# American Institute of Mathematical Sciences

October  2020, 25(10): 4039-4055. doi: 10.3934/dcdsb.2020137

## Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process

 Charles University, Prague, Faculty of Mathematics and Physics, Czech Republic

Received  July 2019 Revised  December 2019 Published  October 2020 Early access  April 2020

Fund Project: This research was supported by GAČR grant no. 19-07140S

An ergodic control problem is studied for controlled linear stochastic equations driven by cylindrical Lévy noise with unbounded control operator in a Hilbert space. A family of optimal controls is shown to consist of those asymptotically achieving the feedback form that employs the corresponding Riccati equation. The formula for optimal cost is given. The general results are applied to stochastic heat equation with boundary control and to stochastic structurally damped plate equations with point control.

Citation: Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137
##### References:
 [1] D. Applebaum and M. Riedle, Cylindrical Lévy Processes in Banach spaces, Proc. Lond. Math. Soc., 101 (2010), 697-726.  doi: 10.1112/plms/pdq004. [2] A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976. [3] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15. [4] G. Da Prato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl., 140 (1985), 209-221.  doi: 10.1007/BF01776850. [5] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Adaptive boundary and point control of linear stochastic distributed parameter systems, SIAM J. Control Optim., 32 (1994), 648-672.  doi: 10.1137/S0363012992228726. [6] T. E. Duncan and B. Pasik-Duncan, Some aspects of the adaptive control of stochastic evolution systems, Proceedings of the 28th Conference on Decision and Control, IEEE, New York, 1-3 (1989), 732-735. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Adaptive boundary control of linear distributed parameter systems described by analytic semigroups, Appl. Math. Optim., 33 (1996), 107-138.  doi: 10.1007/BF01183140. [8] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic control of some stochastic semilinear systems in Hilbert spaces, SIAM J. Control Optim., 36 (1998), 1020-1047.  doi: 10.1137/S0363012996303190. [9] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear-quadratic control for stochastic equations in a Hilbert space with fractional Brownian motions, SIAM J. Control Optim., 50 (2012), 507-531.  doi: 10.1137/110831416. [10] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic control of linear stochastic equations in a Hilbert space with fractional Brownian motions, Stochastic Analysis, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 105 (2015), 91-102.  doi: 10.4064/bc105-0-7. [11] T. E. Duncan, B. Goldys and B. Pasik-Duncan, Adaptive control of linear stochastic evolution systems, Stochastics Stochastics Rep., 36 (1991), 71-90.  doi: 10.1080/17442509108833711. [12] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear stochastic differential equations driven by Gauss-Volterra processes and related linear-quadratic control problems, Appl. Math. Optim., 80 (2019), 369-389.  doi: 10.1007/s00245-017-9468-3. [13] T. Duncan, L. Stettner and B. Pasik-Duncan, On ergodic control of stochastic evolution equations, Stochastic Anal. Appl., 15 (1997), 723-750.  doi: 10.1080/07362999708809504. [14] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, C. R. Math. Acad. Sci. Paris, 350 (2012), 683-688.  doi: 10.1016/j.crma.2012.07.009. [15] B. Goldys and B. Maslowski, Ergodic control of semilinear stochastic equations and Hamilton-Jacobi equations, J. Math. Anal. Appl., 234 (1999), 592-631.  doi: 10.1006/jmaa.1999.6387. [16] I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales. II. Itô formula in Banach spaces, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202. [17] E. Hausenblas, Maximal inequalities of the Itô integral with respect to Poisson random measures or Lévy processes on Banach spaces, Potential Anal., 35 (2011), 223-251.  doi: 10.1007/s11118-010-9210-0. [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [19] K. Kadlec and B. Maslowski, Ergodic Control for Lévy-driven linear stochastic equations in Hilbert spaces, Appl. Math. Optim., 79 (2017), 547-565.  doi: 10.1007/s00245-017-9447-8. [20] I. Lasiecka and R. Triggiani, Numerical approximations of algebraic Riccati equations modelled by analytic semigroups and applications, Math. Comput., 57 (1991), 639–662, S13–S37. doi: 10.1090/S0025-5718-1991-1094953-1. [21] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [22] J.-L. Lions and E. Magenes, Non-homogenous Boundary Value Problems and Applications. I, Springer, Berlin, 1972. [23] R. Sh. Lipster and A. N. Shiryayev, Theory of Martingales., Kluwer Academic Publ., Dobrecht, 1989. doi: 10.1007/978-94-009-2438-3. [24] V. Mandrekar, B. Rüdiger and S. Tappe, Itô's formula for Banach-space-valued jump process driven by Poisson random measures, Seminar on Stochastic Analysis, Random Fields and Applications VII, Progr. Probab., 67 (2013), 171-186. [25] B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. SCi., 22 (1995), 55-93. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations Driven by Lévy Processes., Cambridge University Press, Cambridge, 2006. [28] M. Riedle, Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An $L^2$ approach, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 1450008, 19 pp. doi: 10.1142/S0219025714500088. [29] T. F. Jiang, M. B. Rao, X. X. Wang and D. L. Li, Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments, Stoch. Process. Appl., 44 (1993), 205-219.  doi: 10.1016/0304-4149(93)90025-Y. [30] J. G. Wang, The asymptotic behavior of locally square integrable martingales, Ann. Probab., 23 (1995), 552-585.  doi: 10.1214/aop/1176988279.

