Nonlinear Parabolic Optimal Control 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.

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear elliptic and parabolic optimal control problem. The Schemes use discretizations base on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables was obtained in the sense of some norms. Some numerical experiments are presented to test the theoretical result. Finally, we give some conclusions and future works.

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.

The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay

ABSTRACTIn this paper, we introduce the optimal control problems governed by a new class of impulsive stochastic partial neutral evolution equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, the analytic semigroup theory, fractional powers of closed operators, and suitable fixed point theorems, we prove an existence result of mild solutions for the control systems in the α-norm without the assumptions of compactness. Next, we derive the existence conditions of optimal pairs of these systems. Finally, application to a nonlinear impulsive stochastic parabolic optimal control system is considered.

New a posteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems

In this paper, we investigate residual-based a posteriori error estimates for the hp version of the finite element approximation of nonlinear parabolic optimal control problems. By using the hp finite element approximation for both the state and the co-state variables and the hp discontinuous Galerkin finite element approximation for the control variable, we derive hp residual-based a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, can be used to construct a reliable hp adaptive finite element approximation for the nonlinear parabolic optimal control problems.

Multiple shooting methods for parabolic optimal control problems with control constraints

AbstractIn the context of ordinary differential equations, shooting techniques are a state‐of‐the‐art solver component, whereas their application in the framework of partial differential equations (PDE) is still at an early stage. We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic PDE. Direct and indirect shooting for PDE optimal control stem from the same extended problem formulation. Our approach reveals that they are structurally similar but show major differences in their algorithmic realizations. In the presented numerical examples we cover a nonlinear parabolic optimal control problem with additional control constraints. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Indirect Multiple Shooting for Nonlinear Parabolic Optimal Control Problems with Control Constraints

We discuss the indirect multiple shooting approach for the solution of PDE-based parabolic optimal control problems with control constraints. The method is formulated within an abstract function space setting and uses a space-time Galerkin finite element discretization. The emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the PDE context. Numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail.

A PRIORI ERROR ESTIMATES AND SUPERCONVERGENCE PROPERTY OF VARIATIONAL DISCRETIZATION FOR NONLINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

In this paper, we investigate a priori error estimates and superconvergence of varitional discretization for nonlinear parabolic optimal control problems with control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. We derive a priori error estimates for the control and superconvergence between the numerical solution and elliptic projection for the state and the adjoint state and present a numerical example for illustrating our theoretical results.

Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control

The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.

Multigrid Methods and Sparse-Grid Collocation Techniques for Parabolic Optimal Control Problems with Random Coefficients

An efficient computational framework to solve nonlinear parabolic optimal control problems with random coefficients is presented. This framework allows us to investigate the influence of randomness or uncertainty of problem's parameters values on the control provided by the optimal control theory. The proposed framework combines space-time multigrid methods with sparse-grid collocation techniques. Theoretical and numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented.

On Regularity of Solutions and Lagrange Multipliers of Optimal Control Problems for Semilinear Equations with Mixed Pointwise Control-State Constraints

A class of nonlinear elliptic and parabolic optimal control problems with mixed control-state constraints is considered. Extending a method known for the control of ordinary differential equations to the case of PDEs, the Yosida–Hewitt theorem is applied to show that the Lagrange multipliers are functions of certain $L^p$-spaces. By bootstrapping arguments, under natural assumptions, optimal controls are shown to be Lipschitz continuous in the elliptic case and Hölder continuous for parabolic problems.

Analysis and semidiscrete Galerkin approximation of a class of nonlinear parabolic optimal control problems

This paper presents an optimization problem involving a system governed by a nonlinear partial differential equation with control acting on the domain, in the boundary conditions, or in the initial condition. We describe the Galerkin approximation and prove its convergence to the solutions of the initial optimization problem.

On the variational stability of a class of nonlinear parabolic optimal control problems

AbstractIn this paper we study parametric optimal control problems governed by a nonlinear parabolic equation in divergence form. The parameter appears in all the data of the problem, including the partial differential operator. Using as tools the G-convergence of operators and the Γ-convergence of functionals, we show that the set-valued map of optimal pairs is upper semicontinuous with respect to the parameter and the optimal value function responds continuously to changes of the parameter. Finally in the case of semilinear systems we show that our framework can also incorporate systems with weakly convergent coefficients.

On the Variational Stability of a Class of Nonlinear Parabolic Optimal Control Problems

In this paper we study parametric optimal control problems governed by a non-linear parabolic equation in divergence form. The parameter appears in all the data of the problem, including the partial differential operator. Using as tools the G-convergence of operators and the \Gamma -convergence of functionals, we show that the set-valued map of optimal pairs is upper-semicontinuous with respect to the parameter and the optimal value function responds continuously to changes of the parameter. Finally, in the case of semilinear systems we show that our framework can also incorporate systems with weakly convergent coefficients.

Optimal control of nonlinear evolution equations with nonmonotone nonlinearities

In this paper, we prove the existence of optimal admissible pairs for a large class of strongly nonlinear evolution equations, involving nonmonotone nonlinearities. An example of a nonlinear parabolic optimal control system is also worked out in detail.

Infinite dimensional parametric optimal control problems

In this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parameter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.

Infinite Dimensional Parametric Optimal Control Problems

AbstractIn this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parameter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.

On the optimal control of strongly nonlinear evolution equations

In this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state. First we prove an existence theorem. Then using the penalty method we obtain necessary conditions for optimality. From them we derive a “bang-bang” characterization of optimal controls. Finally an example of a nonlinear parabolic optimal control system is worked out in detail.

Variational stability of infinite-dimensional optimal control problems

In this paper we examine the variational stability of infinite-dimensional optimal control problems governed by nonlinear evolution equations. Our tools are the Kuratowski—Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functional and the convergence of the values of the problems, and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of nonlinear parabolic distributed-parameter optimal control problems