September  2021, 10(3): 471-489. doi: 10.3934/eect.2020076

Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Pengyu Chen

Received  November 2019 Revised  April 2020 Published  September 2021 Early access  June 2020

Fund Project: The first author is supported by NSF of China (No. 11661071), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016); The second author is supported by the Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047) and Project of NWNU-LKQN2019-13

In this paper, we are considered with approximate controllability for a class of non-autonomous stochastic evolution equations of parabolic type with discrete nonlocal initial conditions. Some new results about existence of mild solutions as well as approximate controllability are established under more natural conditions on nonlinear functions and control operator by introducing a new Green function and using the theory of evolution family, Schauder fixed point theorem and the resolvent operator condition. At last, as a sample of application, these results are applied to a class of non-autonomous stochastic partial differential equation of parabolic type with discrete nonlocal initial conditions. The results obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Citation: Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations and Control Theory, 2021, 10 (3) : 471-489. doi: 10.3934/eect.2020076
References:
[1]

P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457. 

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107. 

[3]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.

[4]

P. Balasubramaniam and J. P. Dauer, Controllability of semilinear stochastic evolution equations with nonlocal conditions, Int. J. Pure Appl. Math., 19 (2005), 281-296. 

[5]

L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413-426.  doi: 10.1016/S0362-546X(97)00594-4.

[6]

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.

[7]

P. ChenA. Abdelmonem and Y. Li, Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions, J. Integral Equations Appl., 29 (2017), 325-348.  doi: 10.1216/JIE-2017-29-2-325.

[8]

P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.  doi: 10.1007/s00025-012-0230-5.

[9]

P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.

[10]

P. ChenX. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.  doi: 10.1515/fca-2020-0011.

[11]

P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y.

[12]

P. ChenY. Li and X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.  doi: 10.3934/cpaa.2015.14.1817.

[13]

P. Chen, X. Zhang and Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14 (2017), Art. 226, 16pp. doi: 10.1007/s00009-017-1029-0.

[14]

P. ChenX. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calcu. Appl. Anal., 19 (2016), 1507-1526.  doi: 10.1515/fca-2016-0078.

[15]

P. ChenX. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.

[16]

P. ChenX. Zhang and Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803.  doi: 10.1016/j.camwa.2017.01.009.

[17]

P. ChenX. Zhang and Y. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.  doi: 10.3934/cpaa.2018094.

[18]

P. ChenX. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9.

[19]

P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2119), Art. 84, 17pp. doi: 10.1007/s11784-019-0719-6.

[20]

P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), Art. 118, 14pp. doi: 10.1007/s00009-019-1384-0.

[21]

J. CuiL. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.  doi: 10.1016/j.jkss.2011.10.001.

[22]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.

[23] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[24]

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.

[25]

K. EzzinbiX. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003.

[26]

Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Functional Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023.

[27]

S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.

[28]

W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.

[29]

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

[30]

X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control, 77 (2016), 428-442.  doi: 10.1134/s000511791603005x.

[31]

X. Fu and Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 747-757.  doi: 10.1016/S0252-9602(13)60035-1.

[32]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.

[33]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.

[34]

R. E. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ., 1 (1963), 189-213. 

[35]

J. LiangJ. H. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlnear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.

[36]

J. LiangJ. H. Liu and T. J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529.

[37]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006.

[38]

Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.

[39]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.

[40]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[42]

R. SakthivelaY. RenA. Debbouchec and N. I. Mahmudovd, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.

[43]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991. doi: 10.1007/978-94-011-3712-6.

[44]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.

[45]

I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035, 22pp. doi: 10.1142/S0219199713500351.

[46]

J. WangM. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.

[47]

R. N. WangK. 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.

[48]

R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013) 180–191. doi: 10.1016/j.na.2013.02.026.

[49]

X. ZhangP. ChenA. Abdelmonem and Y. Li, Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.  doi: 10.1080/17442508.2018.1466885.

[50]

X. ZhangP. ChenA. Abdelmonem and Y. Li, Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.  doi: 10.1515/ms-2017-0207.

[51]

H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21 (1983), 551-565.  doi: 10.1137/0321033.

show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457. 

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107. 

[3]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.

[4]

P. Balasubramaniam and J. P. Dauer, Controllability of semilinear stochastic evolution equations with nonlocal conditions, Int. J. Pure Appl. Math., 19 (2005), 281-296. 

[5]

L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413-426.  doi: 10.1016/S0362-546X(97)00594-4.

[6]

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.

[7]

P. ChenA. Abdelmonem and Y. Li, Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions, J. Integral Equations Appl., 29 (2017), 325-348.  doi: 10.1216/JIE-2017-29-2-325.

[8]

P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.  doi: 10.1007/s00025-012-0230-5.

[9]

P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.

[10]

P. ChenX. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.  doi: 10.1515/fca-2020-0011.

[11]

P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y.

[12]

P. ChenY. Li and X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.  doi: 10.3934/cpaa.2015.14.1817.

[13]

P. Chen, X. Zhang and Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14 (2017), Art. 226, 16pp. doi: 10.1007/s00009-017-1029-0.

[14]

P. ChenX. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calcu. Appl. Anal., 19 (2016), 1507-1526.  doi: 10.1515/fca-2016-0078.

[15]

P. ChenX. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.

[16]

P. ChenX. Zhang and Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803.  doi: 10.1016/j.camwa.2017.01.009.

[17]

P. ChenX. Zhang and Y. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.  doi: 10.3934/cpaa.2018094.

[18]

P. ChenX. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9.

[19]

P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2119), Art. 84, 17pp. doi: 10.1007/s11784-019-0719-6.

[20]

P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), Art. 118, 14pp. doi: 10.1007/s00009-019-1384-0.

[21]

J. CuiL. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.  doi: 10.1016/j.jkss.2011.10.001.

[22]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.

[23] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[24]

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.

[25]

K. EzzinbiX. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003.

[26]

Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Functional Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023.

[27]

S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.

[28]

W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.

[29]

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

[30]

X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control, 77 (2016), 428-442.  doi: 10.1134/s000511791603005x.

[31]

X. Fu and Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 747-757.  doi: 10.1016/S0252-9602(13)60035-1.

[32]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.

[33]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.

[34]

R. E. Kalman, Controllablity of linear dynamical systems, Contrib. Diff. Equ., 1 (1963), 189-213. 

[35]

J. LiangJ. H. Liu and T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlnear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.

[36]

J. LiangJ. H. Liu and T. J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529.

[37]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006.

[38]

Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.

[39]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.

[40]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[42]

R. SakthivelaY. RenA. Debbouchec and N. I. Mahmudovd, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.

[43]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991. doi: 10.1007/978-94-011-3712-6.

[44]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.

[45]

I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035, 22pp. doi: 10.1142/S0219199713500351.

[46]

J. WangM. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.

[47]

R. N. WangK. 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.

[48]

R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013) 180–191. doi: 10.1016/j.na.2013.02.026.

[49]

X. ZhangP. ChenA. Abdelmonem and Y. Li, Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.  doi: 10.1080/17442508.2018.1466885.

[50]

X. ZhangP. ChenA. Abdelmonem and Y. Li, Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.  doi: 10.1515/ms-2017-0207.

[51]

H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21 (1983), 551-565.  doi: 10.1137/0321033.

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