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doi: 10.3934/naco.2020038

Approximate controllability of a non-autonomous evolution equation in Banach spaces

1. 

Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641 046, INDIA

2. 

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

* Corresponding author: Manil T. Mohan

Received  April 2020 Revised  September 2020 Published  September 2020

Fund Project: The second author is supported by DST-INSPIRE Faculty Award (IFA17-MA110)

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

Citation: K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020038
References:
[1]

N. AbadaM. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, Journal of Differential Equations, 246 (2009), 3834-3863.  doi: 10.1016/j.jde.2009.03.004.  Google Scholar

[2]

E. Asplund, Averaged norms, Israel Journal of Mathematics, 5 (1967), 227-233.  doi: 10.1007/BF02771611.  Google Scholar

[3]

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

[4]

K. Balachandran and R. Sakthivel, Approximate controllability of integrodifferential systems in Banach spaces, Appl. Math. Comput., 118 (2001), 63-71.  doi: 10.1016/S0096-3003(00)00040-0.  Google Scholar

[5]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, Academic Press, Inc, 190 (1993).  Google Scholar

[6]

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

[7]

J. M. Borwein and J. Vanderwerff, Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.   Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[9]

Y. K. ChangJ. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[10]

Y. K. ChangW. S. Li and J. J. Nieto, Controllability inclusions in Banach spaces, Nonlinear Anal., 67 (2007), 623-632.  doi: 10.1016/j.na.2006.06.018.  Google Scholar

[11]

P. ChenX. Zhang and Y. Li, Approximate controllability of Non-autonomous evolution system with nonlocal conditions, Journal of Dynamical and Control Systems, 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.  Google Scholar

[12]

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

[13]

J. P. Dauer and N. I. Mahmudov, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl., 273 (2002), 310-327.  doi: 10.1016/S0022-247X(02)00225-1.  Google Scholar

[14]

R. Dhayal, M. Malik, S. Abbas, A. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evolution Equations and Control Theory, 2020. doi: 10.3934/eect.2020073.  Google Scholar

[15] I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.   Google Scholar
[16]

Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67.  doi: 10.1016/j.amc.2014.01.051.  Google Scholar

[17]

Z. Fan, Approximate controllability of fractional differential equations via resolvent operators, Advances in Difference Equations, 54 (2014), 1-11.  doi: 10.1186/1687-1847-2014-54.  Google Scholar

[18]

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

[19]

X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, J. Dyn. Control Syst., 15 (2009), 425-443.  doi: 10.1007/s10883-009-9068-x.  Google Scholar

[20]

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

[21]

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.  Google Scholar

[22] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, CRC Press, 1998.   Google Scholar
[23]

R. K. George, Approximate controllability of non-autonomous semilinear systems, Nonlinear Analysis, 24 (1995), 1377-1393.  doi: 10.1016/0362-546X(94)E0082-R.  Google Scholar

[24]

H. Huang and X. Fu, Approximate controllability of semilinear neutral integro-differential equations with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 127-147.  doi: 10.1007/s10883-019-09438-5.  Google Scholar

[25]

J. Klamka, Constrained controllability of semilinear systems with delays, Nonlinear Dynam, 56 (2009), 169-177.  doi: 10.1007/s11071-008-9389-4.  Google Scholar

[26]

J. Klamka, Controllability and Minimum Energy Control, Monograph in Studies in Decision and Control, Springer-Verlag, 2018. doi: 10.1007/978-3-319-92540-0.  Google Scholar

[27]

A. KumarM. C. Joshi and A. K. Pani, On approximation theorems for controllability of non-linear parabolic problems, IMA Journal of Mathematical Control and Information, 24 (2007), 115-136.  doi: 10.1093/imamci/dnl012.  Google Scholar

[28]

S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equ., 252 (2012), 6163-6174.  doi: 10.1016/j.jde.2012.02.014.  Google Scholar

[29]

X. Li, and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser Boston, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[30]

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.  Google Scholar

[31]

N. I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Internat. J. Control, 73 (2000), 144-151.  doi: 10.1080/002071700219849.  Google Scholar

[32]

N. I. Mahmudov, On controllability of linear stochastic systems, IEEE Transactions on Automatic Control, 46 (2001), 724-731.  doi: 10.1109/9.920790.  Google Scholar

[33]

R. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, Springer, New York, Vol. 183, 1998. doi: 10.1007/978-1-4612-0603-3.  Google Scholar

[34]

I. Mishra and M. Sharma, Approximate controllability of a non-autonomous differential equation, Proc. Indian Acad. Sci. (Math. Sci.), 128. doi: 10.1007/s12044-018-0391-6.  Google Scholar

[35]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations, Evolution Equations and Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations in Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

R. SakthivelY. Ren and N. I. Mahmudov, Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern Phys. Lett. -B, 24 (2010), 1559-1572.  doi: 10.1142/S0217984910023359.  Google Scholar

[38]

R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control, 83 (2010), 387-393.  doi: 10.1080/00207170903171348.  Google Scholar

[39]

R. SakthivelJ. J. Nieto and N. I. Mahmudov, Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese. J. Math., 14 (2010), 1777-1797.  doi: 10.11650/twjm/1500406016.  Google Scholar

[40]

R. Sakthivel, Approximate controllability of impulsive stochastic evolution equations, Funkcial. Ekvac, 52 (2009), 425-443.  doi: 10.1619/fesi.52.381.  Google Scholar

[41]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM Journal on Control and Optimization, 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[42]

