Approximation Of Optimal Control Problems Research Articles

A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain L^{2}(H^{1})-L^{2}(L ^{2}) a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain L^{2}(L^{2})-L^{2}(L^{2}) a posteriori error estimates of the approximation solutions for both the state and the control.

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

In this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed/discontinuous approximation) measured in suitable norms are derived, showing optimal orders of convergence.

A set-based model-free reinforcement learning design technique for nonlinear systems

In this study, we propose an extremum-seeking approach for the approximation of optimal control problems for unknown nonlinear dynamical systems. The technique combines a phasor extremum seeking controller with an reinforcement learning strategy. The learning approach is used to estimate the value function of an optimal control problem of interest. The phasor extremum seeking controller implements the approximate optimal controller. The approach is shown to provide reasonable approximations of optimal control problems without the need for a parameterization of the nonlinear control system. A simulation example are provided to demonstrate the effectiveness of the technique.

Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. A symmetric interior penalty Galerkin method in space and a backward Euler method in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimators.

Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t = 0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.

Approximate Optimal Control via Measurement Feedback for a Class of Nonlinear Systems

The approximate optimal control problem via measurement feedback for input-affine nonlinear systems is considered in this paper. In particular, a systematic method is provided for constructing stabilising output feedbacks that approximate - with the optimality loss explicitly quantifiable - the solution of the optimal control problem by requiring only the solution of algebraic equations. In fact, the combination of a classical state estimate with an additional dynamic extension permits the construction of a dynamic control law, without involving the solution of any partial differential equation or inequality. Moreover, provided a given sufficient condition is satisfied, the dynamic control law is guaranteed to be (locally) stabilising. A numerical example illustrating the method is provided.

A stabilized mixed finite element approximation of bilinear state optimal control problems

We propose a stabilized mixed finite element approximation for optimal control problems governed by bilinear state equations. It is proved that the resulting mixed bilinear formulation is coercive and also continuous, which avoids the difficulty in choosing the mixed finite element spaces, i.e., the Ladyzhenkaya–Babuska–Brezzi matching condition for the mixed finite element spaces is unnecessary. Under pointwise bilateral constraint on the control variable, we deduce the optimality conditions at both continuous and discrete levels for the optimal control problems under consideration. Then an a priori error analysis in a weighted norm is discussed, with relatively low regularity requirements for the solutions to the optimal control problems. Finally, numerical experiments are given to confirm the efficiency and reliability of the proposed stabilized mixed finite element method.

Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state

An integral constraint on state optimal control problem governed by the first bi-harmonic equation is analyzed and its Legendre Galerkin spectral approximations are constructed. The optimality conditions of the continuous and the discretized optimal control problems are presented by the Kuhn–Tucker condition, respectively. The a priori error estimates of high accuracy are obtained. Furthermore, an efficient projection algorithm is proposed, and the convergent rate is discussed with precise details. Ample numerical examples are carried out to validate the theoretical results.

Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives

Mathematical formulations of nonlinear optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with controls in the coefficients multiplying the highest derivatives are studied. Finite difference approximations of optimization problems are constructed, and the approximation error is estimated with respect to the state and the cost functional. Weak convergence of the approximations with respect to the control is proved. The approximations are regularized in the sense of Tikhonov.

Optimal control of a batch fermentation process with nonlinear time-delay and free terminal time and cost sensitivity constraint

In this paper, we consider a nonlinear time-delay dynamic system with uncertain system parameters to characterize the process of batch fermentation. Our goal is to design an optimal control scheme to maximize the productivity of 1,3-propanediol (1,3-PD). Accordingly, we introduce an optimal control problem governed by the nonlinear time-delay dynamic system, in which the control variables are the free terminal time of the batch fermentation process and the initial concentrations of biomass and glycerol. The optimal control problem is subject to a cost sensitivity constraint for ensuring that an acceptable level of the system performance is achieved and continuous state inequality constraints for ensuring that the concentrations of biomass, glycerol, 1,3-PD, acetate, ethanol lie within specified limits. Then, the optimal control problem with free terminal time is transformed, via a hybrid time-scaling strategy, into an equivalent problem with fixed terminal time, which is much preferred for numerical computation. Using the constraint transcription and local smoothing approximation techniques, we approximate these continuous state inequality constraints by conventional inequality constraints to yield an approximate optimal control problem. Because of the highly complex nature of this approximate problem, a parallel algorithm based on the filled function method is constructed to solve this approximate problem. Finally, it is observed that the optimal control obtained is satisfactory through numerical simulations.

New a posteriori L ∞(L 2) and L 2(L 2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L∞(J; L2Ω)-norm and L2(J; L2Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.

