# American Institute of Mathematical Sciences

March  2016, 5(1): 105-134. doi: 10.3934/eect.2016.5.105

## The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

 1 Department of Mathematics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, Innsbruck, A 6020, Austria, Austria 2 The American University of Sharjah, Sharjah, United Arab Emirates

Received  November 2015 Revised  January 2016 Published  March 2016

We consider the stochastic linear quadratic optimal control problem for state equations of the Itô-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.
Citation: Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105
##### References:
 [1] E. Arias, V. Hernández, J. Ibanes and J. Peinado, A family of BDF algorithms for solving differential matrix Riccati equations using adaptive techniques,, Procedia Computer Science, 1 (2010), 2569. Google Scholar [2] G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics,, J. Optim. Theory Appl., 91 (1996), 695. doi: 10.1007/BF02190128. Google Scholar [3] H. T. Banks, R. J. Silcox and R. C. Smith, The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples,, ASME J. Vibration Acoustics, 116 (1994), 386. doi: 10.1115/1.2930440. Google Scholar [4] H. T. Banks, R. C. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates and beams,, Quart. Appl. Math, 53 (1995), 353. Google Scholar [5] P. Benner, P. Ezzatti, H. Mena, E. S. Quintana-Ortí and A. Remón, Solving matrix equations on multi-core and many-core architectures,, Algorithms, 6 (2013), 857. doi: 10.3390/a6040857. Google Scholar [6] P. Benner and H. Mena, Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations,, MPI Magdeburg Preprint MPIMD/12-13, (2012), 12. Google Scholar [7] P. Benner and H. Mena, Rosenbrock methods for solving differential Riccati equations,, IEEE Transactions on Automatic Control, 58 (2013), 2950. doi: 10.1109/TAC.2013.2258495. Google Scholar [8] F. E. Benth and T. G. Theting, Some regularity results for the stochastic pressure equation of Wick-type,, Stochastic Analysis and Applications, 20 (2002), 1191. doi: 10.1081/SAP-120015830. Google Scholar [9] J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients,, SIAM J. Control. Optim., 14 (1976), 419. doi: 10.1137/0314028. Google Scholar [10] J.-M. Bismut, Contrôle des systmes linéaires quadratiques: Applications de l'intégrale stochastique,, in Séminaire de Probabilités XII, 649 (1978), 180. Google Scholar [11] K. R. Dahl, S.-E. A. Mohammed, B. Øksendal and E. E. Røse, Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives,, preprint, (). Google Scholar [12] G. Da Prato, Direct solution of a Riccati equation arising in stochastic control theory,, Appl. Math. Optim., 11 (1984), 191. doi: 10.1007/BF01442178. Google Scholar [13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, $2^{nd}$ edition, (2014). doi: 978-1-107-05584-1. Google Scholar [14] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary,, Appl. Math. Optim, 14 (1986), 107. doi: 10.1007/BF01442231. Google Scholar [15] H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, $2^{nd}$ edition, 25 (2006). Google Scholar [16] J. Fisher and R. Bhattacharya, On stochastic LQR design and polynomial chaos,, in American Control Conference, (2008), 95. doi: 10.1109/ACC.2008.4586473. Google Scholar [17] J. Fisher and R. Bhattacharya, Stability analysis of stochastic systems using polynomial chaos,, In American Control Conference, (2008), 4250. doi: 10.1109/ACC.2008.4587161. Google Scholar [18] R. Ghanem and P. D. Spanos, Polynomial chaos in stochastic finite elements,, Journal of Applied Mechanics, 57 (1990), 197. doi: 10.1115/1.2888303. Google Scholar [19] M. Grothaus, Y. G. Kondratiev and G. F. Us, Wick calculus for regular generalized stochastic functionals,, Random Oper. Stochastic Equations, 7 (1999), 263. doi: 10.1515/rose.1999.7.3.263. Google Scholar [20] G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions,, SIAM J. Control Optim., 44 (2005), 159. doi: 10.1137/S0363012903425507. Google Scholar [21] G. Guatteri