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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 & 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.  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
Solution without Control in the Stochastic Case
Solution with Optimal Control, $\sigma = 0.5$
Solution without Control, $\sigma = 1$
Optimal Control
Potential
Solution without Control, σ = 1
Optimal Control, σ = 0.5
Solution with Optimal Control, σ = 0.5
Optimal Control, σ = 1
Solution with Optimal Control, σ = 1
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