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September  2020, 9(3): 845-863. doi: 10.3934/eect.2020036

Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability

Yong-Kui Chang , and  Xiaojing Liu

School of Mathematics and Statistics, Xidian University, Xi'an 710071, Shaanxi, China

* Corresponding author: Y.-K. Chang

Received  May 2019 Revised  January 2020 Published  March 2020

In this paper, we mainly consider a time-varying semi-linear integro-differential inclusion with Clarke sub-differential and a non-local initial condition. By a suitable Green function combined with a resolvent operator, we firstly formulate its mild solutions and show that it admits at least one mild solution which can exist in a well-defined ball with a radius big enough. Through constructing a proper functional, we then derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a sufficient condition for the approximate controllability of the time-varying semi-linear integro-differential inclusion. Lastly, we also consider the finite approximate controllability of the time-varying semi-linear integro-differential inclusion via variational method.

Keywords: Approximate controllability, integro-differential inclusion with Clarke sub-differential, non-local initial conditions, variational method.
Mathematics Subject Classification: Primary: 93B05, 35R09; Secondary: 35R70, 34A60.
Citation: Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations & Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036
References:
[1]

K. Balchandran and J. P. Dauer, Controllability of nonlinear systems in Banach spaces: A survey, J. Optim. Theory Appl., 115 (2002), 7-28.  doi: 10.1023/A:1019668728098.  Google Scholar

[2]

K. Balchandran and J. H. Kim, Remarks on the paper "Controllability of second order differential inclusion in Banach spaces", J. Math. Anal. Appl., 324 (2006), 746-749.  doi: 10.1016/j.jmaa.2005.11.070.  Google Scholar

[3]

A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821.  doi: 10.1137/S036301299732184X.  Google Scholar

[4]

M. Benchohra and M. S. Souid, $L^1$-Solutions for implicit fractional order differential equations with nonlocal conditions, Filomat, 30 (2016), 1485-1492.  doi: 10.2298/FIL1606485B.  Google Scholar

[5]

L. Byszewski, Existence and uniqueness of a classical solutions to a functional differential abstract nonlocal Cauchy problem, J. Math. Appl. Stoch. Anal., 12 (1999), 91-97.  doi: 10.1155/S1048953399000088.  Google Scholar

[6]

Y.-K. Chang and R. Ponce, Uniform exponential stability and its applications to bounded solutions of integro-differential equations in Banach spaces, J. Integral Equations Appl., 30 (2018), 347-369.  doi: 10.1216/JIE-2018-30-3-347.  Google Scholar

[7]

Y.-K. Chang and Y. Pei, Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators, Int. J. Control, (2018). Available from: https://doi.org/10.1080/00207179.2018.1479540. doi: 10.1080/00207179.2018.1479540.  Google Scholar

[8] R. M. Christensen, The Theory of Linear Viscoelasticity: An Introduction, Academic Press, New York, 1982.   Google Scholar
[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[10]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[11]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[12]

K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar

[13]

M. Dieye, M. A. Diopa and K. Ezzinbi, On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Stat. Prob. Lett., 123 (2017), 61-76.  doi: 10.1016/j.spl.2016.10.031.  Google Scholar

[14]

S. Djebali, L. Górniewicz and A. Ouahab, Solutions Set for Differential Equations and Inclusions, De Gruyter Series in Nonlinear Analysis and Applications, 18. Walter de Gruyter & Co., Berlin, 2013. doi: 10.1515/9783110293562.  Google Scholar

[15]

K. Ezzinbi and S. Ghnimi, Existence and regularity of solutions for neutral partial functional integrodifferential equations, Nonlinear Anal. RWA, 11 (2010), 2335-2344.  doi: 10.1016/j.nonrwa.2009.07.007.  Google Scholar

[16]

X. L. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theo., 6 (2017), 517-534.  doi: 10.3934/eect.2017026.  Google Scholar

[17]

R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar

[18]

R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in a Banach space, J. Differential Equations, 50 (1983), 234-259.  doi: 10.1016/0022-0396(83)90076-1.  Google Scholar

[19]

Y. R. Jiang, N.-J. Huang and J.-C. Yao, Solvability and opitimal control of semilinear nonlocal fractional evolution inclusion with Clarke subdifferential, Appl. Anal., 96 (2017), 2349-2366.  doi: 10.1080/00036811.2017.1321111.  Google Scholar

