Solving a Class of Non-Smooth Optimal Control Problems (original) (raw)

Smoothers for control- and state-constrained optimal control problems

Computing and Visualization in Science, 2007

A framework for designing robust smoothing procedures for control-and state-constrained optimal control problems is presented. The focus is on minimization problems governed by elliptic partial differential equations with additional pointwise constraints on the control variable or on the state variable, respectively. The basic principle for the construction of the present smoothers is the solution of the corresponding optimality systems at grid-point level. A new approach is presented to cope with the lack of differentiability due to the presence of the constraints.

A Semi-Analytic Method for Solving a Class of Non-Linear Optimal Control Problems

International Journal of Industrial Electronics, Control and Optimization (IECO), 2020

This paper, proposes an approximate analytical method to solve a class of optimal control problems. This method is an enhancement of the variational iteration method (VIM) that named modified variational iteration method (MVIM) and eliminates all additional calculations in VIM, thus requires less time to do the calculations. In this approach, first, the optimal control problem is converted into a nonlinear two-point boundary value problem via the Pontryagins maximum principle, and then we applied the MVIM method to solve this boundary value problem. This suggested method is suitable for a large class of nonlinear optimal control problems that for the non-linear part of the problem, we used the Taylor series expansion. In the end, three examples are provided to demonstrate the simplicity and efficiency of the method. Numerical results of the proposed method versus other methods is presented in tables. All calculations were carried out using Mathematica software.

Discretization-Optimization Methods for Relaxed Optimal Control Problems

2006

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints. Since this problem may have no classical solutions, it is also formulated in relaxed form. The classical control problem is then discretized by using the implicit midpoint scheme for state approximation, while the controls are approximated by piecewise constant classical ones. We first study the behavior in the limit of properties of discrete optimality, and of discrete admissibility and extremality. We then apply a penalized gradient projection method to each discrete classical problem, and also a corresponding progressively refining discretization-optimization method to the continuous classical problem, thus reducing computing time and memory. We show that accumulation points of sequences generated by these methods are admissible and extremal for the corresponding discrete or continuous, classical or relaxed, problem. For nonconvex problems whose solutions are non-classical, we show that we can apply the above methods to the problem formulated in the Gamkrelidze form. Finally, numerical examples are given.

Numerical solution of nonlinear optimal control problems using nonlinear programming

Applied Mathematics and Computation, 2007

To solving nonlinear control problems and especially nonlinear optimal control problems (NOCP), classical methods are not usually efficient. In this paper we introduce a new approach for solving this class of problems by using Nonlinear Programming Problem (NLPP). First, we transfer the original problem to a new problem in form of calculus of variations. Then we discretize the new problem and solve it by using NLPP packages. The solution of the NLPP is used to obtain the optimal control and states, which are the exact solution of the original problem (NOCP). What is more, a NLPP is transferred to a Linear Programming Problem (LPP) which empower us to use powerful LP softwares. The degree of desirability is described for suboptimal approximate solutions. Also the nonlinear approximate solution and the optimal control are shown as a combination of polynomial functions or periodic functions. Finally, efficiency of our approach is confirmed by some numerical examples.

Numerical Solution of a Class of Nonlinear Optimal Control Problems

2015

In this article, a numerical approach for solving a class of nonlinear optimal control problems is presented. This approach is a combination of a spectral collocation method and the parametric iteration method. As will be shown, the proposed indirect strategy provides good approximations of all variables i.e. control, state and costate as opposed to the many direct methods. Several examples are considered to assess the accuracy and features of the presented method.

Algorithms for Nonlinear Optimal Control Problems Based on the First and Second Order Necessary Conditions

Journal of Mathematical Sciences, 2019

We develop an algorithmic support for some classes of nonlinear optimal control problems. We propose several algorithms based on the first-and second order necessary conditions for solving optimal control problems with constraints on control. Efficiency of the algorithms is confirmed by solving various test and applied optimal control problems. Bibliography: 11 titles. Illustrations: 3 figures.

On the Numerical Solution to Optimal Control Problems with Non-Local Conditions

2019

Optimal control problems involving non-separated multipoint and integral conditions are investigated. For numerical solution to the problem, we propose to use first order optimization methods with application of the formulas for the gradient of the functional obtained in the work. To solve the adjoint boundary problems, we propose an approach. This approach makes it possible to reduce solving initial boundary problems to solving supplementary Cauchy problems and a linear algebraic system of equations. Results of numerical experiments are given.

Discretization Methods for Nonconvex Optimal Control Problems with State Constraints

Numerical Functional Analysis and Optimization, 2005

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Because no convexity assumptions are made, the problem may have no classical solutions, and it is reformulated in relaxed form. The relaxed control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by piecewise constant relaxed controls. We first study the behavior in the limit of properties of discrete relaxed optimality, and of discrete relaxed admissibility and extremality. We then apply a penalized conditional descent method to each discrete relaxed problem, and also a corresponding discrete method to the continuous relaxed problem that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal for the discrete or the continuous problem. Finally, numerical examples are given.

A Discretisation Method for Solving Time Optimal Control Problems

Classical methods are mostly deficient for solving nonlinear time optimal control problems. In this paper an approach to solve this kind of problems is considered. This method was first presented by Badakhshan et al. in [1] and in this paper, this method is expanded to solve time optimal control problems. In this approach the time optimal control problem is changed to a problem in calculus of variations and then is solved by a discretisation method.

A note on non‐smooth programming problems

International Journal of ADVANCED AND APPLIED SCIENCES, 2016

In this paper, we introduce a new approach to obtain a novel numerical solution of nonlinear programming problems (NLP) which the objective function (functions) or constraint function (functions) are non-smooth ones. This technique is based on a new piecewise linearization approach. In fact, we transfer the nonlinear programming problem (NLP) to a variational problem that would reduce the new approximated problem to a linear programming problem (LP). Then, the approximated solution of the original problem would be obtained by the LP problem. Finally, numerical examples are given to show the efficiency of the proposed approach.