and G. Tessitore, Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients,, Appl. Math. Optim., 57 (2008), 207. doi: 10.1007/s00245-007-9020-y. Google Scholar [22] C. Hafizoglu, Linear Quadratic Boundary{/Point Control of Stochastic Partial Differential Equation Systems with Unbounded Coefficients},, Ph.D thesis, (2006). Google Scholar [23] C. Hafizoglu, I. Lasiecka, T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic control problem with singular estimates,, preprint, (2015). Google Scholar [24] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise. An Infinite-Dimensional Calculus,, Mathematics and its Applications, 253 (1993). doi: 10.1007/978-94-017-3680-0. Google Scholar [25] H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A modeling, White Noise Functional Approach,, $2^{nd}$ edition, (2010). doi: 10.1007/978-0-387-89488-1. Google Scholar [26] F. S. Hover and M. S. Triantafyllou, Application of polynomial chaos in stability and control,, Automatica, 42 (2006), 789. doi: 10.1016/j.automatica.2006.01.010. Google Scholar [27] A. Ichikawa, Dynamic programming approach to stochastic evolution equations,, SIAM J. Control. Optim., 17 (1979), 152. doi: 10.1137/0317012. Google Scholar [28] M. Kohlmann and S. Tang, New developments in backward stochastic Riccati equations and their applications,, In: Mathematical Finance (Konstanz 2000), (2000), 194. Google Scholar [29] M. Kohlmann and S. Tang, Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging,, Stoch. Process. Appl., 97 (2002), 255. doi: 10.1016/S0304-4149(01)00133-8. Google Scholar [30] M. Kohlmann and S. Tang, Multidimensional backward stochastic Riccati equations and applications,, SIAM J. Control. Optim., 41 (2003), 1696. doi: 10.1137/S0363012900378760. Google Scholar [31] M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach,, SIAM J. Control. Optim., 38 (2000), 1392. doi: 10.1137/S036301299834973X. Google Scholar [32] H. J. Kushner, Optimal stochastic control,, IRE Trans. Auto. Control, 7 (1962), 120. doi: 10.1109/TAC.1962.1105490. Google Scholar [33] N. Lang, H. Mena and J. Saak, On the benefits of the LDL factorization for large-scale differential matrix equation solvers,, Linear Algebra and its Applications, 480 (2015), 44. doi: 10.1016/j.laa.2015.04.006. Google Scholar [34] 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, 74 (2000). Google Scholar [35] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories II. Abstract Hyperbolic-Like Systems over a Finite Time Horizon,, Encyclopedia of Mathematics and its Applications 75, 75 (2000). doi: 10.1017/CBO9780511574801.002. Google Scholar [36] I. Lasiecka and R. Triggiani, Optimal control and differential Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity,, Advances in Dynamics and Control, 2 (2004), 270. Google Scholar [37] C. Lebdzek and R. Triggiani, Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions,, Control Cybernet, 38 (2009), 1461. Google Scholar [38] T. Levajković and H. Mena, On deterministic and stochastic linear quadratic control problem,, Current Trends in Analysis and Its Applications, (2015), 315. Google Scholar [39] T. Levajković, H. Mena and A. Tuffaha, A Numerical approximation framework for the stochastic linear quadratic regulator on Hilbert spaces,, Appl. Math. Optim., (2016). doi: 10.1007/s00245-016-9339-3. Google Scholar [40] T. Levajković, S. Pilipović and D. Seleši, The stochastic Dirichlet problem driven by the Ornstein-Uhlenbeck operator$:$ Approach by the Fredholm alternative for chaos expansions,, Stoch. Anal. Appl., 29 (2011), 317. doi: 10.1080/07362994.2011.548998. Google Scholar [41] T. Levajković and D. Seleši, Chaos expansion methods for stochastic differential equations involving the Malliavin derivative Part I,, Publ. Inst. Math. (Beograd) (N.S.), 90 (2011), 65. doi: 10.2298/PIM1104065L. Google Scholar [42] T. Levajković, S. Pilipović and D. Seleši, Fundamental equations with higher order Malliavin operators,, Stochastics: An International Journal of Probability and