-
Previous Article
Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation
- EECT Home
- This Issue
-
Next Article
S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations
Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
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.
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. Chen, A. 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. Chen, X. 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. Chen, Y. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Cui, L. 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. Ezzinbi, X. 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. Liang, J. 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. Liang, J. 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. Sakthivela, Y. Ren, A. 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. Wang, M. 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. 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. |
[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. Zhang, P. Chen, A. 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. Zhang, P. Chen, A. 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. Chen, A. 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. Chen, X. 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. Chen, Y. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Chen, X. 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. Cui, L. 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. Ezzinbi, X. 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. Liang, J. 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. Liang, J. 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. Sakthivela, Y. Ren, A. 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. Wang, M. 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. 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. |
[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. Zhang, P. Chen, A. 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. Zhang, P. Chen, A. 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. |
[1] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[2] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[3] |
Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 |
[4] |
V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 |
[5] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
[6] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[7] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[8] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
[9] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[10] |
Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73 |
[11] |
Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021004 |
[12] |
Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 |
[13] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[14] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[15] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[16] |
Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 |
[17] |
Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 |
[18] |
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 |
[19] |
Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021040 |
[20] |
Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]