We propose a time-implicit, finite-element based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+σ(t)]dW(t)$, and prove optimal convergence w.r.t. both, space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work [16], where the error analysis was restricted to additive noise.
We introduce a time-implicit, finite-element based space-time discretization scheme for the backward stochastic heat equation, and for the forward-backward stochastic heat equation from stochastic optimal control, and prove strong rates of convergence. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise.
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our f…
This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multi-parametric disaggregation technique (NMDT), applying a sophisticated discretization to both variables. We refer to this approach as doubly discretized normalized multiparametric disaggregation technique (D-NMDT). In a comprehensive theoretical analysis, we underline the…
This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.
We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We prove theoretically that the randomized Newton's method has a quadratic convergence locally. We also apply this new method to various numerical examples, from one- to high-dimensional differential equations, in order to verify its feasibility and efficienc…
In this work, we propose a feedback control based temporal discretization for linear quadratic optimal control problems (LQ problems) governed by controlled mean-field stochastic differential equations. We firstly decompose the original problem into two problems: a stochastic LQ problem and a deterministic one. Secondly, we discretize both LQ problems one after another relying on Riccati equations and control's feedback representations. Then, we prove the convergence rates for the proposed discretization and present an effective algorithm. Finally, a numerical example is provided to support th…
In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretization are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in [W. Hager, Runge-Kutta method in optimal control and the discrete adjoint system, Numer. Math.,2000, pp. 247-282], direct approach with RK discretization is equivalent with indirect approach based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue…
We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. We prove error bounds for control and state.
We consider a rod that is heated/cooled and sensed at multiple point locations. To stabilize it to a constant temperature we set up a Linear Quadratic Regulator that we explicitly solve by the method of completing the square to find the optimal linear state feedback for the point actuators. But we don't assume that the whole state is mesureable so we construct an infinite dimensional Kalman filter to estimate the whole state from a finite number of noisy point measurements. These two components yield a Linear Quadratic Gaussian (LQG) Synthesis for the heat equation under point actuation and po…
Functional error estimators for the adaptive discretization of inverse problems
Faculty of Mathematics, Universität Duisburg-Essen（杜伊斯堡-埃森大学数学学院）；Institute of Mathematics, Alpen-Adria Universität Klagenfurt（克拉根福阿尔卑斯-亚得里亚大学数学研究所）；Institute of Mathematics, Universität Würzburg（维尔茨堡大学数学研究所）
This paper deals with time-optimal control of nonlinear continuous-time systems based on direct collocation. The underlying discretization grid is variable in time, as the time intervals are subject to optimization. This technique differs from approaches that are usually based on a time transformation. Hermite-Simpson collocation is selected as common representative in the field of optimal control and trajectory optimization. Hereby, quadratic splines approximate the system dynamics. Several splines of different order are suitable for the control parameterization. A comparative analysis reveal…
We consider the variational discretization of a linear-quadratic optimal control problem with pointwise control and state constraints. In order to allow for a Fréchet smooth norm, the problem is reformulated by means of a reflexive Sobolev space instead of the space of continuous functions. The discretization of the state equation yields a family of perturbed optimal control problems, whose solutions can be computed numerically. We apply an implicit multifunction theorem (IMT) to the first order necessary conditions to proof a bound on the perturbation error for these solutions. In order to ve…
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,…
The derivation of second-order ordinary differential equations (ODEs) as continuous-time limits of optimization algorithms has been shown to be an effective tool for the analysis of these algorithms. Additionally, discretizing generalizations of these ODEs can lead to new families of optimization methods. We study discretizations of an Euler-Lagrange equation which generate a large class of accelerated methods whose convergence rate is $O(\frac{1}{t^p})$ in continuous-time, where parameter $p$ is the order of the optimization method. Specifically, we address the question asking why a naive exp…
The linear quadratic regulator problem is central in optimal control and was investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical ``maximum principle`` perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time con…