[20]

J.-L. Lions and E. Zuazua, The cost of controlling unstable systems: Time irreversible systems, Rev. Mat. UCM, 10 (1997), 481-523.   Google Scholar

[21]

J. H. Liu and K. Ezzinbi, Non-autonomous integrodifferential equations with nonlocal conditions, J. Integral Equations Appl., 15 (2003), 79-93.  doi: 10.1216/jiea/1181074946.  Google Scholar

[22]

Z. H. Liu and X. W. Li, Approxiamte controllability for a class of hemivariational inequalities, Nonlinear Anal. RWA, 22 (2015), 581-591.  doi: 10.1016/j.nonrwa.2014.08.010.  Google Scholar

[23]

Z. H. Liu, X. W. Li and D. Motreanu, Approxiamte controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces, SIAM J. Control Optim., 53 (2015), 3228-3244.  doi: 10.1137/140994058.  Google Scholar

[24]

C. Lizama and G. M. N'Guérékata, Bounded mild solutions for semilinear integrodifferential equations in Banach spaces, Integr. Equ. Oper. Theory, 68 (2010), 207-227.  doi: 10.1007/s00020-010-1799-2.  Google Scholar

[25]

C. Lizama and G. M. N'Guérékata, Mild solutions for abstract fractional differential equations, Appl. Anal., 92 (2013), 1731-1754.  doi: 10.1080/00036811.2012.698271.  Google Scholar

[26]

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

[27]

L. Lu, Z. H. Liu, W. Jiang and J. L. Luo, Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities, Math. Methods Appl. Sci., 39 (2016), 5452-5464.  doi: 10.1002/mma.3930.  Google Scholar

[28]

N. I. Mahmudov, Finite-approximate controllability of evolution equations, Appl. Comput. Math., 16 (2017), 159-167.   Google Scholar

[29]

N. I. Mahmudov, Finite-approximate controllability of fractional evolution equations: Variational approach, Fract. Calc. Appl. Anal., 21 (2018), 919-936.  doi: 10.1515/fca-2018-0050.  Google Scholar

[30]

N. I. Mahmudov, Variational approach to finite-approximate controllability of Sobolev-type fractional systems, J. Optim. Theory Appl., 184 (2020), 671-686.  doi: 10.1007/s10957-018-1255-z.  Google Scholar

[31]

L. Mahto, S. Abbas, M. Hafayed and H. M. Srivastava, Approximate controllability of sub-diffusion equation with impulsive condition, Mathematics, 7 (2019), 190. Available from: https://doi.org/10.3390/math7020190. doi: 10.3390/math7020190.  Google Scholar

[32] M. A. Meyers and K. K. Chawla, Mechanical Behavior of Materials, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511810947.  Google Scholar
[33]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[34]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with $p$-Laplacian, Trans. Amer. Math. Soc., 360 (2008), 2527-2545.  doi: 10.1090/S0002-9947-07-04449-2.  Google Scholar

[35]

P. D. Panagiotopoulos, Nonconvex superpotentials in sense of F. H. Clarke and applications, Mech. Res. Comm., 8 (1981), 335-340.  doi: 10.1016/0093-6413(81)90064-1.  Google Scholar

[36]

P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar

[37]

R. Ravi Kumar, Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces, Appl. Math. Comput., 204 (2008), 352-362.  doi: 10.1016/j.amc.2008.06.050.  Google Scholar

[38]

R. Ravi Kumar, Regularity of solutions of evolution integrodifferential equations with deviating argument, Appl. Math. Comput., 217 (2011), 9111-9121.  doi: 10.1016/j.amc.2011.03.136.  Google Scholar

[39]

R. Sakthivel, Y. Ren, A. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar

[40]

R.-N. Wang, K. Ezzinbi and P. X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar

[41]

J.-Z. Xiao and X.-H. Zhu, Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures, Topol. Method Nonlinear Anal., 52 (2018), 353-372.  doi: 10.12775/TMNA.2018.030.  Google Scholar

[42]

Y.-B. Xiao, X. M. Yang and N.-J. Huang, Some equivalence results for well-posedness of hemivariational inequalities, J. Global Optim., 61 (2015), 789-802.  doi: 10.1007/s10898-014-0198-7.  Google Scholar