Stochastic Processes, 88 (2016), 106. doi: 10.1080/17442508.2015.1036434. Google Scholar [43] T. Levajković, S. Pilipović and D. Seleši, Chaos expansion methods in Malliavin calculus: A survey of recent results,, Novi Sad J. Math., 45 (2015), 45. Google Scholar [44] T. Levajković, S. Pilipović, D. Seleši and M. Žigić, Stochastic evolution equations with multiplicative noise,, Electronic Journal of Probability, 20 (2015). doi: 10.1214/EJP.v20-3696. Google Scholar [45] S. Lototsky and B. Rozovskii, Stochastic differential equations: A Wiener chaos approach,, From stochastic calculus to mathematical finance, (2006), 433. doi: 10.1007/978-3-540-30788-4_23. Google Scholar [46] E. A. Kalpinelli, N. E. Frangos and A. N. Yannacopoulos, Numerical methods for hyperbolic SPDEs: A Wiener chaos approach,, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 606. doi: 10.1007/s40072-013-0019-x. Google Scholar [47] H. Matthies, Stochastic finite elements: Computational approaches to stochastic partial differential equations,, Z. Angew. Math. Mech., 88 (2008), 849. doi: 10.1002/zamm.200800095. Google Scholar [48] R. Mikulevicius and B. Rozovskii, On unbiased stochastic Navier-Stokes equations,, Probab. Theory Related Fields, 154 (2012), 787. doi: 10.1007/s00440-011-0384-1. Google Scholar [49] A. Monti, F. Ponci and T. Lovett, A polynomial chaos theory approach to the control design of a power converter,, In Power Electronics Specialists Conference, (6) (2004), 4809. doi: 10.1109/PESC.2004.1354850. Google Scholar [50] D. Nualart, The Malliavin Calculus and Related Topics,, $2^{nd}$ edition, (2006). Google Scholar [51] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences 44, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [52] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control. Optim., 30 (1992), 284. doi: 10.1137/0330018. Google Scholar [53] S. Peng, Open problems on backward stochastic differential equations,, in Control of Distributed Parameter and Stochastic Systems, (1998), 265. Google Scholar [54] S. Pilipović and D. Seleši, Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10 (2007), 79. doi: 10.1142/S0219025707002634. Google Scholar [55] A. Sandu, C. Sandu, B. J. Chan and M. Ahmadian, Control mechanical systems using a parameterized spectral decomposition approach,, in Proceedings of the IMECE04, (2004). Google Scholar [56] W. Schoutens, Stochastic Processes and Orthogonal Polynomials,, Lecture Notes in Statistics 146, 146 (2000). doi: 10.1007/978-1-4612-1170-9. Google Scholar [57] B. A. Templeton, A Polynomial Chaos Approach to Control Design,, Ph.D Thesis, (2009). Google Scholar [58] D. Venturi, X. Wan, R. Mikulevicius, B. L. Rozovskii and G. E. Karniadakis, Wick - Malliavin approximation to nonlinear stochastic partial differential equations: Analysis and simulations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013). doi: 10.1098/rspa.2013.0001. Google Scholar [59] N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897. doi: 10.2307/2371268. Google Scholar [60] W. M. Wonham, On the separation theorem of stochastic control,, SIAM J. Control, 6 (1968), 312. doi: 10.1137/0306023. Google Scholar [61] W. M. Wonham, On a matrix Riccati equation of stochastic control,, SIAM J. Control, 6 (1968), 681. doi: 10.1137/0306044. Google Scholar [62] D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations,, SIAM J. Sci. Comput., 24 (2002), 619. doi: 10.1137/S1064827501387826. Google Scholar [63] J. Yong and X. Y. Zhou, Stochastic Controls - Hamiltonian Systems and HJB Equations,, Applications of Mathematics, 43 (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

show all references

