March  2020, 10(1): 141-156. doi: 10.3934/mcrf.2019033

Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

1. 

College of Information Science and Technology, Donghua University, Shanghai 201620, China

2. 

School of Computer Science, China University of Geosciences, Wuhan 430074, China

3. 

School of Engineering (MESA-Lab), University of California, Merced, CA 95343, USA

4. 

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

* Corresponding author

Received  July 2018 Revised  February 2019 Published  March 2020 Early access  April 2019

Fund Project: The second author is supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences, Wuhan (No.CUG170627), the Natural Science Foundation of China (NSFC, No.KZ18W30084) and the Hubei NSFC (No.2017CFB279).

This paper investigates the regional gradient controllability for ultra-slow diffusion processes governed by the time fractional diffusion systems with a Hadamard-Caputo time fractional derivative. Some necessary and sufficient conditions on regional gradient exact and approximate controllability are first given and proved in detail. Secondly, we propose an approach on how to calculate the minimum number of $\omega-$strategic actuators. Moreover, the existence, uniqueness and the concrete form of the optimal controller for the system under consideration are presented by employing the Hilbert Uniqueness Method (HUM) among all the admissible ones. Finally, we illustrate our results by an interesting example.

Citation: Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control and Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033
References:
[1]

A. Aacute and D. Castillo-Negrete, Fluid limit of the continuous-time random walk with general Levy jump distribution functions, Physical Review E Statistical Nonlinear and Soft Matter Physics, 76 (2007), 041105.

[2]

S. AbbasM. BenchohraJ. E. Lazreg and Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability, Chaos Solitons and Fractals, 102 (2017), 47-71.  doi: 10.1016/j.chaos.2017.03.010.

[3] L. AfifiA. El Jai and E. Zerrik, Regional Analysis of Linear Distributed Parameter Systems, Princeton University Press, Princeton, 2005. 
[4]

B. Ahmad, S. K. Ntouyas and J. Tariboon, et al., Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer International Publishing, 2017. doi: 10.1007/978-3-319-52141-1.

[5]

P. L. ButzerA. A. Kilbas and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, Journal of Mathematical Analysis and Applications, 269 (2002), 387-400.  doi: 10.1016/S0022-247X(02)00049-5.

[6]

L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[7]

R. Garra and F. Polito, On some operators involving Hadamard derivatives, Integral Transforms and Special Functions, 24 (2013), 773-782.  doi: 10.1080/10652469.2012.756875.

[8]

F. Ge, Y. Q. Chen and C. Kou, Regional Analysis of Time-Fractional Diffusion Processes, Springer, 2018. doi: 10.1007/978-3-319-72896-4.

[9]

F. GeY. Q. Chen and C. Kou, Regional gradient controllability of sub-diffusion processes, Journal of Mathematical Analysis and Applications, 440 (2016), 865-884.  doi: 10.1016/j.jmaa.2016.03.051.

[10]

F. GeY. Q. ChenC. Kou and I. Podlubny, On the regional controllability of the sub-diffusion process with Caputo fractional derivative, Fractional Calculus and Applied Analysis, 19 (2016), 1262-1281.  doi: 10.1515/fca-2016-0065.

[11]

F. GeY. Q. Chen and C. Kou, Regional controllability analysis of fractional diffusion equations with Riemann-Liouville time fractional derivatives, Automatica, 76 (2017), 193-199.  doi: 10.1016/j.automatica.2016.10.018.

[12]

F. GeY. Q. Chen and C. Kou, Actuator characterisations to achieve approximate controllability for a class of fractional sub-diffusion equations, International Journal of Control, 90 (2017), 1212-1220.  doi: 10.1080/00207179.2016.1163619.

[13]

Z. Gong, D. Qian and C. Li, et al., On the Hadamard type fractional differential system, Fractional Dynamics and Control. Springer, New York, (2012), 159–171. doi: 10.1007/978-1-4614-0457-6_13.

[14]

V. Govindaraj and R. K. George, Controllability of fractional dynamical systems–-A functional analytic approach, Mathematical Control and Related Fields, 7 (2017), 537-562.  doi: 10.3934/mcrf.2017020.

[15]

J. R. GraefS. R. Grace and E. Tunc, Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivatives, Fractional Calculus and Applied Analysis, 20 (2017), 71-87.  doi: 10.1515/fca-2017-0004.

[16]

J. Hadamard, Essai sur letude des fonctions donnees par leur developpement de Taylor, Journal de Mathematiques Pures et Appliquees, 8 (1892), 101–186 (In French).

[17] A. EI Jai and A. J. Pritchard, Sensors and Controls in the Analysis of Distributed Systems, Halsted Press, 1988. 
[18]

F. JaradT. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012), 1-8.  doi: 10.1186/1687-1847-2012-142.

[19]

Q. Katatbeha and A. Al-Omarib, Existence and uniqueness of mild and classical solutions to fractional order Hadamard-type Cauchy problem, Journal of Nonlinear Science and Applications, 9 (2016), 827-835.  doi: 10.22436/jnsa.009.03.11.

