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June  2021, 11(2): 291-312. doi: 10.3934/mcrf.2020037

A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator

1. 

Department of Mathematics, Sichuan University, Chengdu 610064, China

2. 

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China, and, Department of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author

Received  March 2020 Revised  June 2020 Published  June 2021 Early access  August 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (11471230, 11671282)

In this paper, we first introduce a new spatial-temporal interaction operator to describe the space-time dependent phenomena. Then we consider the stochastic optimal control of a new system governed by a stochastic partial differential equation with the spatial-temporal interaction operator. To solve such a stochastic optimal control problem, we derive an adjoint backward stochastic partial differential equation with spatial-temporal dependence by defining a Hamiltonian functional, and give both the sufficient and necessary (Pontryagin-Bismut-Bensoussan type) maximum principles. Moreover, the existence and uniqueness of solutions are proved for the corresponding adjoint backward stochastic partial differential equations. Finally, our results are applied to study the population growth problems with the space-time dependent phenomena.

Citation: Zhun Gou, Nan-jing Huang, Ming-hui Wang, Yao-jia Zhang. A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator. Mathematical Control and Related Fields, 2021, 11 (2) : 291-312. doi: 10.3934/mcrf.2020037
References:
[1]

N. Agram, A. Hilbert and B. Øksendal, SPDEs with space-mean dynamics, preprint, arXiv: 1807.07303, 2019.

[2]

N. AgramA. Hilbert and B. Øksendal, Singular control of SPDEs with space-mean dynamics, Mathematical Control and Related Fields, 10 (2020), 425-441.  doi: 10.3934/mcrf.2020004.

[3]

N. Agram and B. Øksendal, Stochastic control of memory mean-field processes, Applied Mathematics & Optimization, 79 (2019), 181-204.  doi: 10.1007/s00245-017-9425-1.

[4]

A. Basse-O'ConnorM. S. NielsenJ. Pedersen and V. Rohde, Multivariate stochastic delay differential equations and CAR representations of CARMA processes, Stochastic Processes and Their Applications, 129 (2019), 4119-4143.  doi: 10.1016/j.spa.2018.11.011.

[5] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.
[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[7]

G. Da Prato and J. Zabczyk, Stochastic evolution equations, Mathematical biosciences, 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9.

[8]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM Journal on Control and Optimization, 51 (2013), 4343-4362.  doi: 10.1137/120882433.

[9]

R. DumitrescuB. Øksendal and A. Sulem, Stochastic control for mean-field stochastic partial differential equations with jumps, Journal of Optimization Theory and Applications, 176 (2018), 559-584.  doi: 10.1007/s10957-018-1243-3.

[10]

S. N. EvansP. L. RalphS. J. Schreibe and A. Sen, Stochastic population growth in spatially heterogeneous environments, Journal of Mathematical Biology, 66 (2013), 423-476.  doi: 10.1007/s00285-012-0514-0.

[11]

H. Frankowska and Q. Lü, First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints, Journal of Differential Equations, 268 (2020), 2949-3015.  doi: 10.1016/j.jde.2019.09.045.

[12]

M. Fuhrman and C. Orrieri, Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift, SIAM Journal on Control and Optimization, 54 (2016), 341-371.  doi: 10.1137/15M1012888.

[13]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer Science & Business Media, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[14]

A. HeningD. H. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: the density-dependent case, Journal of Mathematical Biology, 76 (2018), 697-754.  doi: 10.1007/s00285-017-1153-2.

[15]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Approach, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4684-9215-6.

[16]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics and Stochastic Reports, 33 (1990), 159-180.  doi: 10.1080/17442509008833671.

[17]

V. L. Kocic, Generalized attenuant cycles in some discrete periodically forced delay population models, Journal of Difference Equations and Applications, 16 (2010), 1141-1149.  doi: 10.1080/10236190902766850.

[18]

S. LenhartX. TangJ. Xiong and J-m. Yong, Controlled stochastic partial differential equations for rabbits on a grassland, Acta Mathematicae Applicatae Sinica, English Series, 36 (2020), 262-282.  doi: 10.1007/s10255-020-0925-4.

