American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Decay rates for the moore-gibson-thompson equation with memory
  • EECT Home
  • This Issue
  • Next Article
    Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy
doi: 10.3934/eect.2020087

Deterministic control of stochastic reaction-diffusion equations

Wilhelm Stannat and  Lukas Wessels ,

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

*Corresponding author

Received  February 2020 Revised  June 2020 Published  August 2020

Fund Project: The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center 910 "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" (project A10)

We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.

Keywords: Stochastic reaction-diffusion equations, variational approach, optimal control, stochastic Schlögl model, stochastic Nagumo equation, nonlinear conjugate gradient descent.
Mathematics Subject Classification: Primary: 93E20, 49K45; Secondary: 49M05, 60H15, 35K57.
Citation: Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2020087
References:
[1]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems, Journal of the Franklin Institute, 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5.  Google Scholar

[2]

R. Buchholz, H. Engel, E. Kammann and F. Tröltzsch, On the optimal control of the Schlögl-model, Computational Optimization and Applications, 56 (2013), 153-185.  doi: 10.1007/s10589-013-9550-y.  Google Scholar

[3]

R. Buchholz, H. Engel, E. Kammann and F. Tröltzsch, Erratum to: On the optimal control of the Schlögl-model, Computational Optimization and Applications, 56 (2013), 187-188.  doi: 10.1007/s10589-013-9570-7.  Google Scholar

[4]

S. Cerrai, Optimal control problems for stochastic reaction-diffusion systems with non-Lipschitz coefficients, SIAM Journal on Control and Optimization, 39 (2001), 1779-1816.  doi: 10.1137/S0363012999356465.  Google Scholar

[5]

S. Cerrai, Second Order PDEs in Finite and Infinite Dimension: A Probabilistic Approach, Lecture Notes in Mathematics, Springer, 2001. doi: 10.1007/b80743.  Google Scholar

[6]

Z. X. Chen and B. Y. Guo, Analytic solutions of the Nagumo equation, IMA Journal of Applied Mathematics, 48 (1992), 107-115.  doi: 10.1093/imamat/48.2.107.  Google Scholar

[7]

F. Cordoni and L. Di Persio, Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable, Evolution Equations & Control Theory, 7 (2018), 571-585.  doi: 10.3934/eect.2018027.  Google Scholar

[8]

F. Cordoni and L. Di Persio, Optimal control of the FitzHugh-Nagumo stochastic model with nonlinear diffusion, preprint, arXiv: 1912.00683. Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

G. Fabbri, F. Gozzi and A. Swiech, Stochastic Optimal Control in Infinite Dimension, Dynamic programming and HJB equations. Probability Theory and Stochastic Modeling, Springer, 2017. doi: 10.1007/978-3-319-53067-3.  Google Scholar

[11]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory and Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[12]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of partial differential equations driven by white noise, Stochastics and Partial Differential Equations: Analysis and Computations, 6 (2018), 255-285.  doi: 10.1007/s40072-017-0108-3.  Google Scholar

[13]

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.  Google Scholar

[14]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.   Google Scholar

[15]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[16]

H. Lisei, Existence of optimal and $\varepsilon$-optimal controls for the stochastic Navier-Stokes equation, Nonlinear Analysis: Theory, Methods and Applications, 51 (2002), 95-118.  doi: 10.1016/S0362-546X(01)00814-8.  Google Scholar

[17]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[18] G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, 2014.  doi: 10.1017/CBO9781139017329.  Google Scholar
[19]

C. Marinelli and L. Scarpa, Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: A variational approach, Potential Analysis, 52 (2020), 69-103.  doi: 10.1007/s11118-018-9731-5.  Google Scholar

[20]

F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces, SIAM Journal on Control and Optimization, 47 (2008), 251-300.  doi: 10.1137/050632725.  Google Scholar

[21]

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

[22]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar

[23]

C. Ryll, Optimal Control of Patterns in Some Reaction-Diffusion Systems, Ph.D thesis, Technische Universität Berlin, 2016. Google Scholar

[24]

C. Ryll, J. Löber, S. Martens, H. Engel and F. Tröltzsch, Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction-diffusion systems, in Control of self-organizing nonlinear systems, (eds. E. Schöll, S. H. L. Klapp and P. Hövel), Springer, (2016), 189–210.  Google Scholar