[20]

F. A. KhodjaF. Chouly and M. Duprez, Partial null controllability of parabolic linear systems, Mathematical Control and Related Fields, 6 (2016), 185-216.  doi: 10.3934/mcrf.2016001.

[21]

F. A. KhodjaA. BenabdallahM. G. Burgos and L. Teresa, Recent results on the controllability of linear coupled parabolic problems–-A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[22]

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

[23]

A. A. Kilbas, Hadamard-type fractional calculus, Journal of the Korean Mathematical Society, 38 (2001), 1191-1204. 

[24]

H. LeivaN. Merentes and J. L. Sanchez, Approximate controllability of semilinear reaction diffusion equations, Mathematical Control and Related Fields, 2 (2012), 171-182.  doi: 10.3934/mcrf.2012.2.171.

[25]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Vol. 170, Springer Verlag, 1971.

[26]

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

[27]

Y. Liu, Survey and new results on boundary-value problems of singular fractional differential equations with impulse effects, Electronic Journal of Differential Equations, 296 (2016), 1-177. 

[28]

Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Mathematics of Control Signals and Systems, 28 (2016), Art. 10, 21 pp. doi: 10.1007/s00498-016-0162-9.

[29]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, Physics, (2007), 312–350.

[30]

F. Mainardi, A. Mura and G. Pagnini, et al., Sub-diffusion equations of fractional order and their fundamental solutions, Mathematical Methods in Engineering. Springer, (2007), 23–55.

[31]

T. Mur and H. R. Henriquez, Relative controllability of linear systems of fractional order with delay, Mathematical Control and Related Fields, 5 (2015), 845-858.  doi: 10.3934/mcrf.2015.5.845.

[32] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. 
[33]

Y. Povstenko, Fractional Thermoelasticity, Springer International Publishing, 2015. doi: 10.1007/978-3-319-15335-3.

[34]

A. J. Pritchard and A. Wirth, Unbounded control and observation systems and their duality, SIAM Journal on Control and Optimization, 16 (1978), 535-545.  doi: 10.1137/0316036.

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[36]

J. Tariboon, S. K. Ntouyas and C. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Advances in Mathematical Physics, 2014 (2014), Art. ID 372749, 15 pp. doi: 10.1155/2014/372749.

[37]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

[38]

C. Zeng and Y. Q. Chen, Optimal random search, fractional dynamics and fractional calculus, Fractional Calculus and Applied Analysis, 17 (2014), 321-332.  doi: 10.2478/s13540-014-0171-7.

[39]

E. ZerrikA. Boutoulout and A. Kamal, Regional gradient controllability of parabolic systems, International Journal of Applied Mathematics and Computer Science, 9 (1999), 767-787. 

[40]

E. ZerrikA. Kamal and A. Boutoulout, Regional gradient controllability and actuators, International Journal of Systems Science, 33 (2002), 239-246.  doi: 10.1080/00207720110073163.

[41]

E. Zerrik and F. Ghafrani, Regional gradient-constrained control problem. Approaches and simulations, Journal of Dynamical and Control Systems, 9 (2003), 585-599.  doi: 10.1023/A:1025652520034.

[42]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers and Mathematics with Applications, 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

show all references

References:
[1]

A. Aacute and D. Castillo-Negrete, Fluid limit of the continuous-time random walk with general Levy jump distribution functions, Physical Review E Statistical Nonlinear and Soft Matter Physics, 76 (2007), 041105.

[2]

S. AbbasM. BenchohraJ. E. Lazreg and Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability, Chaos Solitons and Fractals, 102 (2017), 47-71.  doi: 10.1016/j.chaos.2017.03.010.

[3] L. AfifiA. El Jai and E. Zerrik, Regional Analysis of Linear Distributed Parameter Systems, Princeton University Press, Princeton, 2005. 
[4]

B. Ahmad, S. K. Ntouyas and J. Tariboon, et al., Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer International Publishing, 2017. doi: 10.1007/978-3-319-52141-1.

[5]

P. L. ButzerA. A. Kilbas and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, Journal of Mathematical Analysis and Applications, 269 (2002), 387-400.  doi: 10.1016/S0022-247X(02)00049-5.

[6]

L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[7]

R. Garra and F. Polito, On some operators involving Hadamard derivatives, Integral Transforms and Special Functions, 24 (2013), 773-782.  doi: 10.1080/10652469.2012.756875.

[8]

F. Ge, Y. Q. Chen and C. Kou, Regional Analysis of Time-Fractional Diffusion Processes, Springer, 2018. doi: 10.1007/978-3-319-72896-4.

[9]

F. GeY. Q. Chen and C. Kou, Regional gradient controllability of sub-diffusion processes, Journal of Mathematical Analysis and Applications, 440 (2016), 865-884.  doi: 10.1016/j.jmaa.2016.03.051.

[10]

F. GeY. Q. ChenC. Kou and I. Podlubny, On the regional controllability of the sub-diffusion process with Caputo fractional derivative, Fractional Calculus and Applied Analysis, 19 (2016), 1262-1281.  doi: 10.1515/fca-2016-0065.