##### References:
 [1] E. Arias, V. Hernández, J. Ibanes and J. Peinado, A family of BDF algorithms for solving differential matrix Riccati equations using adaptive techniques,, Procedia Computer Science, 1 (2010), 2569. Google Scholar [2] G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics,, J. Optim. Theory Appl., 91 (1996), 695. doi: 10.1007/BF02190128. Google Scholar [3] H. T. Banks, R. J. Silcox and R. C. Smith, The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples,, ASME J. Vibration Acoustics, 116 (1994), 386. doi: 10.1115/1.2930440. Google Scholar [4] H. T. Banks, R. C. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates and beams,, Quart. Appl. Math, 53 (1995), 353. Google Scholar [5] P. Benner, P. Ezzatti, H. Mena, E. S. Quintana-Ortí and A. Remón, Solving matrix equations on multi-core and many-core architectures,, Algorithms, 6 (2013), 857. doi: 10.3390/a6040857. Google Scholar [6] P. Benner and H. Mena, Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations,, MPI Magdeburg Preprint MPIMD/12-13, (2012), 12. Google Scholar [7] P. Benner and H. Mena, Rosenbrock methods for solving differential Riccati equations,, IEEE Transactions on Automatic Control, 58 (2013), 2950. doi: 10.1109/TAC.2013.2258495. Google Scholar [8] F. E. Benth and T. G. Theting, Some regularity results for the stochastic pressure equation of Wick-type,, Stochastic Analysis and Applications, 20 (2002), 1191. doi: 10.1081/SAP-120015830. Google Scholar [9] J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients,, SIAM J. Control. Optim., 14 (1976), 419. doi: 10.1137/0314028. Google Scholar [10] J.-M. Bismut, Contrôle des systmes linéaires quadratiques: Applications de l'intégrale stochastique,, in Séminaire de Probabilités XII, 649 (1978), 180. Google Scholar [11] K. R. Dahl, S.-E. A. Mohammed, B. Øksendal and E. E. Røse, Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives,, preprint, (). Google Scholar [12] G. Da Prato, Direct solution of a Riccati equation arising in stochastic control theory,, Appl. Math. Optim., 11 (1984), 191. doi: 10.1007/BF01442178. Google Scholar [13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, $2^{nd}$ edition, (2014). doi: 978-1-107-05584-1. Google Scholar [14] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary,, Appl. Math. Optim, 14 (1986), 107. doi: 10.1007/BF01442231. Google Scholar [15] H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, $2^{nd}$ edition, 25 (2006). Google Scholar [16] J. Fisher and R. Bhattacharya, On stochastic LQR design and polynomial chaos,, in American Control Conference, (2008), 95. doi: 10.1109/ACC.2008.4586473. Google Scholar [17] J. Fisher and R. Bhattacharya, Stability analysis of stochastic systems using polynomial chaos,, In American Control Conference, (2008), 4250. doi: 10.1109/ACC.2008.4587161. Google Scholar [18] R. Ghanem and P. D. Spanos, Polynomial chaos in stochastic finite elements,, Journal of Applied Mechanics, 57 (1990), 197. doi: 10.1115/1.2888303. Google Scholar [19] M. Grothaus, Y. G. Kondratiev and G. F. Us, Wick calculus for regular generalized stochastic functionals,, Random Oper. Stochastic Equations, 7 (1999), 263. doi: 10.1515/rose.1999.7.3.263. Google Scholar [20] G. Guatteri and G. Tessitore, On the backward stochastic Riccati equation in infinite dimensions,, SIAM J. Control Optim., 44 (2005), 159. doi: 10.1137/S0363012903425507. Google Scholar [21] G. Guatteri and G. Tessitore, Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients,, Appl. Math. Optim., 57 (2008), 207. doi: 10.1007/s00245-007-9020-y. Google Scholar [22] C. Hafizoglu, Linear Quadratic Boundary{/Point Control of Stochastic Partial Differential Equation Systems with Unbounded Coefficients},, Ph.D thesis, (2006). Google Scholar [23] C. Hafizoglu, I. Lasiecka, T. Levajković, H. Mena and A. Tuffaha, The stochastic linear quadratic control problem with singular estimates,, preprint, (2015). Google Scholar [24] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise. An Infinite-Dimensional Calculus,, Mathematics and its Applications, 253 (1993). doi: 10.1007/978-94-017-3680-0. Google Scholar [25] H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A modeling, White Noise Functional Approach,, $2^{nd}$ edition, (2010). doi: 10.1007/978-0-387-89488-1. Google Scholar [26] F. S. Hover and M. S. Triantafyllou, Application of polynomial chaos in stability