February  2022, 11(1): 177-197. doi: 10.3934/eect.2020107

Optimal control problems for a neutral integro-differential system with infinite delay

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China

* Corresponding author: Xianlong Fu

Received  May 2020 Revised  September 2020 Published  February 2022 Early access  December 2020

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000)

This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.

Citation: Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations and Control Theory, 2022, 11 (1) : 177-197. doi: 10.3934/eect.2020107
References:
[1]

N. U. Ahmed, Partially observed stochastic evolution equations on Banach spaces and their optimal Lipschitz feedback control law, SIAM J. Control Optim., 57 (2019), 3101-3117.  doi: 10.1137/19M1243282.

[2]

P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.

[3]

C. D'ApiceM. P. D'ArienzoP. I. Kogut and R. Manzo, On boundary optimal control problem for an arterial system: Existence of feasible solutions, J. Evol. Equ., 18 (2018), 1745-1786.  doi: 10.1007/s00028-018-0460-4.

[4]

M. A. DialloK. Ezzinbi and A. Sène, Optimal control problem for some integrodifferential equations in Banach spaces, Optim. Control Appl. Methods., 39 (2018), 563-574.  doi: 10.1002/oca.2359.

[5]

M. A. DiopT. Caraballo and A. A. Ndiaye, Exponential behavior of solutions to stochastic integrodifferential equations with distributed delays, Stoch. Anal. Appl., 33 (2015), 399-412.  doi: 10.1080/07362994.2014.1000070.

[6]

J. P. C. Dos Santos, Existence results for a partial neutral integro-differential equation with state-dependent delay, Electr. J. Qual. Theory Diff. Equ., 29 (2010), 1-12.  doi: 10.14232/ejqtde.2010.1.29.

[7]

J. P. C. Dos SantosH. Henríquez and E. Hernández, Existence results for neutral integrodifferential equations with unbounded delay, J. Integral Equ. Appl., 23 (2011), 289-330.  doi: 10.1216/JIE-2011-23-2-289.

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.

[9]

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

[10]

R. C. Grimmer and F. Kappel, Series expansions of volterra integrodifferential equations in Banach space, SIAM J. Math. Anal., 15 (1984), 595-604.  doi: 10.1137/0515045.

[11]

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

[12]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funk. Ekvac., 21 (1978), 11-41. 

[13]

A. HarratJ. J. Nieto and A. Debbouche, Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential, J. Comput. Appl. Math., 344 (2018), 725-737.  doi: 10.1016/j.cam.2018.05.031.

[14]

H. R. Henríquez and J. P. C. Dos Santos, Differentiability of solutions of abstract neutral integro-differential equations, J. Integral Equ. Appl., 25 (2013), 47-77.  doi: 10.1216/JIE-2013-25-1-47.

[15]

E. Hernández and D. O'Regan, On a new class of abstract neutral integro-differential equations and applications, Acta. Appl. Math., 149 (2017), 125-137.  doi: 10.1007/s10440-016-0090-1.

[16]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer, 1991. doi: 10.1007/BFb0084432.

[17]

K. Jeet and N. Sukavanam, Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique, Appl. Math. Comput., 364 (2020), 124690, 15pp. doi: 10.1016/j.amc.2019.124690.

[18]

J.-M. Jeong and H.-J. Hwang, Optimal control problems for semilinear retarded functional differential equations, J. Optim. Theory Appl., 167 (2015), 49-67.  doi: 10.1007/s10957-015-0726-8.

[19]

J.-M. Jeong and S.-J. Son, Time optimal control of semilinear control systems involving time delays, J. Optim. Theory Appl., 165 (2015), 793-811.  doi: 10.1007/s10957-014-0639-y.

[20]

Y. Jiang and N. Huang, Solvability and optimal controls of fractional delay evolution inclusions with Clarke subdifferential, Math. Methods Appl. Sci., 40 (2017), 3026-3039.  doi: 10.1002/mma.4218.

[21]

V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Diff. Equ., 230 (2006), 634-660.  doi: 10.1016/j.jde.2006.07.018.

[22]

S. Kumar, Mild solution and fractional optimal control of semilinear system with fixed delay, J. Optim. Theory Appl., 174 (2017), 108-121.  doi: 10.1007/s10957-015-0828-3.

[23]

T. LevajkovićH. Mena and A. Tuffaha, The stochastic linear quadratic optimal control problem in Hilbert spaces: A chaos expansion approach, Evol. Equ. Control Theory., 5 (2016), 105-134.  doi: 10.3934/eect.2016.5.105.

[24]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkh$\ddot{a}$user, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[25]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[26]

Q. Meng and Y. Shen, Optimal control for stochastic delay evolution equations, Appl. Math. Optim., 74 (2016), 53-89.  doi: 10.1007/s00245-015-9308-2.

