order $ \alpha\in (1,2) $. Evolution Equations and Control Theory, 2021, 10 (3) : 491-509. doi: 10.3934/eect.2020077
References:
[1]

R. P. AgarwalD. BaleanuJ. J. NietoD. F. M. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Applied Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems (Second Edition), Birkhauser Verlag, 2011. doi: 10.1007/978-3-0348-5075-9.

[3]

K. BalachandranV. GovindarajL. Rodríguez-Germa and J. J. Trujillo, Controllability results for nonlinear fractional-order dynamical systems, J. Optimization Theory, Appl., 156 (2013), 33-44.  doi: 10.1007/s10957-012-0212-5.

[4]

M. BonforteY. Sire and J. L. Vázquez, Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal., 153 (2017), 142-168.  doi: 10.1016/j.na.2016.08.027.

[5]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.

[6]

L. A. Caffarelli and Y. Sire, Minimal surfaces and free boundaries: Recent developments, Bull. Amer. Math. Soc., 57 (2020), 91-106.  doi: 10.1090/bull/1673.

[7]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

[8]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland, Elsevier, 1985.

[9]

E. Fernández-CaraQ. Lü and E. Zuazua, Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54 (2016), 2009-2019.  doi: 10.1137/15M1044291.

[10]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Part. Diff. Eq., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

[11] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985. 
[12]

J. W. Hanneken, D. M. Vaught and B. N. Narahari Achar, Enumeration of the Real Zeros of the Mittag-Leffler Function $E_\alpha(z)$, $1 < \alpha < 2$, in Advances in Fractional Calculus, Springer, Dordrecht, 2007, 15–26. doi: 10.1007/978-1-4020-6042-7_2.

[13]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[15]

I. KimK. H. Kim and S. Lim, $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.

[16]

V. Komornik, Exact controllability in short time for the wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153-164.  doi: 10.1016/S0294-1449(16)30327-4.

[17]

K. LiJ. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.  doi: 10.1016/j.jfa.2012.04.011.

[18]

K. LiJ. Peng and J. Gao, Controllability of nonlocal fractional differential systems of order $\alpha\in (1, 2]$ in Banach spaces, Rep. Math. Phys., 71 (2013), 33-43.  doi: 10.1016/S0034-4877(13)60020-8.

[19]

Y. Li, Regularity of mild Solutions for fractional abstract Cauchy problem with order $\alpha\in (1, 2)$, Z. Angew. Math. Phys., 66 (2015), 3283-3298.  doi: 10.1007/s00033-015-0577-z.

[20]

Y. LiH. Sun and Z. Feng, Fractional abstract Cauchy problem with order $\alpha\in (1, 2)$, Dyn. Partial Differ. Equ., 13 (2016), 155-177.  doi: 10.4310/DPDE.2016.v13.n2.a4.

[21]

L. LiJ. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Diff. Equa., 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[22]

C. Lin and G. Nakamura, Unique continuation property for multi-terms time fractional diffusion equations, Math. Ann., 373 (2019), 929-952.  doi: 10.1007/s00208-018-1710-z.

[23]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[24] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010.  doi: 10.1142/9781848163300.
[25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[26]

I. Podlubny, Fractional-order systems and $PI^\lambda D^\mu$ controller, IEEE Trans. Auto. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[27]

X. B. Shu and Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2$, Comput. Math. Appl., 64 (2012), 2100-2110.  doi: 10.1016/j.camwa.2012.04.006.

[28]

C. C. Travis and G. F. Webb, Cosine families and abstractnonlinear second order differential equations, Acta Math. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.

[29]

V. V. Vasil'evS. G. Krein and S. I. Piskarev, Semigroups of operators, cosine operator functions, and linear differential equations, J. Soviet Math., 54 (1991), 1042-1129. 

[30]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equ., 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[31]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

[32] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press, Elsevier, 2016. 
[33]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.

show all references

References:
[1]

R. P. AgarwalD. BaleanuJ. J. NietoD. F. M. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Applied Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems (Second Edition), Birkhauser Verlag, 2011. doi: 10.1007/978-3-0348-5075-9.

[3]

K. BalachandranV. GovindarajL. Rodríguez-Germa and J. J. Trujillo, Controllability results for nonlinear fractional-order dynamical systems, J. Optimization Theory, Appl., 156 (2013), 33-44.  doi: 10.1007/s10957-012-0212-5.

[4]

M. BonforteY. Sire and J. L. Vázquez, Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal., 153 (2017), 142-168.  doi: 10.1016/j.na.2016.08.027.

[5]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.

[6]

L. A. Caffarelli and Y. Sire, Minimal surfaces and free boundaries: Recent developments, Bull. Amer. Math. Soc., 57 (2020), 91-106.  doi: 10.1090/bull/1673.

[7]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

[8]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland, Elsevier, 1985.

[9]

E. Fernández-CaraQ. Lü and E. Zuazua, Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54 (2016), 2009-2019.  doi: 10.1137/15M1044291.

[10]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Part. Diff. Eq., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

[11] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985. 
[12]

J. W. Hanneken, D. M. Vaught and B. N. Narahari Achar, Enumeration of the Real Zeros of the Mittag-Leffler Function $E_\alpha(z)$, $1 < \alpha < 2$, in Advances in Fractional Calculus, Springer, Dordrecht, 2007, 15–26. doi: 10.1007/978-1-4020-6042-7_2.

[13]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[15]

I. KimK. H. Kim and S. Lim, $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.

[16]

V. Komornik, Exact controllability in short time for the wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153-164.  doi: 10.1016/S0294-1449(16)30327-4.

[17]

K. LiJ. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.  doi: 10.1016/j.jfa.2012.04.011.

[18]

K. LiJ. Peng and J. Gao, Controllability of nonlocal fractional differential systems of order $\alpha\in (1, 2]$ in Banach spaces, Rep. Math. Phys., 71 (2013), 33-43.  doi: 10.1016/S0034-4877(13)60020-8.

[19]

Y. Li, Regularity of mild Solutions for fractional abstract Cauchy problem with order $\alpha\in (1, 2)$, Z. Angew. Math. Phys., 66 (2015), 3283-3298.  doi: 10.1007/s00033-015-0577-z.

[20]

Y. LiH. Sun and Z. Feng, Fractional abstract Cauchy problem with order $\alpha\in (1, 2)$, Dyn. Partial Differ. Equ., 13 (2016), 155-177.  doi: 10.4310/DPDE.2016.v13.n2.a4.

[21]

L. LiJ. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Diff. Equa., 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[22]

C. Lin and G. Nakamura, Unique continuation property for multi-terms time fractional diffusion equations, Math. Ann., 373 (2019), 929-952.  doi: 10.1007/s00208-018-1710-z.

[23]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[24] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010.  doi: 10.1142/9781848163300.
[25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[26]

I. Podlubny, Fractional-order systems and $PI^\lambda D^\mu$ controller, IEEE Trans. Auto. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[27]

X. B. Shu and Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2$, Comput. Math. Appl., 64 (2012), 2100-2110.  doi: 10.1016/j.camwa.2012.04.006.

[28]

C. C. Travis and G. F. Webb, Cosine families and abstractnonlinear second order differential equations, Acta Math. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.

[29]

V. V. Vasil'evS. G. Krein and S. I. Piskarev, Semigroups of operators, cosine operator functions, and linear differential equations, J. Soviet Math., 54 (1991), 1042-1129. 

[30]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equ., 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[31]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

[32] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press, Elsevier, 2016. 
[33]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.

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