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February  2022, 11(1): 95-124. doi: 10.3934/eect.2020104

Complete controllability for a class of fractional evolution equations with uncertainty

1. 

Faculty of Natural Science, Hanoi Metropolitan University, Hanoi, Vietnam

2. 

Department of Mathematics, Hanoi Pedagogical University 2, Hanoi, Vietnam

3. 

Vietnam National University, Hanoi, Vietnam

4. 

Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

5. 

Faculty of Information Technology, People's Police University of Technology and Logistics, Bac Ninh, Vietnam

* Corresponding author: Hoang Viet Long

Received  May 2019 Revised  August 2020 Published  February 2022 Early access  December 2020

Fund Project: This work is supported by NAFOSTED - Vietnam under grant contract 101.02-2018.311

In this paper, we study the complete controllability for a class of fractional evolution equations with a common type of fuzzy uncertainty. By using Hausdorff measure of noncompactness and Krasnoselskii's fixed point theorem in complete semilinear metric space, we give some sufficient conditions of the controllability for the fuzzy fractional evolution equations without involving the compactness of strongly continuous semigroup and the perturbation function. In addition, the controllable problem is considered in a subspace of fuzzy numbers in which the gH-differences always exist, that guarantees the satisfaction of hypotheses of the problem. An application example related to electrical circuit is given to illustrate the effectiveness of theoretical results.

Citation: Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations and Control Theory, 2022, 11 (1) : 95-124. doi: 10.3934/eect.2020104
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. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[2]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, vol. 55, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

[3]

T. Allahviranloo and M. B. Ahmadi, Fuzzy Laplace transforms, Soft Comput., 14 (2010), Art. no. 235. doi: 10.1007/s00500-008-0397-6.

[4]

C. T. Anh and T. D. Ke, On nonlocal problems for retarded fractional differential equations in Banach spaces, Fixed Point Theory, 15 (2014), 373-392. 

[5]

G. Arthi and K. Balachandran, Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions, J. Control Theory Appl., 11 (2013), 186-192.  doi: 10.1007/s11768-013-1084-4.

[6]

P. Balasubramaniam, Controllability for the nonlinear fuzzy neutral functional differential equations, Far East J. Appl. Math., 9 (2002), 31-48. 

[7]

D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-0457-6.

[8]

B. Bede, Mathematics of Fuzzy Sets and Fuzzy Logic, Studies in Fuzziness and Soft Computing, vol. 295, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35221-8.

[9]

B. Bede and S. G. Gal, Generalizations of the differential of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.

[10]

B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.  doi: 10.1016/j.fss.2012.10.003.

[11]

A. ChaddhaS. N. Bora and R. Sakthivel, Approximate controllability of impulsive stochastic fractional differential equations with nonlocal conditions, Dyn. Syst. Appl., 27 (2018), 1-29. 

[12]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[13]

T. DonchevA. Nosheen and V. Lupulescu, Fuzzy integro-differential equations with compactness type conditions, Hacet. J. Math. Stat., 43 (2014), 249-257. 

[14]

X. Fu, Controllability of abstract neutral functional differential systems with unbounded delay, Appl. Math. Comput., 151 (2004), 299-314.  doi: 10.1016/S0096-3003(03)00342-4.

[15]

C. S. Gal and S. G. Gal, Semigroup of mappings on spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, J. Fuzzy Math., 13 (2005), 647-682. 

[16]

R. GaneshR. Sakthivel and N. Mahmudov, Approximate controllability of fractional functional equations with infinite delay, Topol. Methods Nonlinear Anal., 43 (2014), 345-364.  doi: 10.12775/TMNA.2014.020.

[17]

J. H. JeongJ. S. KimH. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, J. Comput. Anal. Appl., 23 (2017), 1056-1069. 

[18]

S. JiG. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217 (2011), 6981-6989.  doi: 10.1016/j.amc.2011.01.107.

[19]

O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35 (1990), 389-396.  doi: 10.1016/0165-0114(90)90010-4.