[27]

R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 313-332.  doi: 10.1016/0022-247X(78)90234-2.

[28]

F. Z. Mokkedem and X. Fu, Optimal control problems for a semilinear evolution system with infinite delay, Appl. Math. Optim., 79 (2019), 41-67.  doi: 10.1007/s00245-017-9420-6.

[29]

B. S. MordukhovichD. Wang and L. Wang, Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions, Nonl. Anal., 71 (2009), 2740-2749.  doi: 10.1016/j.na.2009.06.022.

[30]

S. Nakagiri, Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl., 120 (1986), 169-210.  doi: 10.1016/0022-247X(86)90210-6.

[31]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007%2F978-1-4612-5561-1.

[32]

B. Radhakrishnan and K. Balachandran, Controllability of neutral evolution integrodifferential systems with state dependent delay, J. Optim. Theory Appl., 153 (2012), 85-97.  doi: 10.1007/s10957-011-9934-z.

[33]

C. RavichandranN. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.

[34]

R. SakthivelQ. H. Choi and S. M. Anthoni, Controllability of nonlinear neutral evolution integrodifferential systems, J. Math. Anal. Appl., 275 (2002), 402-417.  doi: 10.1016/S0022-247X(02)00375-X.

[35]

D. Sforza, Existence in the large for a semilinear integrodifferential equation with infinite delay, J. Diff. Equ., 120 (1995), 289-303.  doi: 10.1006/jdeq.1995.1113.

[36]

M. TucsnakJ. Valein and C. Wu, Finite dimensional approximations for a class of infinite dimensional time optimal control problems, Int. J. Control., 92 (2019), 132-144.  doi: 10.1080/00207179.2016.1228122.

[37]

V. Vijayakumar, Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IMA J. Math. Control Inf., 35 (2018), 297-314.  doi: 10.1093/imamci/dnw049.

[38]

J. WangY. Zhou and M. Medved, On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. Optim. Theory Appl., 152 (2012), 31-50.  doi: 10.1007/s10957-011-9892-5.

[39]

Z. Yan and F. Lu, Existence of an optimal control for fractional stochastic partial neutral integro-differential equations with infinite delay, J. Nonl. Science Appl., 8 (2015), 557-577.  doi: 10.22436/jnsa.008.05.10.

[40]

Z. Yan and F. Lu, The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay, Int. J. Control., 89 (2016), 1592-1612.  doi: 10.1080/00207179.2016.1140229.

show all references

References:
[1]

N. U. Ahmed, Partially observed stochastic evolution equations on Banach spaces and their optimal Lipschitz feedback control law, SIAM J. Control Optim., 57 (2019), 3101-3117.  doi: 10.1137/19M1243282.

[2]

P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.

[3]

C. D'ApiceM. P. D'ArienzoP. I. Kogut and R. Manzo, On boundary optimal control problem for an arterial system: Existence of feasible solutions, J. Evol. Equ., 18 (2018), 1745-1786.  doi: 10.1007/s00028-018-0460-4.

[4]

M. A. DialloK. Ezzinbi and A. Sène, Optimal control problem for some integrodifferential equations in Banach spaces, Optim. Control Appl. Methods., 39 (2018), 563-574.  doi: 10.1002/oca.2359.

[5]

M. A. DiopT. Caraballo and A. A. Ndiaye, Exponential behavior of solutions to stochastic integrodifferential equations with distributed delays, Stoch. Anal. Appl., 33 (2015), 399-412.  doi: 10.1080/07362994.2014.1000070.

[6]

J. P. C. Dos Santos, Existence results for a partial neutral integro-differential equation with state-dependent delay, Electr. J. Qual. Theory Diff. Equ., 29 (2010), 1-12.  doi: 10.14232/ejqtde.2010.1.29.

[7]

J. P. C. Dos SantosH. Henríquez and E. Hernández, Existence results for neutral integrodifferential equations with unbounded delay, J. Integral Equ. Appl., 23 (2011), 289-330.  doi: 10.1216/JIE-2011-23-2-289.

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.

[9]

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

[10]

R. C. Grimmer and F. Kappel, Series expansions of volterra integrodifferential equations in Banach space, SIAM J. Math. Anal., 15 (1984), 595-604.  doi: 10.1137/0515045.

[11]

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

[12]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funk. Ekvac., 21 (1978), 11-41. 

[13]

A. HarratJ. J. Nieto and A. Debbouche, Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential, J. Comput. Appl. Math., 344 (2018), 725-737.  doi: 10.1016/j.cam.2018.05.031.

