Method For Optimal Control Problems Research Articles

Dual Control and Online Optimal Experimental Design

Dual control refers to strategies that strike a balance between control and estimation. Combined with nonlinear model predictive control, dual control offers advanced feedback methods for optimal control problems under uncertainties. We present dual control from a new perspective, namely, the interplay between the performance control task and the information gain task in connection with optimal experimental design. A new approach to dual control is proposed in which the covariance matrix of the estimates is weighted by the derivatives of the nominal objective value with respect to unknown parameters and initial states. This quantity can be interpreted as a predictive variance. Applying the idea of biobjective formulation, we use a weighted sum of the original objective function and the predictive variance as a modified objective function. Most notably, we discuss the theoretical background of this approach and provide probabilistic bounds for the controller performance with respect to the original objective function. As applications, we investigate a moon lander problem and a batch bioreactor problem. Numerical results demonstrate the superiority of dual control over nominal control. We also carry out an analysis of the relationship between the performance control task and the information gain task in order to assess when the extra effort for dual control is justified.

An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates are obtained.

A sharp regularization error estimate for bang‐bang solutions for an iterative Bregman regularization method for optimal control problems

AbstractIn the present work, we present numerical results for an iterative method for solving an optimal control problem with inequality contraints. The method is based on generalized Bregman distances. Under a combination of a source condition and a regularity condition on the active sets convergence results are presented. Furthermore we show by numerical examples that the provided a‐priori estimate is sharp in the bang‐bang case. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations

AbstractIn this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive L2 and H–1-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control

A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

Summary An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l1 exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non-differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.

Modified homotopy perturbation method for optimal control problems using the Padé approximant

In this study, we propose a hybrid method that combines the homotopy perturbation method (HPM) and Padé technique to obtain the approximate analytic solution of the Hamilton–Jacobi–Bellman equation. The truncated series solution for the HPM is suitable but only in a small domain when the exact solution is not obtained. To improve the accuracy and enlarge the convergence domain, the Padé technique is applied to the series solution for the HPM. Three examples are given to illustrate the applicability, simplicity, and efficiency of the proposed method. The results obtained are then compared with the exact solution and basic HPM. We demonstrate that this hybrid method provides an approximate analytic solution with higher accuracy than the classic HPM.

A spectral method for optimal control problems governed by the abnormal diffusion equation with integral constraint on state

In this paper, we study the space-time spectral approximation to an optimal control problem governed by the time fractional diffusion equation with an integral constraint on the state variable. The optimality conditions of the exact and discrete optimal control systems are derived in virtue of the Kuhn-Tucker condition. Then some a priori error estimates showing the spectral accuracy are obtained. Furthermore, an iterative algorithm based on the gradient projection optimization method using Galerkin spectral approximation is proposed to solve the discrete system. Finally, some numerical examples are performed to verify the theoretical results.

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

This paper analyzes the h p version of the finite element method for optimal control problems constrained by elliptic partial differential equations with random inputs. The main result is that the h p error bound for the control problems subject to stochastic partial differential equations leads to an exponential rate of convergence with respect to p as for the corresponding direct problems. Numerical examples are used to confirm the theoretical results.

Symplectic Irregular Interpolation Algorithms for Optimal Control Problems

Symplectic numerical methods for optimal control problems with irregular interpolation schemes are developed and the comparisons between irregular interpolation schemes and equidistant scheme are made in this paper. The irregular interpolation points, which are the collocation points usually adopted by pseudospectral (PS) methods, such as Legendre–Gauss, Legendre–Gauss–Radau, Legendre–Gauss–Lobatto and Chebyshev–Gauss–Lobatto points, are taken into consideration in this study. The symplectic numerical method with irregular points is proposed firstly. Then, several examples with different complexities highlight the differences in performance between different kinds of interpolation schemes. The numerical results show that the convergence of the present symplectic numerical methods can be obtained by increasing the number of sub-intervals or the number of interpolation points. Moreover, the comparison results show that the convergence of the symplectic numerical methods are generally independent on the type of interpolation points and the computational efficiency is not sensitive to the choice of interpolation points in general. Thus, the symplectic numerical methods with different interpolation schemes have obvious difference with the PS methods.

A stabilized finite element method for the convection dominated diffusion optimal control problem

In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results.