[25]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

show all references

References:
[1]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems, Journal of the Franklin Institute, 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5.  Google Scholar

[2]

R. Buchholz, H. Engel, E. Kammann and F. Tröltzsch, On the optimal control of the Schlögl-model, Computational Optimization and Applications, 56 (2013), 153-185.  doi: 10.1007/s10589-013-9550-y.  Google Scholar

[3]

R. Buchholz, H. Engel, E. Kammann and F. Tröltzsch, Erratum to: On the optimal control of the Schlögl-model, Computational Optimization and Applications, 56 (2013), 187-188.  doi: 10.1007/s10589-013-9570-7.  Google Scholar

[4]

S. Cerrai, Optimal control problems for stochastic reaction-diffusion systems with non-Lipschitz coefficients, SIAM Journal on Control and Optimization, 39 (2001), 1779-1816.  doi: 10.1137/S0363012999356465.  Google Scholar

[5]

S. Cerrai, Second Order PDEs in Finite and Infinite Dimension: A Probabilistic Approach, Lecture Notes in Mathematics, Springer, 2001. doi: 10.1007/b80743.  Google Scholar

[6]

Z. X. Chen and B. Y. Guo, Analytic solutions of the Nagumo equation, IMA Journal of Applied Mathematics, 48 (1992), 107-115.  doi: 10.1093/imamat/48.2.107.  Google Scholar

[7]

F. Cordoni and L. Di Persio, Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable, Evolution Equations & Control Theory, 7 (2018), 571-585.  doi: 10.3934/eect.2018027.  Google Scholar

[8]

F. Cordoni and L. Di Persio, Optimal control of the FitzHugh-Nagumo stochastic model with nonlinear diffusion, preprint, arXiv: 1912.00683. Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

G. Fabbri, F. Gozzi and A. Swiech, Stochastic Optimal Control in Infinite Dimension, Dynamic programming and HJB equations. Probability Theory and Stochastic Modeling, Springer, 2017. doi: 10.1007/978-3-319-53067-3.  Google Scholar

[11]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory and Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[12]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of partial differential equations driven by white noise, Stochastics and Partial Differential Equations: Analysis and Computations, 6 (2018), 255-285.  doi: 10.1007/s40072-017-0108-3.  Google Scholar

[13]

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.  Google Scholar

[14]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.   Google Scholar

[15]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[16]

H. Lisei, Existence of optimal and $\varepsilon$-optimal controls for the stochastic Navier-Stokes equation, Nonlinear Analysis: Theory, Methods and Applications, 51 (2002), 95-118.  doi: 10.1016/S0362-546X(01)00814-8.  Google Scholar

[17]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[18] G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, 2014.  doi: 10.1017/CBO9781139017329.  Google Scholar
[19]

C. Marinelli and L. Scarpa, Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: A variational approach, Potential Analysis, 52 (2020), 69-103.  doi: 10.1007/s11118-018-9731-5.  Google Scholar

[20]

F. Masiero, Stochastic optimal control problems and parabolic equations in Banach spaces, SIAM Journal on Control and Optimization, 47 (2008), 251-300.  doi: 10.1137/050632725.  Google Scholar

[21]

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

[22]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser/Springer Basel AG, Basel, 2013.  Google Scholar

[23]

C. Ryll, Optimal Control of Patterns in Some Reaction-Diffusion Systems, Ph.D thesis, Technische Universität Berlin, 2016. Google Scholar

[24]

C. Ryll, J. Löber, S. Martens, H. Engel and F. Tröltzsch, Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction-diffusion systems, in Control of self-organizing nonlinear systems, (eds. E. Schöll, S. H. L. Klapp and P. Hövel), Springer, (2016), 189–210.  Google Scholar

[25]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

Figure 1.  Solution without Control in the Stochastic Case
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Solution with Optimal Control, $ \sigma = 0.5 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Solution without Control, $ \sigma = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Optimal Control
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Potential
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Solution without Control, σ = 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Optimal Control, σ = 0.5
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Solution with Optimal Control, σ = 0.5
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Optimal Control, σ = 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Solution with Optimal Control, σ = 1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[2]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[3]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[4]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[5]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[6]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[7]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[8]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[9]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[10]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[11]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[12]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[13]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[14]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[15]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[16]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054
[17]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[18]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[19]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[20]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

2019 Impact Factor: 0.953

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences