September  2021, 11(3): 461-485. 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 2021 Early access  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 and Optimization, 2021, 11 (3) : 461-485. 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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[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.

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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[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.

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