scholarly journals Stability Analysis of Optimal Trajectory for Nonlinear Optimal Control Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Hongyong Deng ◽  
Wei Zhang ◽  
Changchun Shen
Keyword(s):  
Optimal Control ◽  
Optimal Trajectory ◽  
Optimal Control Problems ◽  
Baire Category ◽  
The State ◽  
Optimal Trajectories ◽  
Control Problems ◽  
State Equations ◽  
Right Hand ◽  
The Right

Due to the need for numerical calculation and mathematical modelling, this paper focuses on the stability of optimal trajectories for optimal control problems. The basic ideas and techniques are based on the compactness of the optimal trajectory set and set-valued mapping theorem. Through lack of optimal control stability, the result of generic stability for optimal trajectories is obtained under the perturbations of the right-hand side functions of the state equations; in the sense of Baire category, the right-hand side functions of the state equations of optimal control can be approximated by other functions.

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

2016 ◽  
Vol 8 (6) ◽  
pp. 1050-1071 ◽  
Author(s):  
Tianliang Hou ◽  
Li Li
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
Elliptic Equations ◽  
Optimal Control Problems ◽  
Mixed Finite Element ◽  
Control Variable ◽  
The State ◽  
Control Problems ◽  
General Elliptic

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.

scholarly journals Optimal control problems with elastic collisions

1989 ◽  
Vol 30 (4) ◽  
pp. 470-482
Author(s):  
J. M. Murray
Keyword(s):  
Optimal Control ◽  
State Vector ◽  
State Constraints ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
The State ◽  
Control Problems ◽  
Elastic Collisions ◽  
Linear State

AbstractIn this paper consider we optimal control problems with linear state constraints where the states can be discontinuous at the boundary. In fact the state vector models the cause the position and velocity of a particle where the collisions with the boundary that cause the discontinuities are elastic. Necessary conditions are derived by looking at limits of approximate problems that are unconstrained.

A Posteriori Error Estimates of Semidiscrete Mixed Finite Element Methods for Parabolic Optimal Control Problems

2015 ◽  
Vol 5 (1) ◽  
pp. 85-108 ◽  
Author(s):  
Yanping Chen ◽  
Zhuoqing Lin
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
Finite Element Methods ◽  
Optimal Control Problems ◽  
Mixed Finite Element ◽  
Mixed Finite Element Methods ◽  
The State ◽  
Control Problems ◽  
A Posteriori

AbstractA posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

scholarly journals Numerical Solution of Some Types of Fractional Optimal Control Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Tamer Mostafa Al-Ajami ◽  
Ronald H. W. Hoppe
Keyword(s):  
Optimal Control ◽  
Numerical Solution ◽  
Optimal Control Problems ◽  
Ritz Method ◽  
Necessary Optimality Conditions ◽  
State Equation ◽  
The State ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm “optimize first, then discretize” and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.

scholarly journals Approximate Solutions to Nonlinear Optimal Control Problems in Astrodynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Francesco Topputo ◽  
Franco Bernelli-Zazzera
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Approximate Solutions ◽  
Nonlinear Optimal Control ◽  
The State ◽  
Control Problems ◽  
Linear Quadratic ◽  
Approximating Sequence

A method to solve nonlinear optimal control problems is proposed in this work. The method implements an approximating sequence of time-varying linear quadratic regulators that converge to the solution of the original, nonlinear problem. Each subproblem is solved by manipulating the state transition matrix of the state-costate dynamics. Hard, soft, and mixed boundary conditions are handled. The presented method is a modified version of an algorithm known as “approximating sequence of Riccati equations.” Sample problems in astrodynamics are treated to show the effectiveness of the method, whose limitations are also discussed.

scholarly journals Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xenia Kerkhoff ◽  
Sandra May
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Burgers Equation ◽  
Adjoint Equation ◽  
State Equation ◽  
Space Time ◽  
The State ◽  
Control Problems ◽  
Viscous Term ◽  
Distributed Optimal Control Problems

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Solving optimal control problems using the Picard’s iteration method

2020 ◽  
Vol 54 (5) ◽  
pp. 1419-1435
Author(s):  
Abderrahmane Akkouche ◽  
Mohamed Aidene
Keyword(s):  
Optimal Control ◽  
Iteration Method ◽  
Optimal Trajectory ◽  
Optimal Control Problems ◽  
Approximate Analytical Solution ◽  
Minimum Principle ◽  
Necessary Optimality Conditions ◽  
Control Law ◽  
Control Problems ◽  
Objective Functional

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

scholarly journals Analysis of one class of optimal control problems for distributed-parameter systems

2021 ◽  
Vol 5 (4 (113)) ◽  
pp. 26-33
Author(s):  
Kamil Mamtiyev ◽  
Tarana Aliyeva ◽  
Ulviyya Rzayeva
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Optimal Control ◽  
Approximate Solution ◽  
Boundary Conditions ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Method Of Straight Lines ◽  
The Right ◽  
Straight Lines

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems

2010 ◽  
Vol 10 (3) ◽  
pp. 283-301 ◽  
Author(s):  
E. Laitinen ◽  
A. Lapin ◽  
S. Lapin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Function ◽  
Elliptic Boundary ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
Finite Dimensional ◽  
The Right

AbstractIterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

Sign in / Sign up

Export Citation Format

Share Document