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December  2021, 10(4): 701-722. doi: 10.3934/eect.2020087

Deterministic control of stochastic reaction-diffusion equations

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

*Corresponding author

Received  February 2020 Revised  June 2020 Published  December 2021 Early access  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.

Citation: Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. 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.

[2]

R. BuchholzH. EngelE. 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.

[3]

R. BuchholzH. EngelE. 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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.
[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.

[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.

[12]

M. FuhrmanY. 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.

[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.

[14]

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

[15]

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

[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.

[17]

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

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[25]

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

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.

[2]

R. BuchholzH. EngelE. 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.

[3]

R. BuchholzH. EngelE. 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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.
[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.

[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.

[12]

M. FuhrmanY. 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.

[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.

[14]

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

[15]

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

[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.

[17]

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

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[25]

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

Figure 1.  Solution without Control in the Stochastic Case
Figure 2.  Solution with Optimal Control, $ \sigma = 0.5 $
Figure 3.  Solution without Control, $ \sigma = 1 $
Figure 4.  Optimal Control
Figure 5.  Potential
Figure 6.  Solution without Control, σ = 1
Figure 7.  Optimal Control, σ = 0.5
Figure 8.  Solution with Optimal Control, σ = 0.5
Figure 9.  Optimal Control, σ = 1
Figure 10.  Solution with Optimal Control, σ = 1
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