[11]

F. GeY. Q. Chen and C. Kou, Regional controllability analysis of fractional diffusion equations with Riemann-Liouville time fractional derivatives, Automatica, 76 (2017), 193-199.  doi: 10.1016/j.automatica.2016.10.018.

[12]

F. GeY. Q. Chen and C. Kou, Actuator characterisations to achieve approximate controllability for a class of fractional sub-diffusion equations, International Journal of Control, 90 (2017), 1212-1220.  doi: 10.1080/00207179.2016.1163619.

[13]

Z. Gong, D. Qian and C. Li, et al., On the Hadamard type fractional differential system, Fractional Dynamics and Control. Springer, New York, (2012), 159–171. doi: 10.1007/978-1-4614-0457-6_13.

[14]

V. Govindaraj and R. K. George, Controllability of fractional dynamical systems–-A functional analytic approach, Mathematical Control and Related Fields, 7 (2017), 537-562.  doi: 10.3934/mcrf.2017020.

[15]

J. R. GraefS. R. Grace and E. Tunc, Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivatives, Fractional Calculus and Applied Analysis, 20 (2017), 71-87.  doi: 10.1515/fca-2017-0004.

[16]

J. Hadamard, Essai sur letude des fonctions donnees par leur developpement de Taylor, Journal de Mathematiques Pures et Appliquees, 8 (1892), 101–186 (In French).

[17] A. EI Jai and A. J. Pritchard, Sensors and Controls in the Analysis of Distributed Systems, Halsted Press, 1988. 
[18]

F. JaradT. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012), 1-8.  doi: 10.1186/1687-1847-2012-142.

[19]

Q. Katatbeha and A. Al-Omarib, Existence and uniqueness of mild and classical solutions to fractional order Hadamard-type Cauchy problem, Journal of Nonlinear Science and Applications, 9 (2016), 827-835.  doi: 10.22436/jnsa.009.03.11.

[20]

F. A. KhodjaF. Chouly and M. Duprez, Partial null controllability of parabolic linear systems, Mathematical Control and Related Fields, 6 (2016), 185-216.  doi: 10.3934/mcrf.2016001.

[21]

F. A. KhodjaA. BenabdallahM. G. Burgos and L. Teresa, Recent results on the controllability of linear coupled parabolic problems–-A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[22]

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

[23]

A. A. Kilbas, Hadamard-type fractional calculus, Journal of the Korean Mathematical Society, 38 (2001), 1191-1204. 

[24]

H. LeivaN. Merentes and J. L. Sanchez, Approximate controllability of semilinear reaction diffusion equations, Mathematical Control and Related Fields, 2 (2012), 171-182.  doi: 10.3934/mcrf.2012.2.171.

[25]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Vol. 170, Springer Verlag, 1971.

[26]

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

[27]

Y. Liu, Survey and new results on boundary-value problems of singular fractional differential equations with impulse effects, Electronic Journal of Differential Equations, 296 (2016), 1-177. 

[28]

Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Mathematics of Control Signals and Systems, 28 (2016), Art. 10, 21 pp. doi: 10.1007/s00498-016-0162-9.

[29]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, Physics, (2007), 312–350.

[30]

F. Mainardi, A. Mura and G. Pagnini, et al., Sub-diffusion equations of fractional order and their fundamental solutions, Mathematical Methods in Engineering. Springer, (2007), 23–55.

[31]

T. Mur and H. R. Henriquez, Relative controllability of linear systems of fractional order with delay, Mathematical Control and Related Fields, 5 (2015), 845-858.  doi: 10.3934/mcrf.2015.5.845.

[32] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. 
[33]

Y. Povstenko, Fractional Thermoelasticity, Springer International Publishing, 2015. doi: 10.1007/978-3-319-15335-3.

[34]

A. J. Pritchard and A. Wirth, Unbounded control and observation systems and their duality, SIAM Journal on Control and Optimization, 16 (1978), 535-545.  doi: 10.1137/0316036.

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[36]

J. Tariboon, S. K. Ntouyas and C. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Advances in Mathematical Physics, 2014 (2014), Art. ID 372749, 15 pp. doi: 10.1155/2014/372749.

[37]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

[38]

C. Zeng and Y. Q. Chen, Optimal random search, fractional dynamics and fractional calculus, Fractional Calculus and Applied Analysis, 17 (2014), 321-332.  doi: 10.2478/s13540-014-0171-7.

[39]

E. ZerrikA. Boutoulout and A. Kamal, Regional gradient controllability of parabolic systems, International Journal of Applied Mathematics and Computer Science, 9 (1999), 767-787. 

[40]

E. ZerrikA. Kamal and A. Boutoulout, Regional gradient controllability and actuators, International Journal of Systems Science, 33 (2002), 239-246.  doi: 10.1080/00207720110073163.

[41]

E. Zerrik and F. Ghafrani, Regional gradient-constrained control problem. Approaches and simulations, Journal of Dynamical and Control Systems, 9 (2003), 585-599.  doi: 10.1023/A:1025652520034.

[42]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers and Mathematics with Applications, 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

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