[19]

S. LenhartJ. Xiong and J. Yong, Optimal controls for stochastic partial differential equations with an application in population modeling, SIAM Journal on Control and Optimization, 54 (2016), 495-535.  doi: 10.1137/15M1010233.

[20]

J. Liu and C. A. Tudor, Analysis of the density of the solution to a semilinear SPDE with fractional noise, Stochastics, 88 (2016), 959-979.  doi: 10.1080/17442508.2016.1177056.

[21]

Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM Journal on Control and Optimizatio, 53 (2015), 3457-3482.  doi: 10.1137/151002605.

[22]

Q. Lü and X. Zhang, Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application, Mathematical Control and Related Fields, 8 (2018), 337-381.  doi: 10.3934/mcrf.2018014.

[23]

Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Mathematical Control and Related Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529.

[24]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.

[25]

Q. Meng and Y. Shen, Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach, Journal of Computational and Applied Mathematics, 279 (2015), 13-30.  doi: 10.1016/j.cam.2014.10.011.

[26]

J. B. Mijena and E. Nane, Intermittence and space-time fractional stochastic partial differential equations, Potential Analysis, 44 (2016), 295-312.  doi: 10.1007/s11118-015-9512-3.

[27]

S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics Ⅵ (eds. Decreusefond L., Øksendal B., Gjerde J. and Üstünel A. S.), Birkhäuser, Boston, (1998), 1–77.

[28]

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

[29]

B. Øksendal, Optimal control of stochastic partial differential equations, Stochastic Analysis and Applications, 23 (2005), 165-179.  doi: 10.1081/SAP-200044467.

[30]

B. ØksendalA. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.

[31]

B. Øksendal, A. Sulem and T. Zhang, Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs, in Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan (eds. T. Zhang and X. Zhou), World Scientific Publishing, Hackensack, (2012), 355–383. doi: 10.1142/9789814383585_0018.

[32]

T. Reichenbach, M. Mobilia and E. Frey, Noise and correlations in a spatial population model with cyclic competition, Physical Review Letters, 99 (2007), 238105. doi: 10.1103/PhysRevLett.99.238105.

[33]

S. J. Schreiber and J. O. Lloyd-Smith, Invasion dynamics in spatially heterogeneous environments, Applied Mathematics & Optimization, 174 (2009), 490-505.  doi: 10.1086/605405.

[34] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
[35]

H.-N. Wu and X.-M. Zhang, Boundary static output feedback control for nonlinear stochastic parabolic partial differential systems via fuzzy-model-based approach, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2941698.

show all references

References:
[1]

N. Agram, A. Hilbert and B. Øksendal, SPDEs with space-mean dynamics, preprint, arXiv: 1807.07303, 2019.

[2]

N. AgramA. Hilbert and B. Øksendal, Singular control of SPDEs with space-mean dynamics, Mathematical Control and Related Fields, 10 (2020), 425-441.  doi: 10.3934/mcrf.2020004.

[3]

N. Agram and B. Øksendal, Stochastic control of memory mean-field processes, Applied Mathematics & Optimization, 79 (2019), 181-204.  doi: 10.1007/s00245-017-9425-1.

[4]

A. Basse-O'ConnorM. S. NielsenJ. Pedersen and V. Rohde, Multivariate stochastic delay differential equations and CAR representations of CARMA processes, Stochastic Processes and Their Applications, 129 (2019), 4119-4143.  doi: 10.1016/j.spa.2018.11.011.

[5] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.
[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[7]

G. Da Prato and J. Zabczyk, Stochastic evolution equations, Mathematical biosciences, 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9.

[8]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM Journal on Control and Optimization, 51 (2013), 4343-4362.  doi: 10.1137/120882433.

[9]

R. DumitrescuB. Øksendal and A. Sulem, Stochastic control for mean-field stochastic partial differential equations with jumps, Journal of Optimization Theory and Applications, 176 (2018), 559-584.  doi: 10.1007/s10957-018-1243-3.