K. Yosida, Functional Analysis, Springer-Verlag, Heidelberg, New York, 1978.  Google Scholar

[43]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 3, Elsevier Science, Amsterdam, (2006), 527–621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

N. AbadaM. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, Journal of Differential Equations, 246 (2009), 3834-3863.  doi: 10.1016/j.jde.2009.03.004.  Google Scholar

[2]

E. Asplund, Averaged norms, Israel Journal of Mathematics, 5 (1967), 227-233.  doi: 10.1007/BF02771611.  Google Scholar

[3]

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

[4]

K. Balachandran and R. Sakthivel, Approximate controllability of integrodifferential systems in Banach spaces, Appl. Math. Comput., 118 (2001), 63-71.  doi: 10.1016/S0096-3003(00)00040-0.  Google Scholar

[5]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, Academic Press, Inc, 190 (1993).  Google Scholar

[6]

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

[7]

J. M. Borwein and J. Vanderwerff, Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.   Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[9]

Y. K. ChangJ. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[10]

Y. K. ChangW. S. Li and J. J. Nieto, Controllability inclusions in Banach spaces, Nonlinear Anal., 67 (2007), 623-632.  doi: 10.1016/j.na.2006.06.018.  Google Scholar

[11]

P. ChenX. Zhang and Y. Li, Approximate controllability of Non-autonomous evolution system with nonlocal conditions, Journal of Dynamical and Control Systems, 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.  Google Scholar

[12]

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

[13]

J. P. Dauer and N. I. Mahmudov, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl., 273 (2002), 310-327.  doi: 10.1016/S0022-247X(02)00225-1.  Google Scholar

[14]

R. Dhayal, M. Malik, S. Abbas, A. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evolution Equations and Control Theory, 2020. doi: 10.3934/eect.2020073.  Google Scholar

[15] I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.   Google Scholar
[16]

Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67.  doi: 10.1016/j.amc.2014.01.051.  Google Scholar

[17]

Z. Fan, Approximate controllability of fractional differential equations via resolvent operators, Advances in Difference Equations, 54 (2014), 1-11.  doi: 10.1186/1687-1847-2014-54.  Google Scholar

[18]

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

[19]

X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, J. Dyn. Control Syst., 15 (2009), 425-443.  doi: 10.1007/s10883-009-9068-x.  Google Scholar

[20]

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

[21]

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.  Google Scholar

[22] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, CRC Press, 1998.   Google Scholar
[23]

R. K. George, Approximate controllability of non-autonomous semilinear systems, Nonlinear Analysis, 24 (1995), 1377-1393.  doi: 10.1016/0362-546X(94)E0082-R.  Google Scholar

[24]

H. Huang and X. Fu, Approximate controllability of semilinear neutral integro-differential equations with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 127-147.  doi: 10.1007/s10883-019-09438-5.  Google Scholar

[25]

J. Klamka, Constrained controllability of semilinear systems with delays, Nonlinear Dynam, 56 (2009), 169-177.  doi: 10.1007/s11071-008-9389-4.  Google Scholar

[26]

J. Klamka, Controllability and Minimum Energy Control, Monograph in Studies in Decision and Control, Springer-Verlag, 2018. doi: 10.1007/978-3-319-92540-0.  Google Scholar

[27]

A. KumarM. C. Joshi and A. K. Pani, On approximation theorems for controllability of non-linear parabolic problems, IMA Journal of Mathematical Control and Information, 24 (2007), 115-136.  doi: 10.1093/imamci/dnl012.  Google Scholar

[28]

S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equ., 252 (2012), 6163-6174.  doi: 10.1016/j.jde.2012.02.014.  Google Scholar

[29]

X. Li, and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser Boston, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[30]

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.  Google Scholar

[31]

N. I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Internat. J. Control, 73 (2000), 144-151.  doi: 10.1080/002071700219849.  Google Scholar

[32]

N. I. Mahmudov, On controllability of linear stochastic systems, IEEE Transactions on Automatic Control, 46 (2001), 724-731.  doi: 10.1109/9.920790.  Google Scholar

[33]

R. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, Springer, New York, Vol. 183, 1998. doi: 10.1007/978-1-4612-0603-3.  Google Scholar

[34]

I. Mishra and M. Sharma, Approximate controllability of a non-autonomous differential equation, Proc. Indian Acad. Sci. (Math. Sci.), 128. doi: 10.1007/s12044-018-0391-6.  Google Scholar

[35]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations, Evolution Equations and Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations in Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

R. SakthivelY. Ren and N. I. Mahmudov, Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern Phys. Lett. -B, 24 (2010), 1559-1572.  doi: 10.1142/S0217984910023359.  Google Scholar

[38]

R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control, 83 (2010), 387-393.  doi: 10.1080/00207170903171348.  Google Scholar

[39]

R. SakthivelJ. J. Nieto and N. I. Mahmudov, Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese. J. Math., 14 (2010), 1777-1797.  doi: 10.11650/twjm/1500406016.  Google Scholar

[40]

R. Sakthivel, Approximate controllability of impulsive stochastic evolution equations, Funkcial. Ekvac, 52 (2009), 425-443.  doi: 10.1619/fesi.52.381.  Google Scholar

[41]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM Journal on Control and Optimization, 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[42]

K. Yosida, Functional Analysis, Springer-Verlag, Heidelberg, New York, 1978.  Google Scholar

[43]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 3, Elsevier Science, Amsterdam, (2006), 527–621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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