[43]

Z. M. Yan and X. M. Jia, Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay, Collect. Math., 66 (2015), 93-124.  doi: 10.1007/s13348-014-0109-8.  Google Scholar

[44]

M. Yang and Q. R. Wang, Existence of mild soltions for a class of Hilfer fractional evolution equations with nonlocal conditions, Fract. Calc. Appl. Anal., 20 (2017), 679-705.  doi: 10.1515/fca-2017-0036.  Google Scholar

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

show all references

References:
[1]

K. Balchandran and J. P. Dauer, Controllability of nonlinear systems in Banach spaces: A survey, J. Optim. Theory Appl., 115 (2002), 7-28.  doi: 10.1023/A:1019668728098.  Google Scholar

[2]

K. Balchandran and J. H. Kim, Remarks on the paper "Controllability of second order differential inclusion in Banach spaces", J. Math. Anal. Appl., 324 (2006), 746-749.  doi: 10.1016/j.jmaa.2005.11.070.  Google Scholar

[3]

A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821.  doi: 10.1137/S036301299732184X.  Google Scholar

[4]

M. Benchohra and M. S. Souid, $L^1$-Solutions for implicit fractional order differential equations with nonlocal conditions, Filomat, 30 (2016), 1485-1492.  doi: 10.2298/FIL1606485B.  Google Scholar

[5]

L. Byszewski, Existence and uniqueness of a classical solutions to a functional differential abstract nonlocal Cauchy problem, J. Math. Appl. Stoch. Anal., 12 (1999), 91-97.  doi: 10.1155/S1048953399000088.  Google Scholar

[6]

Y.-K. Chang and R. Ponce, Uniform exponential stability and its applications to bounded solutions of integro-differential equations in Banach spaces, J. Integral Equations Appl., 30 (2018), 347-369.  doi: 10.1216/JIE-2018-30-3-347.  Google Scholar

[7]

Y.-K. Chang and Y. Pei, Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators, Int. J. Control, (2018). Available from: https://doi.org/10.1080/00207179.2018.1479540. doi: 10.1080/00207179.2018.1479540.  Google Scholar

[8] R. M. Christensen, The Theory of Linear Viscoelasticity: An Introduction, Academic Press, New York, 1982.   Google Scholar
[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[10]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[11]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[12]

K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar

[13]

M. Dieye, M. A. Diopa and K. Ezzinbi, On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Stat. Prob. Lett., 123 (2017), 61-76.  doi: 10.1016/j.spl.2016.10.031.  Google Scholar

[14]

S. Djebali, L. Górniewicz and A. Ouahab, Solutions Set for Differential Equations and Inclusions, De Gruyter Series in Nonlinear Analysis and Applications, 18. Walter de Gruyter & Co., Berlin, 2013. doi: 10.1515/9783110293562.  Google Scholar

[15]

K. Ezzinbi and S. Ghnimi, Existence and regularity of solutions for neutral partial functional integrodifferential equations, Nonlinear Anal. RWA, 11 (2010), 2335-2344.  doi: 10.1016/j.nonrwa.2009.07.007.  Google Scholar

[16]

X. L. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theo., 6 (2017), 517-534.  doi: 10.3934/eect.2017026.  Google Scholar

[17]

R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar

[18]

R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in a Banach space, J. Differential Equations, 50 (1983), 234-259.  doi: 10.1016/0022-0396(83)90076-1.  Google Scholar

[19]

Y. R. Jiang, N.-J. Huang and J.-C. Yao, Solvability and opitimal control of semilinear nonlocal fractional evolution inclusion with Clarke subdifferential, Appl. Anal., 96 (2017), 2349-2366.  doi: 10.1080/00036811.2017.1321111.  Google Scholar

[20]

J.-L. Lions and E. Zuazua, The cost of controlling unstable systems: Time irreversible systems, Rev. Mat. UCM, 10 (1997), 481-523.   Google Scholar

[21]

J. H. Liu and K. Ezzinbi, Non-autonomous integrodifferential equations with nonlocal conditions, J. Integral Equations Appl., 15 (2003), 79-93.  doi: 10.1216/jiea/1181074946.  Google Scholar

[22]

