A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces

We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solution of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.

[1]  J. Bismut.  Linear Quadratic Optimal Stochastic Control with Random Coefficients , 1976 .

[2]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[3]  Akira Ichikawa,et al.  Dynamic Programming Approach to Stochastic Evolution Equations , 1979 .

[4]  Jacinto Javier Ibáñez,et al.  A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques , 2010, ICCS.

[5]  W. Wonham.  On a Matrix Riccati Equation of Stochastic Control , 1968 .

[6]  Franco Flandoli,et al.  Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary , 1986 .

[7]  Peter Benner,et al.  Rosenbrock Methods for Solving Riccati Differential Equations , 2013, IEEE Transactions on Automatic Control.

[8]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[9]  J. Bismut,et al.  Controle des systemes lineaires quadratiques : Applications de l’integrale stochastique , 1978 .

[10]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[11]  Irena Lasiecka,et al.  Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems , 2004 .

[12]  Shanjian Tang,et al.  Global Adapted Solution of One-Dimensional Backward Stochastic Riccati Equations, with Application to the Mean-Variance Hedging , 2002 .

[13]  W. Wonham.  On the Separation Theorem of Stochastic Control , 1968 .

[14]  Michael Kohlmann,et al.  Multidimensional Backward Stochastic Riccati Equations and Applications , 2002, SIAM J. Control. Optim..

[15]  Gianmario Tessitore,et al.  Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients , 2008 .

[16]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems (Systems & Control: Foundations & Applications) , 2006 .

[17]  Irena Lasiecka,et al.  Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction , 2009, Syst. Control. Lett..

[18]  F. Tröltzsch,et al.  Fast solution of optimal control problems in the selective cooling of steel , 2001 .

[19]  Irena Lasiecka,et al.  Differential riccati equation for the active control of a problem in structural acoustics , 1996 .

[20]  Karl Kunisch,et al.  The linear regulator problem for parabolic systems , 1984 .

[21]  G. Da Prato.  Direct solution of a riccati equation arising in stochastic control theory , 1984 .

[22]  Irena Lasiecka,et al.  Optimal Control and Differential Riccati Equations under Singular Estimates for eAtB in the Absence of Analyticity , 2004 .

[23]  S. Pilipovic,et al.  Stochastic evolution equations with multiplicative noise , 2015, 2303.06225.

[24]  Xun Yu Zhou,et al.  Relationship Between Backward Stochastic Differential Equations and Stochastic Controls: A Linear-Quadratic Approach , 2000, SIAM J. Control. Optim..

[25]  Hermann Mena,et al.  On Deterministic and Stochastic Linear Quadratic Control Problems , 2015 .

[26]  Gianmario Tessitore,et al.  On the Backward Stochastic Riccati Equation in Infinite Dimensions , 2005, SIAM J. Control. Optim..

[27]  K. Kunisch,et al.  Convergence rates for the feedback operators arising in the linear quadratic regulator problem governed by parabolic equations , 1991 .

[28]  Michael Kohlmann,et al.  New Developments in Backward Stochastic Riccati Equations and Their Applications , 2001 .

[29]  Harold J. Kushner,et al.  Optimal stochastic control , 1962 .

[30]  H. Mena,et al.  The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach , 2016 .

[31]  Enrique S. Quintana-Ortí,et al.  Solving Matrix Equations on Multi-Core and Many-Core Architectures , 2013, Algorithms.

[32]  J. S. Gibson,et al.  The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces , 1979 .

[33]  Hermann Mena,et al.  On the benefits of the LDLT factorization for large-scale differential matrix equation solvers , 2015 .

[34]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[35]  V. Dragan,et al.  Mathematical Methods in Robust Control of Linear Stochastic Systems , 2006 .

[36]  Amjad Tuffaha,et al.  The Stochastic Linear Quadratic Control Problem with Singular Estimates , 2017, SIAM J. Control. Optim..