Method For Optimal Control Problems Research Articles

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

. A priori and a posteriori error estimates of hp spectral element method for an elliptic optimal control problem with L2-norm control constraint are investigated in this paper. We then present the optimality conditions and set up hp spectral element discretization scheme. Based on some important interpolation operators and suitable immediate variable, a priori error estimates for this problem in hp spectral element discretization are established carefully. Furthermore, using some interpolation operators, a rigorous posteriori error estimates of hp spectral element approximation are also proved for control and the coupled state approximation in L2−H1-norm and L2−L2-norm, respectively. Such estimators can be used to construct reliable adaptive spectral element methods for optimal control problems. Finally, the error analysis results are confirmed by numerical results.

Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove the existence and uniqueness of the stochastic optimal control solution and then derive the stochastic optimality system. For the numerical approximation, we use the stochastic Galerkin spectral methods, which apply the stochastic Galerkin method for the discretization of random variables and employ the spectral-Galerkin approach for the approximation of spatial variables. To solve the large coupled saddle-point system resulted from discretization, we adopt the most commonly used MINRES method and a more effective PPCG method in low-rank matrix iteration format. Specially, we develop the efficient preconditioners based on the matrix decomposition method and the virtual variable method. We also study the eigenvalue distribution of the corresponding preconditioned matrix. Besides, for the approximation of the discretized state equation, we use the mean-based approximation and the Ullmann approximation to handle different values of the variance of the random inputs. Finally, we present numerical experiments to demonstrate the effectiveness of our solvers and preconditioners.

Tractable global solutions to chance-constrained Bayesian optimal experiment design for arbitrary prior and noise distributions

Optimal experiment design (OED) aims to optimize the information content of experimental observations by designing the experimental conditions. In Bayesian OED for parameter estimation, the design selection is based on an expected utility metric that accounts for the joint probability distribution of the uncertain parameters and the observations. This work presents solution methods for two approximate formulations of the Bayesian OED problem based on Kullback–Leibler divergence for the particular case of Gaussian prior and observation noise distributions and the general case of arbitrary prior distributions and arbitrary observation noise distributions when the observation noise corresponds to arbitrary functions of the states and random variables with an arbitrary multivariate distribution. The proposed methods also allow satisfying path constraints with a specified probability. The solution approach relies on the reformulation of the approximate Bayesian OED problem as an optimal control problem (OCP), for which a parsimonious input parameterization is adopted to reduce the number of decision variables. An efficient global solution method for OCPs via sum-of-squares polynomials and parallel computing is then applied, which is based on approximating the cost of the OCP by a polynomial function of the decision variables and solving the resulting polynomial optimization problem to global optimality in a tractable way via semidefinite programming. It is established that the difference between the cost obtained by solving the polynomial optimization problem and the globally optimal cost of the OCP is bounded and depends on the polynomial approximation error.

Residual-based a posteriori error estimates for the [formula omitted] version of the finite element discretization of the elliptic Robin boundary control problem

Optimal control problems governed by partial differential equations have become a very active and successful research area. So, in this paper, we analyzed a priori and a posteriori error estimates for the hp finite element discretization of elliptic Robin boundary control problems. With the discrete and continuous optimality conditions of the problem, we constructed the error estimators. Based on the residual of the model equations for the coupled state and control approximations, the upper error bound is proved using Scott–Zhang-type quasi interpolation estimates. In order to provide the optimality, lower error bound is shown using some polynomial inverse estimates in weighted Sobolev spaces. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

An Adaptive Moving Mesh Method for Solving Optimal Control Problems in Viscous Incompressible Fluid

An adaptive moving mesh method for optimal control problems in viscous incompressible fluid is proposed with the incompressible Navier–Stokes system used to describe the motion of the fluid. The moving distance of nodes in the adopted mesh moving strategy is found by solving a diffusion equation with source terms, and an algorithm that fully considers the characteristics of the control problem is given with symmetry reduction to the incompressible Navier–Stokes equations. Numerical examples are provided to show that the proposed algorithm can solve the optimal control problem stably and efficiently on the premise of ensuring high precision.

A spectral Petrov-Galerkin method for optimal control problem governed by a fractional ordinary differential equation

In this paper, a spectral Petrov-Galerkin method is investigated for an optimal control problem with a fractional ordinary differential equation constraint. With the study of its well-posedness and the optimality condition in standard Sobolev space, the regularity for the optimal control problem is established in the framework of weighted Sobolev space, which is superior to the counterpart analyzed in standard Sobolev space. Based on the obtained regularity results, the error estimate of the spectral Petrov-Galerkin method is presented, and optimal convergence orders in Lω−α,02 -norm or Lω0,−α2 -norm are determined. Finally, numerical examples with both smooth inputs and rough inputs are given to verify the theoretical prediction.

