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

Abstract

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.

Fourier-splitting method for solving hyperbolic LQR problems

“…The stochastic variant of the infinite dimensional LQR problem was studied in the same setting in [21,15,18,33]. Recently, a theoretical framework for this problem has been laid for singular estimates control systems in the presence of noise in the control and in the case of finite time penalization in the performance index, see [19].…”

Numerical solution of the finite horizon stochastic linear quadratic control problem

“…where F is replaced, for example, by A k in Equation 9, Ã k in Equation 10, Â k in Equations 13 and 14, orǍ k+1 in Equation 16. Typically, the right-hand side is assumed to be nonpositive definite, that is, Y ⩾ 0 and the solution X is required to be nonnegative definite, that is, X ⩾ 0.…”

“…The infinite dimensional linear quadratic regulator (LQR) with random coefficients has been investigated in two studies along with the associated backward SRE. In one study, a novel approach for solving the stochastic LQR based on the concept of chaos expansion from white noise analysis is proposed. For a class of control systems known as singular estimate control systems the stochastic analog of the linear quadratic problem has been first treated by Hafizoglu .…”

A splitting/polynomial chaos expansion approach for stochastic evolution equations

“…The list of references is long, here we mention just a few [20], [28], [32], [33]. In [25], [26], [27] this approach has been recently applied to the stochastic optimal regulator control problem [13]. Practical application of the Wiener polynomial chaos involves two truncations, truncation with respect to the number of the random variables and truncation with respect to the order of the orthogonal Askey polynomials used (in the particular case considered, the Legendre polynomials), see e.g.…”