Z. H. Liu and X. W. Li, Approxiamte controllability for a class of hemivariational inequalities, Nonlinear Anal. RWA, 22 (2015), 581-591.  doi: 10.1016/j.nonrwa.2014.08.010.  Google Scholar

[23]

Z. H. Liu, X. W. Li and D. Motreanu, Approxiamte controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces, SIAM J. Control Optim., 53 (2015), 3228-3244.  doi: 10.1137/140994058.  Google Scholar

[24]

C. Lizama and G. M. N'Guérékata, Bounded mild solutions for semilinear integrodifferential equations in Banach spaces, Integr. Equ. Oper. Theory, 68 (2010), 207-227.  doi: 10.1007/s00020-010-1799-2.  Google Scholar

[25]

C. Lizama and G. M. N'Guérékata, Mild solutions for abstract fractional differential equations, Appl. Anal., 92 (2013), 1731-1754.  doi: 10.1080/00036811.2012.698271.  Google Scholar

[26]

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

[27]

L. Lu, Z. H. Liu, W. Jiang and J. L. Luo, Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities, Math. Methods Appl. Sci., 39 (2016), 5452-5464.  doi: 10.1002/mma.3930.  Google Scholar

[28]

N. I. Mahmudov, Finite-approximate controllability of evolution equations, Appl. Comput. Math., 16 (2017), 159-167.   Google Scholar

[29]

N. I. Mahmudov, Finite-approximate controllability of fractional evolution equations: Variational approach, Fract. Calc. Appl. Anal., 21 (2018), 919-936.  doi: 10.1515/fca-2018-0050.  Google Scholar

[30]

N. I. Mahmudov, Variational approach to finite-approximate controllability of Sobolev-type fractional systems, J. Optim. Theory Appl., 184 (2020), 671-686.  doi: 10.1007/s10957-018-1255-z.  Google Scholar

[31]

L. Mahto, S. Abbas, M. Hafayed and H. M. Srivastava, Approximate controllability of sub-diffusion equation with impulsive condition, Mathematics, 7 (2019), 190. Available from: https://doi.org/10.3390/math7020190. doi: 10.3390/math7020190.  Google Scholar

[32] M. A. Meyers and K. K. Chawla, Mechanical Behavior of Materials, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511810947.  Google Scholar
[33]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[34]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with $p$-Laplacian, Trans. Amer. Math. Soc., 360 (2008), 2527-2545.  doi: 10.1090/S0002-9947-07-04449-2.  Google Scholar

[35]

P. D. Panagiotopoulos, Nonconvex superpotentials in sense of F. H. Clarke and applications, Mech. Res. Comm., 8 (1981), 335-340.  doi: 10.1016/0093-6413(81)90064-1.  Google Scholar

[36]

P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar

[37]

R. Ravi Kumar, Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces, Appl. Math. Comput., 204 (2008), 352-362.  doi: 10.1016/j.amc.2008.06.050.  Google Scholar

[38]

R. Ravi Kumar, Regularity of solutions of evolution integrodifferential equations with deviating argument, Appl. Math. Comput., 217 (2011), 9111-9121.  doi: 10.1016/j.amc.2011.03.136.  Google Scholar

[39]

R. Sakthivel, Y. Ren, A. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar

[40]

R.-N. Wang, K. Ezzinbi and P. X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar

[41]

J.-Z. Xiao and X.-H. Zhu, Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures, Topol. Method Nonlinear Anal., 52 (2018), 353-372.  doi: 10.12775/TMNA.2018.030.  Google Scholar

[42]

Y.-B. Xiao, X. M. Yang and N.-J. Huang, Some equivalence results for well-posedness of hemivariational inequalities, J. Global Optim., 61 (2015), 789-802.  doi: 10.1007/s10898-014-0198-7.  Google Scholar

[43]

Z. M. Yan and X. M. Jia, Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay, Collect. Math., 66 (2015), 93-124.  doi: 10.1007/s13348-014-0109-8.  Google Scholar

[44]

M. Yang and Q. R. Wang, Existence of mild soltions for a class of Hilfer fractional evolution equations with nonlocal conditions, Fract. Calc. Appl. Anal., 20 (2017), 679-705.  doi: 10.1515/fca-2017-0036.  Google Scholar

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