and control,, Automatica, 42 (2006), 789. doi: 10.1016/j.automatica.2006.01.010. Google Scholar [27] A. Ichikawa, Dynamic programming approach to stochastic evolution equations,, SIAM J. Control. Optim., 17 (1979), 152. doi: 10.1137/0317012. Google Scholar [28] M. Kohlmann and S. Tang, New developments in backward stochastic Riccati equations and their applications,, In: Mathematical Finance (Konstanz 2000), (2000), 194. Google Scholar [29] M. Kohlmann and S. Tang, Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging,, Stoch. Process. Appl., 97 (2002), 255. doi: 10.1016/S0304-4149(01)00133-8. Google Scholar [30] M. Kohlmann and S. Tang, Multidimensional backward stochastic Riccati equations and applications,, SIAM J. Control. Optim., 41 (2003), 1696. doi: 10.1137/S0363012900378760. Google Scholar [31] M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach,, SIAM J. Control. Optim., 38 (2000), 1392. doi: 10.1137/S036301299834973X. Google Scholar [32] H. J. Kushner, Optimal stochastic control,, IRE Trans. Auto. Control, 7 (1962), 120. doi: 10.1109/TAC.1962.1105490. Google Scholar [33] N. Lang, H. Mena and J. Saak, On the benefits of the LDL factorization for large-scale differential matrix equation solvers,, Linear Algebra and its Applications, 480 (2015), 44. doi: 10.1016/j.laa.2015.04.006. Google Scholar [34] 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, 74 (2000). Google Scholar [35] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories II. Abstract Hyperbolic-Like Systems over a Finite Time Horizon,, Encyclopedia of Mathematics and its Applications 75, 75 (2000). doi: 10.1017/CBO9780511574801.002. Google Scholar [36] I. Lasiecka and R. Triggiani, Optimal control and differential Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity,, Advances in Dynamics and Control, 2 (2004), 270. Google Scholar [37] C. Lebdzek and R. Triggiani, Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions,, Control Cybernet, 38 (2009), 1461. Google Scholar [38] T. Levajković and H. Mena, On deterministic and stochastic linear quadratic control problem,, Current Trends in Analysis and Its Applications, (2015), 315. Google Scholar [39] T. Levajković, H. Mena and A. Tuffaha, A Numerical approximation framework for the stochastic linear quadratic regulator on Hilbert spaces,, Appl. Math. Optim., (2016). doi: 10.1007/s00245-016-9339-3. Google Scholar [40] T. Levajković, S. Pilipović and D. Seleši, The stochastic Dirichlet problem driven by the Ornstein-Uhlenbeck operator$:$ Approach by the Fredholm alternative for chaos expansions,, Stoch. Anal. Appl., 29 (2011), 317. doi: 10.1080/07362994.2011.548998. Google Scholar [41] T. Levajković and D. Seleši, Chaos expansion methods for stochastic differential equations involving the Malliavin derivative Part I,, Publ. Inst. Math. (Beograd) (N.S.), 90 (2011), 65. doi: 10.2298/PIM1104065L. Google Scholar [42] T. Levajković, S. Pilipović and D. Seleši, Fundamental equations with higher order Malliavin operators,, Stochastics: An International Journal of Probability and Stochastic Processes, 88 (2016), 106. doi: 10.1080/17442508.2015.1036434. Google Scholar [43] T. Levajković, S. Pilipović and D. Seleši, Chaos expansion methods in Malliavin calculus: A survey of recent results,, Novi Sad J. Math., 45 (2015), 45. Google Scholar [44] T. Levajković, S. Pilipović, D. Seleši and M. Žigić, Stochastic evolution equations with multiplicative noise,, Electronic Journal of Probability, 20 (2015). doi: 10.1214/EJP.v20-3696. Google Scholar [45] S. Lototsky and B. Rozovskii, Stochastic differential equations: A Wiener chaos approach,, From stochastic calculus to mathematical finance, (2006), 433. doi: 10.1007/978-3-540-30788-4_23. Google Scholar [46] E. A. Kalpinelli, N. E. Frangos and A. N. Yannacopoulos, Numerical methods for hyperbolic SPDEs: A Wiener chaos approach,, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 606. doi: 10.1007/s40072-013-0019-x. Google Scholar [47] H. Matthies, Stochastic finite elements: Computational