[14]

H. R. Henríquez and J. P. C. Dos Santos, Differentiability of solutions of abstract neutral integro-differential equations, J. Integral Equ. Appl., 25 (2013), 47-77.  doi: 10.1216/JIE-2013-25-1-47.

[15]

E. Hernández and D. O'Regan, On a new class of abstract neutral integro-differential equations and applications, Acta. Appl. Math., 149 (2017), 125-137.  doi: 10.1007/s10440-016-0090-1.

[16]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer, 1991. doi: 10.1007/BFb0084432.

[17]

K. Jeet and N. Sukavanam, Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique, Appl. Math. Comput., 364 (2020), 124690, 15pp. doi: 10.1016/j.amc.2019.124690.

[18]

J.-M. Jeong and H.-J. Hwang, Optimal control problems for semilinear retarded functional differential equations, J. Optim. Theory Appl., 167 (2015), 49-67.  doi: 10.1007/s10957-015-0726-8.

[19]

J.-M. Jeong and S.-J. Son, Time optimal control of semilinear control systems involving time delays, J. Optim. Theory Appl., 165 (2015), 793-811.  doi: 10.1007/s10957-014-0639-y.

[20]

Y. Jiang and N. Huang, Solvability and optimal controls of fractional delay evolution inclusions with Clarke subdifferential, Math. Methods Appl. Sci., 40 (2017), 3026-3039.  doi: 10.1002/mma.4218.

[21]

V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Diff. Equ., 230 (2006), 634-660.  doi: 10.1016/j.jde.2006.07.018.

[22]

S. Kumar, Mild solution and fractional optimal control of semilinear system with fixed delay, J. Optim. Theory Appl., 174 (2017), 108-121.  doi: 10.1007/s10957-015-0828-3.

[23]

T. LevajkovićH. Mena and A. Tuffaha, The stochastic linear quadratic optimal control problem in Hilbert spaces: A chaos expansion approach, Evol. Equ. Control Theory., 5 (2016), 105-134.  doi: 10.3934/eect.2016.5.105.

[24]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkh$\ddot{a}$user, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[25]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[26]

Q. Meng and Y. Shen, Optimal control for stochastic delay evolution equations, Appl. Math. Optim., 74 (2016), 53-89.  doi: 10.1007/s00245-015-9308-2.

[27]

R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 313-332.  doi: 10.1016/0022-247X(78)90234-2.

[28]

F. Z. Mokkedem and X. Fu, Optimal control problems for a semilinear evolution system with infinite delay, Appl. Math. Optim., 79 (2019), 41-67.  doi: 10.1007/s00245-017-9420-6.

[29]

B. S. MordukhovichD. Wang and L. Wang, Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions, Nonl. Anal., 71 (2009), 2740-2749.  doi: 10.1016/j.na.2009.06.022.

[30]

S. Nakagiri, Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl., 120 (1986), 169-210.  doi: 10.1016/0022-247X(86)90210-6.

[31]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007%2F978-1-4612-5561-1.

[32]

B. Radhakrishnan and K. Balachandran, Controllability of neutral evolution integrodifferential systems with state dependent delay, J. Optim. Theory Appl., 153 (2012), 85-97.  doi: 10.1007/s10957-011-9934-z.

[33]

C. RavichandranN. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.

[34]

R. SakthivelQ. H. Choi and S. M. Anthoni, Controllability of nonlinear neutral evolution integrodifferential systems, J. Math. Anal. Appl., 275 (2002), 402-417.  doi: 10.1016/S0022-247X(02)00375-X.

[35]

D. Sforza, Existence in the large for a semilinear integrodifferential equation with infinite delay, J. Diff. Equ., 120 (1995), 289-303.  doi: 10.1006/jdeq.1995.1113.

[36]

M. TucsnakJ. Valein and C. Wu, Finite dimensional approximations for a class of infinite dimensional time optimal control problems, Int. J. Control., 92 (2019), 132-144.  doi: 10.1080/00207179.2016.1228122.

[37]

V. Vijayakumar, Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IMA J. Math. Control Inf., 35 (2018), 297-314.  doi: 10.1093/imamci/dnw049.

[38]

J. WangY. Zhou and M. Medved, On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. Optim. Theory Appl., 152 (2012), 31-50.  doi: 10.1007/s10957-011-9892-5.

[39]

Z. Yan and F. Lu, Existence of an optimal control for fractional stochastic partial neutral integro-differential equations with infinite delay, J. Nonl. Science Appl., 8 (2015), 557-577.  doi: 10.22436/jnsa.008.05.10.

[40]

Z. Yan and F. Lu, The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay, Int. J. Control., 89 (2016), 1592-1612.  doi: 10.1080/00207179.2016.1140229.

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