[10]

S. N. EvansP. L. RalphS. J. Schreibe and A. Sen, Stochastic population growth in spatially heterogeneous environments, Journal of Mathematical Biology, 66 (2013), 423-476.  doi: 10.1007/s00285-012-0514-0.

[11]

H. Frankowska and Q. Lü, First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints, Journal of Differential Equations, 268 (2020), 2949-3015.  doi: 10.1016/j.jde.2019.09.045.

[12]

M. Fuhrman and C. Orrieri, Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift, SIAM Journal on Control and Optimization, 54 (2016), 341-371.  doi: 10.1137/15M1012888.

[13]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer Science & Business Media, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[14]

A. HeningD. H. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: the density-dependent case, Journal of Mathematical Biology, 76 (2018), 697-754.  doi: 10.1007/s00285-017-1153-2.

[15]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Approach, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4684-9215-6.

[16]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics and Stochastic Reports, 33 (1990), 159-180.  doi: 10.1080/17442509008833671.

[17]

V. L. Kocic, Generalized attenuant cycles in some discrete periodically forced delay population models, Journal of Difference Equations and Applications, 16 (2010), 1141-1149.  doi: 10.1080/10236190902766850.

[18]

S. LenhartX. TangJ. Xiong and J-m. Yong, Controlled stochastic partial differential equations for rabbits on a grassland, Acta Mathematicae Applicatae Sinica, English Series, 36 (2020), 262-282.  doi: 10.1007/s10255-020-0925-4.

[19]

S. LenhartJ. Xiong and J. Yong, Optimal controls for stochastic partial differential equations with an application in population modeling, SIAM Journal on Control and Optimization, 54 (2016), 495-535.  doi: 10.1137/15M1010233.

[20]

J. Liu and C. A. Tudor, Analysis of the density of the solution to a semilinear SPDE with fractional noise, Stochastics, 88 (2016), 959-979.  doi: 10.1080/17442508.2016.1177056.

[21]

Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM Journal on Control and Optimizatio, 53 (2015), 3457-3482.  doi: 10.1137/151002605.

[22]

Q. Lü and X. Zhang, Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application, Mathematical Control and Related Fields, 8 (2018), 337-381.  doi: 10.3934/mcrf.2018014.

[23]

Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Mathematical Control and Related Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529.

[24]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.

[25]

Q. Meng and Y. Shen, Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach, Journal of Computational and Applied Mathematics, 279 (2015), 13-30.  doi: 10.1016/j.cam.2014.10.011.

[26]

J. B. Mijena and E. Nane, Intermittence and space-time fractional stochastic partial differential equations, Potential Analysis, 44 (2016), 295-312.  doi: 10.1007/s11118-015-9512-3.

[27]

S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics Ⅵ (eds. Decreusefond L., Øksendal B., Gjerde J. and Üstünel A. S.), Birkhäuser, Boston, (1998), 1–77.

[28]

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

[29]

B. Øksendal, Optimal control of stochastic partial differential equations, Stochastic Analysis and Applications, 23 (2005), 165-179.  doi: 10.1081/SAP-200044467.

[30]

B. ØksendalA. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.

[31]

B. Øksendal, A. Sulem and T. Zhang, Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs, in Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan (eds. T. Zhang and X. Zhou), World Scientific Publishing, Hackensack, (2012), 355–383. doi: 10.1142/9789814383585_0018.

[32]

T. Reichenbach, M. Mobilia and E. Frey, Noise and correlations in a spatial population model with cyclic competition, Physical Review Letters, 99 (2007), 238105. doi: 10.1103/PhysRevLett.99.238105.

[33]

S. J. Schreiber and J. O. Lloyd-Smith, Invasion dynamics in spatially heterogeneous environments, Applied Mathematics & Optimization, 174 (2009), 490-505.  doi: 10.1086/605405.

[34] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
[35]

H.-N. Wu and X.-M. Zhang, Boundary static output feedback control for nonlinear stochastic parabolic partial differential systems via fuzzy-model-based approach, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2941698.

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