show all references

##### References:
 [1] D. Applebaum and M. Riedle, Cylindrical Lévy Processes in Banach spaces, Proc. Lond. Math. Soc., 101 (2010), 697-726.  doi: 10.1112/plms/pdq004. [2] A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976. [3] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15. [4] G. Da Prato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl., 140 (1985), 209-221.  doi: 10.1007/BF01776850. [5] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Adaptive boundary and point control of linear stochastic distributed parameter systems, SIAM J. Control Optim., 32 (1994), 648-672.  doi: 10.1137/S0363012992228726. [6] T. E. Duncan and B. Pasik-Duncan, Some aspects of the adaptive control of stochastic evolution systems, Proceedings of the 28th Conference on Decision and Control, IEEE, New York, 1-3 (1989), 732-735. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Adaptive boundary control of linear distributed parameter systems described by analytic semigroups, Appl. Math. Optim., 33 (1996), 107-138.  doi: 10.1007/BF01183140. [8] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic control of some stochastic semilinear systems in Hilbert spaces, SIAM J. Control Optim., 36 (1998), 1020-1047.  doi: 10.1137/S0363012996303190. [9] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear-quadratic control for stochastic equations in a Hilbert space with fractional Brownian motions, SIAM J. Control Optim., 50 (2012), 507-531.  doi: 10.1137/110831416. [10] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic control of linear stochastic equations in a Hilbert space with fractional Brownian motions, Stochastic Analysis, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 105 (2015), 91-102.  doi: 10.4064/bc105-0-7. [11] T. E. Duncan, B. Goldys and B. Pasik-Duncan, Adaptive control of linear stochastic evolution systems, Stochastics Stochastics Rep., 36 (1991), 71-90.  doi: 10.1080/17442509108833711. [12] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear stochastic differential equations driven by Gauss-Volterra processes and related linear-quadratic control problems, Appl. Math. Optim., 80 (2019), 369-389.  doi: 10.1007/s00245-017-9468-3. [13] T. Duncan, L. Stettner and B. Pasik-Duncan, On ergodic control of stochastic evolution equations, Stochastic Anal. Appl., 15 (1997), 723-750.  doi: 10.1080/07362999708809504. [14] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, C. R. Math. Acad. Sci. Paris, 350 (2012), 683-688.  doi: 10.1016/j.crma.2012.07.009. [15] B. Goldys and B. Maslowski, Ergodic control of semilinear stochastic equations and Hamilton-Jacobi equations, J. Math. Anal. Appl., 234 (1999), 592-631.  doi: 10.1006/jmaa.1999.6387. [16] I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales. II. Itô formula in Banach spaces, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202. [17] E. Hausenblas, Maximal inequalities of the Itô integral with respect to Poisson random measures or Lévy processes on Banach spaces, Potential Anal., 35 (2011), 223-251.  doi: 10.1007/s11118-010-9210-0. [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [19] K. Kadlec and B. Maslowski, Ergodic Control for Lévy-driven linear stochastic equations in Hilbert spaces, Appl. Math. Optim., 79 (2017), 547-565.  doi: 10.1007/s00245-017-9447-8. [20] I. Lasiecka and R. Triggiani, Numerical approximations of algebraic Riccati equations modelled by analytic semigroups and applications, Math. Comput., 57 (1991), 639–662, S13–S37. doi: 10.1090/S0025-5718-1991-1094953-1. [21] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. [22] J.-L. Lions and E. Magenes, Non-homogenous Boundary Value Problems and Applications. I, Springer, Berlin, 1972. [23] R. Sh. Lipster and A. N. Shiryayev, Theory of Martingales., Kluwer Academic Publ., Dobrecht, 1989. doi: 10.1007/978-94-009-2438-3. [24] V. Mandrekar, B. Rüdiger and S. Tappe, Itô's formula for Banach-space-valued jump process driven by Poisson random measures, Seminar on Stochastic Analysis, Random Fields and Applications VII, Progr. Probab., 67 (2013), 171-186. [25] B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. SCi., 22 (1995), 55-93. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations Driven by Lévy Processes., Cambridge University Press, Cambridge, 2006. [28] M. Riedle, Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An $L^2$ approach, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 1450008, 19 pp. doi: 10.1142/S0219025714500088. [29] T. F. Jiang, M. B. Rao, X. X. Wang and D. L. Li, Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments, Stoch. Process. Appl., 44 (1993), 205-219.  doi: 10.1016/0304-4149(93)90025-Y. [30] J. G. Wang, The asymptotic behavior of locally square integrable martingales, Ann. Probab., 23 (1995), 552-585.  doi: 10.1214/aop/1176988279.
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