[20]

R. E. Kalman, Lectures on Controllability and Observability, Edizioni Cremonese, Rome, Italy, 1968.

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006.

[22]

A. Khastan, A new representation for inverse fuzzy transform and its applications, Soft Comput., 21 (2017), 3503-3512.  doi: 10.1007/s00500-017-2555-1.

[23]

A. Khastan, New solutions for first order linear fuzzy difference equations, J. Comput. Appl. Math., 312 (2017), 156-166.  doi: 10.1016/j.cam.2016.03.004.

[24]

A. Khastan and R. Rodríguez-López, An existence and uniqueness result for fuzzy Goursat partial differential equation, Fuzzy Sets and Systems, 375 (2019), 141-160.  doi: 10.1016/j.fss.2019.02.011.

[25]

Y. C. KwunJ. S. KimH. E. Youm and J. H. Park, Approximate controllability for fuzzy differential equations driven by Liu process, J. Comput. Anal. Appl., 15 (2013), 163-175. 

[26]

V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Series in Mathematical Analysis and Applications, Taylor & Francis Group, London, 2003. doi: 10.1201/9780203011386.

[27]

J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Appl. Math. Comput., 254 (2015), 20-29. 

[28]

H. V. Long and N. P. Dong, An extension of Krasnoselskii's fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, J. Fixed Point Theory Appl., 20 (2018), Paper no. 37, 27 pp. doi: 10.1007/s11784-018-0507-8.

[29]

H. V. LongN. T. K. Son and H. T. T. Tam, The solvability of fuzzy partial differential equations under Caputo gH-differentiability, Fuzzy Sets and Systems, 309 (2017), 35-63.  doi: 10.1016/j.fss.2016.06.018.

[30]

V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 265 (2015), 63-85.  doi: 10.1016/j.fss.2014.04.005.

[31]

M. MuslimA. Kumar and R. Sakthivel, Exact and trajectory controllability of second-order systems with impulsive and and deviated arguments, Math. Methods Appl. Sci., 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.

[32]

S. Narayanamoorthy and S. Sowmiya, Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, Adv. Difference Equ., 121 (2015), 16 pp. doi: 10.1186/s13662-015-0454-2.

[33]

I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999.

[34]

R. SakthivelN. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093.

[35]

S. SalahshourT. Allahviranloo and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372-1381.  doi: 10.1016/j.cnsns.2011.07.005.

[36]

S. Salahshour, T. Allahviranloo, S. Abbasbandy and D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty, Adv. Difference Equ., (2012), Art. no. 112, 12 pp. doi: 10.1186/1687-1847-2012-112.

[37]

N. T. K. Son, A foundation on semigroup of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations, Fuzzy Sets and Systems, 347 (2018), 1-28.  doi: 10.1016/j.fss.2018.02.003.

[38]

N. T. K. Son and N. P. Dong, Asymptotic behavior of $C^0$-solutions of evolution equations with uncertainties, J. Fixed Point Theory Appl., 20, (2018), Paper no. 153, 30 pp. doi: 10.1007/s11784-018-0633-3.

[39]

L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311-1328.  doi: 10.1016/j.na.2008.12.005.

[40]

L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, 161 (2010), 1564-1584.  doi: 10.1016/j.fss.2009.06.009.

[41]

S. TomasielloS. K. Khattri and J. Awrejcewicz, Differential quadrature-based simulation of a class of fuzzy damped fractional dynamical systems, Int. J. Numer. Anal. Model., 14 (2017), 63-75. 

[42]

S. Tomasiello and J. E. Macias-Diaz, Note on a Picard-like method for Caputo fuzzy fractional differential equations, Appl. Math. Inf. Sci., 11 (2017), 281-287.  doi: 10.18576/amis/110134.

[43]

J. R. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.

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. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[2]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, vol. 55, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

[3]

T. Allahviranloo and M. B. Ahmadi, Fuzzy Laplace transforms, Soft Comput., 14 (2010), Art. no. 235. doi: 10.1007/s00500-008-0397-6.

