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  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 & 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.  Google Scholar

[2]

A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[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.  Google Scholar

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G. Da Prato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl., 140 (1985), 209-221.  doi: 10.1007/BF01776850.  Google Scholar

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T. E. DuncanB. 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.  Google Scholar

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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.   Google Scholar

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T. E. DuncanB. 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.  Google Scholar

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T. E. DuncanB. 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.  Google Scholar

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T. E. DuncanB. 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.  Google Scholar

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T. E. DuncanB. 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.  Google Scholar

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T. E. DuncanB. Goldys and B. Pasik-Duncan, Adaptive control of linear stochastic evolution systems, Stochastics Stochastics Rep., 36 (1991), 71-90.  doi: 10.1080/17442509108833711.  Google Scholar

[12]

T. E. DuncanB. 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.  Google Scholar

[13]

T. DuncanL. Stettner and B. Pasik-Duncan, On ergodic control of stochastic evolution equations, Stochastic Anal. Appl., 15 (1997), 723-750.  doi: 10.1080/07362999708809504.  Google Scholar

[14]

M. FuhrmanY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[22]

J.-L. Lions and E. Magenes, Non-homogenous Boundary Value Problems and Applications. I, Springer, Berlin, 1972. Google Scholar

[23]

R. Sh. Lipster and A. N. Shiryayev, Theory of Martingales., Kluwer Academic Publ., Dobrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[24]

V. MandrekarB. 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.   Google Scholar

[25]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. SCi., 22 (1995), 55-93.   Google Scholar

[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.  Google Scholar

[27] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations Driven by Lévy Processes., Cambridge University Press, Cambridge, 2006.   Google Scholar
[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.  Google Scholar

[29]

T. F. JiangM. B. RaoX. 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.  Google Scholar

[30]

J. G. Wang, The asymptotic behavior of locally square integrable martingales, Ann. Probab., 23 (1995), 552-585.  doi: 10.1214/aop/1176988279.  Google Scholar

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.  Google Scholar

[2]

A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[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.  Google Scholar

[4]

G. Da Prato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl., 140 (1985), 209-221.  doi: 10.1007/BF01776850.  Google Scholar

[5]

T. E. DuncanB. 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.  Google Scholar

[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.   Google Scholar

[7]

T. E. DuncanB. 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.  Google Scholar

[8]

T. E. DuncanB. 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.  Google Scholar

[9]

T. E. DuncanB. 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.  Google Scholar

[10]

T. E. DuncanB. 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.  Google Scholar

[11]

T. E. DuncanB. Goldys and B. Pasik-Duncan, Adaptive control of linear stochastic evolution systems, Stochastics Stochastics Rep., 36 (1991), 71-90.  doi: 10.1080/17442509108833711.  Google Scholar

[12]

T. E. DuncanB. 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.  Google Scholar

[13]

T. DuncanL. Stettner and B. Pasik-Duncan, On ergodic control of stochastic evolution equations, Stochastic Anal. Appl., 15 (1997), 723-750.  doi: 10.1080/07362999708809504.  Google Scholar

[14]

M. FuhrmanY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[22]

J.-L. Lions and E. Magenes, Non-homogenous Boundary Value Problems and Applications. I, Springer, Berlin, 1972. Google Scholar

[23]

R. Sh. Lipster and A. N. Shiryayev, Theory of Martingales., Kluwer Academic Publ., Dobrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[24]

V. MandrekarB. 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.   Google Scholar

[25]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. SCi., 22 (1995), 55-93.   Google Scholar

[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.  Google Scholar

[27] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations Driven by Lévy Processes., Cambridge University Press, Cambridge, 2006.   Google Scholar
[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.  Google Scholar

[29]

T. F. JiangM. B. RaoX. 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.  Google Scholar

[30]

J. G. Wang, The asymptotic behavior of locally square integrable martingales, Ann. Probab., 23 (1995), 552-585.  doi: 10.1214/aop/1176988279.  Google Scholar

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