A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms

In this paper, we study the numerical solution of optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms, arising from chemical processes. The symmetric interior penalty Galerkin (SIPG) method with upwinding for the convection term is used as a discretization method. We use a residual-based error estimator for the state and the adjoint variables. An adaptive mesh refinement indicated by a posteriori error estimates is applied. The arising saddle point system is solved using a suitable preconditioner. Numerical results are presented to illustrate the performance of the proposed error estimator.

A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Stochastic Elliptic PDE with Constrained Control

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The objective is to minimize the expectation of a cost functional with the deterministic constrained control. We represent the random elliptic PDE in term of the generalized polynomial chaos expansion and obtain the deterministic optimal problem. By applying the well-known Lions' Lemma to the reduced optimal problem, we obtain the necessary and sufficient optimality conditions. We establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Stochastic Galerkin method. Then a priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

High accuracy analysis of finite element methods for optimal control problems and its application

In this paper, we briefly review the recent advances of superconvergence analysis and related efficient finite element algorithms for PDE-constrained optimal control problems. The emphasis is on the superconvergence of finite element approximations to optimal control problems governed by elliptic and parabolic equations, the superconvergence of mixed finite element approximations to optimal controls of elliptic and Stokes equations, the recovery type a posteriori error estimates and the extrapolation and defect correction methods for optimal controls. We give a survey on the above topics and perspectives for future work are included.

Expediting the transition from non-renewable to renewable energy via optimal control

Much recent climate research suggests that the transition from non-renewable to renewable energy should be expedited. To address this issue we use an optimal control model, based on an integrated assessment model of climate change that includes two types energy production. After setting up the model, we derive necessary optimality conditions in the form of a Pontryagin type Maximum Principle. We use a numerical discretization method for optimal control problems to explore various policy scenarios. The algorithm allows to compute both state and co-state variables by providing a consistent numerical approximation for the adjoint variables of the various scenarios. Our numerical method applies to control and state-constrained control problems as well as to delayed control problems. In the policy scenarios, we explore ways how the transition from non-renewable to renewable energy can be expedited.

A Fast and Stable Preconditioned Iterative Method for Optimal Control Problem of Wave Equations

In this paper, we develop a new central finite difference scheme in terms of both time and space for solving the first-order necessary optimality systems that characterize the optimal control of wave equations. The obtained new scheme is proved to be unconditionally convergent with a second-order accuracy, without the requirement of the Courant--Friedrichs--Lewy condition on the corresponding grid ratio. An efficient preconditioned iterative method is further developed for solving the discretized sparse linear system based on the relationship between the resultant matrix structure and the coupled PDE optimality system. Numerical examples are presented to verify the theoretical analysis and to demonstrate the high efficiency of the proposed preconditioned iterative solver.

Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls

In this paper, the existence and stability of solutions of nonlinear optimal control problems with $1$-mean equicontinuous controls are discussed. In particular, a new existence theorem is obtained without convexity assumption. We investigate the stability of the optimal control problem with respect to the right-hand side functions, which is important in computational methods for optimal control problems when the function is approximated by a new function. Due to lack of uniqueness of solutions for an optimal control problem, the stability results for a class of optimal control problems with the measurable admissible control set is given based on the theory of set-valued mappings and the definition of essential solutions for optimal control problems. We show that the optimal control problems, whose solutions are all essential, form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem.

Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs

We develop a numerical method for the solution to linear adjoint equations arising, for example, in optimization problems governed by hyperbolic systems of nonlinear conservation and balance laws in one space dimension. Formally, the solution requires one to numerically solve the hyperbolic system forward in time and a corresponding linear adjoint system backward in time. Numerical results for the control problem constrained by either the Euler equations of gas dynamics or isothermal gas dynamics equations are presented. Both smooth and discontinuous prescribed terminal states are considered.

Scalability of a FETI‐DP Method for Optimal Control Problems

AbstractA parallel FETI‐DP (Finite Element Tearing and Interconnecting) domain decomposition method is applied to optimal control problems with control constraints. We show parallel scalability for up to 1024 cores of a Cray XT6. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Penalty Method for Optimal Control Problem with Phase Constraint

We study control problems with phase constraint by replacing the phase constraint with a penalty function in the target functional. We establish the existence of an optimal control and prove the convergence of phase variables and controls as the penalty coefficient unboundedly increases. Bibliography: 12 titles.