[4]

C. T. Anh and T. D. Ke, On nonlocal problems for retarded fractional differential equations in Banach spaces, Fixed Point Theory, 15 (2014), 373-392. 

[5]

G. Arthi and K. Balachandran, Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions, J. Control Theory Appl., 11 (2013), 186-192.  doi: 10.1007/s11768-013-1084-4.

[6]

P. Balasubramaniam, Controllability for the nonlinear fuzzy neutral functional differential equations, Far East J. Appl. Math., 9 (2002), 31-48. 

[7]

D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-0457-6.

[8]

B. Bede, Mathematics of Fuzzy Sets and Fuzzy Logic, Studies in Fuzziness and Soft Computing, vol. 295, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35221-8.

[9]

B. Bede and S. G. Gal, Generalizations of the differential of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.

[10]

B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.  doi: 10.1016/j.fss.2012.10.003.

[11]

A. ChaddhaS. N. Bora and R. Sakthivel, Approximate controllability of impulsive stochastic fractional differential equations with nonlocal conditions, Dyn. Syst. Appl., 27 (2018), 1-29. 

[12]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[13]

T. DonchevA. Nosheen and V. Lupulescu, Fuzzy integro-differential equations with compactness type conditions, Hacet. J. Math. Stat., 43 (2014), 249-257. 

[14]

X. Fu, Controllability of abstract neutral functional differential systems with unbounded delay, Appl. Math. Comput., 151 (2004), 299-314.  doi: 10.1016/S0096-3003(03)00342-4.

[15]

C. S. Gal and S. G. Gal, Semigroup of mappings on spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, J. Fuzzy Math., 13 (2005), 647-682. 

[16]

R. GaneshR. Sakthivel and N. Mahmudov, Approximate controllability of fractional functional equations with infinite delay, Topol. Methods Nonlinear Anal., 43 (2014), 345-364.  doi: 10.12775/TMNA.2014.020.

[17]

J. H. JeongJ. S. KimH. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, J. Comput. Anal. Appl., 23 (2017), 1056-1069. 

[18]

S. JiG. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217 (2011), 6981-6989.  doi: 10.1016/j.amc.2011.01.107.

[19]

O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35 (1990), 389-396.  doi: 10.1016/0165-0114(90)90010-4.

[20]

R. E. Kalman, Lectures on Controllability and Observability, Edizioni Cremonese, Rome, Italy, 1968.

[21]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006.

[22]

A. Khastan, A new representation for inverse fuzzy transform and its applications, Soft Comput., 21 (2017), 3503-3512.  doi: 10.1007/s00500-017-2555-1.

[23]

A. Khastan, New solutions for first order linear fuzzy difference equations, J. Comput. Appl. Math., 312 (2017), 156-166.  doi: 10.1016/j.cam.2016.03.004.

[24]

A. Khastan and R. Rodríguez-López, An existence and uniqueness result for fuzzy Goursat partial differential equation, Fuzzy Sets and Systems, 375 (2019), 141-160.  doi: 10.1016/j.fss.2019.02.011.

[25]

Y. C. KwunJ. S. KimH. E. Youm and J. H. Park, Approximate controllability for fuzzy differential equations driven by Liu process, J. Comput. Anal. Appl., 15 (2013), 163-175. 

[26]

V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Series in Mathematical Analysis and Applications, Taylor & Francis Group, London, 2003. doi: 10.1201/9780203011386.

[27]

J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Appl. Math. Comput., 254 (2015), 20-29. 

[28]

H. V. Long and N. P. Dong, An extension of Krasnoselskii's fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty, J. Fixed Point Theory Appl., 20 (2018), Paper no. 37, 27 pp. doi: 10.1007/s11784-018-0507-8.

[29]

H. V. LongN. T. K. Son and H. T. T. Tam, The solvability of fuzzy partial differential equations under Caputo gH-differentiability, Fuzzy Sets and Systems, 309 (2017), 35-63.  doi: 10.1016/j.fss.2016.06.018.

