Method For Optimal Control Problems Research Articles

On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations

This paper focuses on nonsmooth Newton methods of optimal control problems governed by mixed control–state constraints with differential algebraic equations. In contrast to previous results, we analyze lifting operators involved in nonsmooth Newton methods and establish corresponding convergence results. We also give sufficient conditions for regularity of generalized derivatives of systems of nonsmooth operator equations associated with optimal control problems.

Mixed Methods for Optimal Control Problems

In this paper, we investigate a posteriori error estimates of amixed finite elementmethod for elliptic optimal control problems with an integral constraint. The gradient for ourmethod belongs to the square integrable space instead of the classical H(div; Ω) space. The state and co-state are approximated by the P 0 2 -P1 (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. Using duality argument method and energy method, we derive the residual a posteriori error estimates for all variables.

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

ABSTRACTIn this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.

Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients

Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in $$L^2$$ -norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.

A quantum retrograde canon: complete population inversion in n2-state systems

We present a novel approach for analytically reducing a family of time-dependent multi-state quantum control problems to two-state systems. The presented method translates between related n2-state systems and two-state systems, such that the former undergo complete population inversion (CPI) if and only if the latter reach specific states. For even n, the method translates any two-state CPI scheme to a family of CPI schemes in n2-state systems. In particular, facilitating CPI in a four-state system via real time-dependent nearest-neighbors couplings is reduced to facilitating CPI in a two-level system. Furthermore, we show that the method can be used for operator control, and provide conditions for producing several universal gates for quantum computation as an example. In addition, we indicate a basis for utilizing the method in optimal control problems.

A priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations

In this paper, we investigate a priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Based on two new elliptic projections, we derive a priori error estimates both for the control variable, the state variable and the co-state variable. The related a priori error estimates for the new projections error are also established. Moreover, a posteriori error estimates for all variables are derived via energy method. Such a posteriori error estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation

ABSTRACTThis paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretization and a finite difference method for time discretization. Lagrange interpolating basis polynomials are used to approximate the state, and the differentiation matrix is derived to discrete the spatial derivative. We also discuss the fully discrete scheme for the control problem. A finite difference method developed in Lin and Xu [Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), pp. 1533–1552] is used to discretize the time-fractional derivative. A fully discrete first-order optimality condition is developed based on the ‘first discretize, then optimize’ approach. Furthermore, we design the projected gradient algorithm based on the fully discrete optimality conditions. Numerical examples are given to illustrate the feasibility of the proposed method.

Mixed methods for optimal control problems

Mixed methods for optimal control problems

Hybrid stochastic optimization method for optimal control problems of chemical processes

In the paper, the optimal control problem of chemical process systems is considered. In general, it is very difficult to solve this problem analytically due to its nonlinear nature and the existence of control input constraints. To obtain the numerical solution, based on the time-scaling transformation technology and the control parameterization method, the problem is transformed into a parameter optimization problem with some variable bounds, which can be efficiently solved using the improved conjugate gradient algorithm developed by us. However, in spite of the improved conjugate gradient algorithm is very efficient for local search, the solution obtained is usually a local extremum for non-convex optimal control problems. In order to escape from the local extremum, a novel stochastic search method is developed. A large number of numerical experiments show that the novel stochastic search method is excellent in exploration, while bad in exploitation. In order to improve the exploitation, we propose a hybrid stochastic optimization approach to solve the problem based on the novel stochastic search method and the improved conjugate gradient algorithm. Convergence results indicate that any global optimal solution of the approximate problem is also a global optimal solution of the original problem. Finally, four chemical process system optimal control problems illustrate that the hybrid numerical optimization algorithm proposed by us is low CPU time and obtains a better cost function value than the existing approaches.

Lagrangian methods for optimal control problems governed by a mixed quasi-variational inequality

This paper studies an optimal control problem where the state of the system is defined by a mixed quasi-variational inequality. Several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem are obtained by using nonlinear Lagrangian methods. Our results are applied to an example where the mixed quasi-variational inequality leads to a bilateral obstacle problem.

Heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients

In this paper, we investigate the heterogeneous multiscale method (HMM) for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients. The state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, while the control variable is discretized by the piecewise constants. By applying the well-known Lions’ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates are derived for the state, co-state and the control with uniform bounded constants. Finally, numerical results are presented to illustrate our theoretical findings.

Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint

In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis.

New elliptic projections and a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations

In this paper, we discuss a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Both semidiscrete and fully discrete schemes are considered. Based on some new elliptic projections, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. The related a priori error estimates for the new projections error are also established. A numerical example is given to demonstrate the theoretical results.

Convergence and Quasi-Optimality of an Adaptive Finite Element Method for Optimal Control Problems on $$L^{2}$$ L 2 Errors

Convergence and Quasi-Optimality of an Adaptive Finite Element Method for Optimal Control Problems on $$L^{2}$$ L 2 Errors

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017

A priori and a posteriori error estimates of H1‐Galerkin mixed finite element method for parabolic optimal control problems

SummaryIn this exposition, we study both a priori and a posteriori error analysis for the H1‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and convergence properties for the state variables and the control variable are obtained. In addition, L2(L2)‐norm a posteriori error estimates for the state and control variables and ‐norm for the flux variable are also derived.

A priori stopping rule for an iterative Bregman method for optimal control problems

In this article we continue our investigation of the iterative regularization method for optimization problems based on Bregman distances. The optimization problems are subject to pointwise inequality constraints in . We provide an estimate for the noise error for perturbed data, which can be used to construct an a priori stopping rule. Furthermore we show how to implement our method with a semi-smooth Newton method using finite elements and present numerical results for the stopping rule.

A composite pseudospectral method for optimal control problems with piecewise smooth solutions

In this paper, we develop a composite collocation approximation scheme for solving optimal control problems governed by ordinary differential equations with piecewise smooth solutions. For this purpose, we divide the time interval of the problem into some nonequal subintervals and define a piecewise interpolating polynomial on the base of transformed Legendre-Gauss nodes in subintervals. According to the weak representations approach, we derive the corresponding operational matrix of derivative. Using the Legendre-Gauss quadrature formula and the obtained operational matrix, the optimal control problem is discretized as a nonlinear programming problem. In this approach, the time locations in which corners happen in the state and control functions, are considered as unknown parameters. Therefore, the problem can be solved as a nonlinear programming problem with respect to these parameters. Four examples are investigated to demonstrate the validity and applicability of the proposed technique.

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.