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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  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 & Control Theory, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

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

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

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