[30]

V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 265 (2015), 63-85.  doi: 10.1016/j.fss.2014.04.005.

[31]

M. MuslimA. Kumar and R. Sakthivel, Exact and trajectory controllability of second-order systems with impulsive and and deviated arguments, Math. Methods Appl. Sci., 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.

[32]

S. Narayanamoorthy and S. Sowmiya, Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, Adv. Difference Equ., 121 (2015), 16 pp. doi: 10.1186/s13662-015-0454-2.

[33]

I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999.

[34]

R. SakthivelN. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093.

[35]

S. SalahshourT. Allahviranloo and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372-1381.  doi: 10.1016/j.cnsns.2011.07.005.

[36]

S. Salahshour, T. Allahviranloo, S. Abbasbandy and D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty, Adv. Difference Equ., (2012), Art. no. 112, 12 pp. doi: 10.1186/1687-1847-2012-112.

[37]

N. T. K. Son, A foundation on semigroup of operators defined on the set of triangular fuzzy numbers and its application to fuzzy fractional evolution equations, Fuzzy Sets and Systems, 347 (2018), 1-28.  doi: 10.1016/j.fss.2018.02.003.

[38]

N. T. K. Son and N. P. Dong, Asymptotic behavior of $C^0$-solutions of evolution equations with uncertainties, J. Fixed Point Theory Appl., 20, (2018), Paper no. 153, 30 pp. doi: 10.1007/s11784-018-0633-3.

[39]

L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311-1328.  doi: 10.1016/j.na.2008.12.005.

[40]

L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, 161 (2010), 1564-1584.  doi: 10.1016/j.fss.2009.06.009.

[41]

S. TomasielloS. K. Khattri and J. Awrejcewicz, Differential quadrature-based simulation of a class of fuzzy damped fractional dynamical systems, Int. J. Numer. Anal. Model., 14 (2017), 63-75. 

[42]

S. Tomasiello and J. E. Macias-Diaz, Note on a Picard-like method for Caputo fuzzy fractional differential equations, Appl. Math. Inf. Sci., 11 (2017), 281-287.  doi: 10.18576/amis/110134.

[43]

J. R. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.

Figure 1.  The level sets $ [u]^\alpha $ of a triangular fuzzy number $ u $
Figure 2.  The gH-differences $ w = u\ominus_{gH}v $ and $ z = v\ominus_{gH}u $ of fuzzy numbers $ u = (3, 6, 9) $ and $ v = (0, 1, 2)$
Figure 3.  The electrical circuit diagram
Figure 4.  The fuzzy solutions of the electrical circuit model without control input solved by $ \mathtt{fde12.m} $ and $\mathtt{flmm2.m} $
Table 1.  The parameters of the electrical circuit
Parameters Description The value
$ R_1 $ The first resistance $ 1 $ $ \Omega $
$ R_2 $ The second resistance $ 2 $ $ \Omega $
$ R_3 $ The third resistance $ 1 $ $ \Omega $
$ L_1 $ The inductance of the wire 1 $ 0.5 $ H
$ L_2 $ The inductance of the wire 2 $ 1 $ H
$ b $ The amplitude ("approximately $ 0.2 $") $ (0.15, 0.2, 0.25) $
$ \beta $ The fractional order $ \frac{1}{2} $
$ [0, T] $ Time $ [0, 1] $
Parameters Description The value
$ R_1 $ The first resistance $ 1 $ $ \Omega $
$ R_2 $ The second resistance $ 2 $ $ \Omega $
$ R_3 $ The third resistance $ 1 $ $ \Omega $
$ L_1 $ The inductance of the wire 1 $ 0.5 $ H
$ L_2 $ The inductance of the wire 2 $ 1 $ H
$ b $ The amplitude ("approximately $ 0.2 $") $ (0.15, 0.2, 0.25) $
$ \beta $ The fractional order $ \frac{1}{2} $
$ [0, T] $ Time $ [0, 1] $
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