Distributed and boundary optimal control of the Allen–Cahn equation with regular potential and dynamic boundary conditions

ABSTRACTThis paper is concerned with an optimal control problem (P) (both distributed control as well as boundary control) for the nonlinear phase-field (Allen–Cahn) equation, involving a regular potential and dynamic boundary condition. A family of approximate optimal control problems (Pϵ) is introduced and results for the existence of an optimal control for problems (P) and (Pϵ) are proven. Furthermore, the convergence result of the optimal solution of problem (Pϵ) to the optimal solution of problem (P) is proved. Besides the existence of an optimal control in problem (Pϵ), necessary optimality conditions (Pontryagin's principle) as well as a conceptual gradient-type algorithm to approximate the optimal control, were established in the end.

A priori error analysis of stabilized mixed finite element method for reaction-diffusion optimal control problems

In this paper, we propose a novel stabilized mixed finite element method for the approximation of optimal control problems governed by reaction-diffusion equations. Compared with the classical mixed finite element methods, the main contributions of this paper are as follows. First, the novel method uses only one stabilization parameter which is not mesh-dependent, and, the new mixed bilinear formulation is coercive and continuous. Second, the novel method is easy to be implemented on a computer using the standard Lagrange finite element. Third, the solutions of the novel method to the optimal control problems require low regularities. Fourth, the Ladyzhenkaya-Babuska-Brezzi (LBB) or the inf-sup condition for the mixed element spaces is unnecessary. Based on the novel method, we derive both continuous and discrete optimality systems for the corresponding constrained optimal control problems, and then a priori error analysis in a weighted norm is discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the novel stabilized method.

Approximation of Optimal Control Problems in the Coefficient for the $p$-Laplace Equation. I. Convergence Result

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted $p$-Laplace problem. The coefficient of the $p$-Laplacian, the weight $u$, we take as a control in $BV(\Omega)\cap L^\infty(\Omega)$. In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the $p$-Laplacian, we use a regularization, sometimes referred to as the $\varepsilon$-$p$-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted $\varepsilon$-$p$-Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by $k$. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each $(\varepsilon,k)$-level as the parameters tend to zero and infinity, respectively.

Sensitivity analysis approach for reduced-order approximations of optimal control problems governed by Burgers equation

Summary The reduced-order model of the optimal control problem governed by Burgers equation is derived using the proper orthogonal decomposition (POD) method. The reduced-order solution depending on parameters, which are different from the nominal values, may not be accurate if the POD basis functions depending on the nominal values are used to derive the reduced-order model. It is known that Burgers equation is sensitive to the perturbations in the diffusion term, so we use the sensitivity information to improve the robustness of the POD solution by generating two new bases: extrapolated and expanded POD basis. We compare these different bases in terms of accuracy, robustness, and computational time. Copyright © 2015 John Wiley & Sons, Ltd.

Finite element approximation of optimal control problems governed by time fractional diffusion equation

In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoint state, while the control is discretized by variational discretization method. A priori error estimates for the semi-discrete approximations of the state, adjoint state and control are derived. Furthermore, we also discuss the fully discrete scheme for the control problems. A finite difference method developed in Lin and Xu (2007) is used to discretize the time fractional derivative. Fully discrete first order optimality condition is developed based on ‘first discretize, then optimize’ approach. The stability and truncation error of the fully discrete scheme are analyzed. Numerical example is given to illustrate the theoretical findings.

Scalable Dynamic Bid Prices for Network Revenue Management in Continuous Time

This study develops an approximate optimal control problem to produce time‐dependent bid prices for the airline network revenue management problem. The main contributions of our study are the analysis of time‐dependent bid prices in continuous time and the use of splines to modify the problem into an approximate second‐order cone program (ASOCP). The spline representation of bid prices permits the number of variables to depend solely on the number of resources and not on the size of the booking horizon. The advantage of this framework is the ASOCP's scalability, which we demonstrate by solving for bid prices on an industrial‐sized network. The numerical experiments highlight the ASOCP's ability to solve industrial sized problems in seconds.

A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux $$\sigma $$?. We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya---Babuska---Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretization-then-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

We consider the optimal control problem governed by diffusion–convection–reaction equation without control constraints. The proper orthogonal decomposition (POD) method is used to reduce the dimension of the problem. The POD method may lack accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. To increase the accuracy and the robustness of the basis, we compute five bases additional to the baseline POD in case of the perturbation of the diffusion term, a parameter in the convection field, the reaction term and Tikhonov regularization term. For the first two bases, we use the sensitivity information to extrapolate and expand the baseline POD basis. The other one is based on the subspace angle interpolation method. Multiple snapshot sets are used to derive the last two bases. A-posteriori error estimator is used to analyse the difference between the suboptimal control, computed using the POD basis, and the optimal control. We compare these different bases in terms of accuracy and complexity, investigate the advantages and main drawbacks of them.

Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with control in matching boundary conditions

Mathematical formulations of nonlinear optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with control in matching conditions are examined. Finite difference approximations of optimization problems are constructed, and the approximation error is estimated with respect to the state and the cost functional. Weak convergence of the approximations with respect to the control is proved. The approximations are regularized using Tikhonov regularization.