A priori error estimate of perturbation method for optimal control problem governed by elliptic PDEs with small uncertainties

A priori error estimate of perturbation method for optimal control problem governed by elliptic PDEs with small uncertainties

Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

Superconvergence of Semidiscrete Splitting Positive Definite Mixed Finite Elements for Hyperbolic Optimal Control Problems

In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.

МЕТОДЫ СЛАБОГО УЛУЧШЕНИЯ НА ПЛАВАЮЩЕЙ СЕТИ ОПЕРАТОРОВ В ЗАДАЧЕ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ

The paper proposes first and second order weak improvement methods for optimal control problems on a floating network of operators. The algorithm derivation technique is based on the theory of V.F. Krotov. Conditions for the unimprovability of control, which are closely related to necessary and sufficient conditions for a weak local minimum, are given.

Algorithm of the successive approximation method for optimal control problems with phase restrictions for mechanics tasks

The algorithm of the method of successive approximations for problems of optimal control in the presence of arbitrary restrictions on control and phase variables is proposed. The approach is based on the procedures of consistent satisfaction of the necessary conditions of optimality in the form of Pontryagin's maximum principle. The algorithm application for the problems of weight optimization of power elements of structures in the presence of constraints of strength, rigidity, and technological requirements is demonstrated.

A sharp error analysis for the DG method of optimal control problems

<abstract><p>In this paper, we are concerned with a nonlinear optimal control problem of ordinary differential equations. We consider a discretization of the problem with the discontinuous Galerkin method with arbitrary order $ r \in \mathbb{N}\cup \{0\} $. Under suitable regularity assumptions on the cost functional and solutions of the state equations, we first show the existence of a local solution to the discretized problem. We then provide sharp estimates for the $ L^2 $-error of the approximate solutions. The convergence rate of the error depends on the regularity of the optimal solution $ \bar{u} $ and its adjoint state with the degree of piecewise polynomials. Numerical experiments are presented supporting the theoretical results.</p></abstract>

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

AbstractWe introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two‐ (2D) and three‐dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D spacial grids. Using the diagonalization of the arising matrix‐valued functions in the eigenbasis of the one‐dimensional Sturm–Liouville operators, we construct the rank‐structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right‐hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low‐rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an ‐threshold. This method reduces the numerical cost for solving the control problem to (plus the quadratic term with a small weight), which outperforms traditional approaches with complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by . This essentially outperforms the traditional methods operating with fully populated matrices and vectors in . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation.

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

Adaptive Virtual Element Method for Optimal Control Problem Governed by General Elliptic Equation

Adaptive Virtual Element Method for Optimal Control Problem Governed by General Elliptic Equation

Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise

Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme with three-factor coarsening to solve the corresponding discretized control problem is presented. On staggered grids, a three-factor coarsening strategy helps in simplifying the inter-grid transfer operators and reduction in computation (CPU time). For smoothing, a distributive Gauss–Seidel scheme with a line search strategy is employed. To validate the proposed multigrid staggered grid framework, numerical results are presented with white noise at the end.

Augmented Lagrangian methods for optimal control problems governed by mixed quasi‐equilibrium problems with applications

AbstractThe main goal of this article is to study a general augmented Lagrangian method for optimal control problems governed by mixed (quasi‐)equilibrium problems. We establish zero duality gap properties between a primal problem and its augmented Lagrangian dual problem. As a consequence, we give some existence results for optimal control problem governed by evolutionary quasi‐variational inequalities. An application to optimal control of obstacle problems described by quasi‐hemivariational inequalities is studied. The results obtained in this article are new and improves considerably many recent results in literature.

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

A modified optimization method for optimal control problems of continuous stirred tank reactor

‎Continuous stirred tank reactor (CSTR) is an important and constructive part in various chemical and process industries and therefore it is necessary to control the process in optimal temperature and concentration conditions. Because of the nonlinear nature and limits of the control input‎, ‎solving this problem is very difficult. To achieve a sub-optimal control policy for chemical processes‎, ‎we focused on a new construction model‎. Then‎, ‎a two-phase algorithm‎, ‎denoted as modified sequential general variable neighborhood search (MSGVNS) algorithm based on three local searches that use efficient neighborhood interchange has been employed to solve CSTR problems numerically. The results of the proposed method show that its convergence to the exact solution is achieved by the accuracy comparable to other numerical algorithms in few times‎.

A modified backward integration method for optimal control problems with degenerate equilibrium points

AbstractWe present a new numerical indirect method to solve optimal control problems with degenerate equilibrium points. When the optimal control problem is autonomous, the corresponding canonical system becomes an autonomous system of ordinary differential equations. Combining the well‐known blow‐up technique of vector fields with the classic backward integration method, and focusing on a specific type of degenerate singular points, we develop a new method that is both easy to implement and able to deal with complex problems.