approaches to stochastic partial differential equations,, Z. Angew. Math. Mech., 88 (2008), 849. doi: 10.1002/zamm.200800095. Google Scholar [48] R. Mikulevicius and B. Rozovskii, On unbiased stochastic Navier-Stokes equations,, Probab. Theory Related Fields, 154 (2012), 787. doi: 10.1007/s00440-011-0384-1. Google Scholar [49] A. Monti, F. Ponci and T. Lovett, A polynomial chaos theory approach to the control design of a power converter,, In Power Electronics Specialists Conference, (6) (2004), 4809. doi: 10.1109/PESC.2004.1354850. Google Scholar [50] D. Nualart, The Malliavin Calculus and Related Topics,, $2^{nd}$ edition, (2006). Google Scholar [51] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences 44, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [52] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control. Optim., 30 (1992), 284. doi: 10.1137/0330018. Google Scholar [53] S. Peng, Open problems on backward stochastic differential equations,, in Control of Distributed Parameter and Stochastic Systems, (1998), 265. Google Scholar [54] S. Pilipović and D. Seleši, Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10 (2007), 79. doi: 10.1142/S0219025707002634. Google Scholar [55] A. Sandu, C. Sandu, B. J. Chan and M. Ahmadian, Control mechanical systems using a parameterized spectral decomposition approach,, in Proceedings of the IMECE04, (2004). Google Scholar [56] W. Schoutens, Stochastic Processes and Orthogonal Polynomials,, Lecture Notes in Statistics 146, 146 (2000). doi: 10.1007/978-1-4612-1170-9. Google Scholar [57] B. A. Templeton, A Polynomial Chaos Approach to Control Design,, Ph.D Thesis, (2009). Google Scholar [58] D. Venturi, X. Wan, R. Mikulevicius, B. L. Rozovskii and G. E. Karniadakis, Wick - Malliavin approximation to nonlinear stochastic partial differential equations: Analysis and simulations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013). doi: 10.1098/rspa.2013.0001. Google Scholar [59] N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897. doi: 10.2307/2371268. Google Scholar [60] W. M. Wonham, On the separation theorem of stochastic control,, SIAM J. Control, 6 (1968), 312. doi: 10.1137/0306023. Google Scholar [61] W. M. Wonham, On a matrix Riccati equation of stochastic control,, SIAM J. Control, 6 (1968), 681. doi: 10.1137/0306044. Google Scholar [62] D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations,, SIAM J. Sci. Comput., 24 (2002), 619. doi: 10.1137/S1064827501387826. Google Scholar [63] J. Yong and X. Y. Zhou, Stochastic Controls - Hamiltonian Systems and HJB Equations,, Applications of Mathematics, 43 (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar
 [1] Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 [2] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [3] Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607 [4] Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941 [5] Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 [6] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [7] Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83 [8] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613 [9] Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117 [10] Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 [11] Georg Vossen, Stefan Volkwein. Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 465-485. doi: 10.3934/naco.2012.2.465 [12] Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353 [13] Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261 [14] Olivier P. Le Maître, Lionel Mathelin, Omar M. Knio, M. Yousuff Hussaini. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199 [15] Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056 [16] Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032 [17] Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019 [18] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [19] Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028 [20] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

2018 Impact Factor: 1.048