scholarly journals Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations

2021 ◽  
Vol 11 (15) ◽  
pp. 7096
Author(s):  
Askhat Diveev ◽  
Elena Sofronova ◽  
Sergey Konstantinov
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Solution ◽  
Piecewise Linear ◽  
Control Object ◽  
Quality Criterion ◽  
Piecewise Linear Approximation ◽  
Control Synthesis ◽  
Evolutionary Computations

Two approaches to the numerical solution of the optimal control problem are studied. The direct approach is based on the reduction of the optimal control problem to a nonlinear programming problem. Another approach is so-called synthesized optimal control, and it includes the solution of the control synthesis problem and stabilization at some point in the state space, followed by the search of stabilization points and movement of the control object along these points. The comparison of these two approaches was carried out as the solution of the optimal control problem as a time function cannot be directly used in the control system, although the obtained discretized control can be embedded. The control object was a group of interacting mobile robots. Dynamic and static constraints were included in the quality criterion. Implemented methods were evolutionary algorithms and a random parameter search of piecewise linear approximation and coordinates of stabilization points, along with a multilayer network operator for control synthesis.

Numerical algorithms for solving an elliptic optimal control problem with a power-law nonlinearity

2021 ◽  
Vol 26 (jai2021.26(2)) ◽  
pp. 64-76
Author(s):  
Hart L ◽  
◽  
Yatsechko N ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Piecewise Linear ◽  
Boundary Value ◽  
Method Of Lines ◽  
The Method Of Lines ◽  
Gradient Type ◽  
Elliptic Optimal Control Problem ◽  
Elliptic Optimal Control

The paper is devoted to the development and analysis of approximation-iteration algorithms based on the method of grids and the method of lines for solving an elliptic optimal control problem with a power-law nonlinearity. For the numerical solution of the main boundary value problem and the adjoint one, the second order of accuracy difference schemes are applied using the implicit method of simple iteration. Computational schemes of the method of lines for solving the above-mentioned elliptic boundary value problems are implemented in combination with the shooting method for the approximate solution of boundary value problems for the corresponding ordinary differential equations systems arising in the considered domain after lattice approximation. To minimize the objective functional, well-known gradient-type methods (gradient projection and conditional gradient methods) of constrained optimization are used. The essence of the proposed approximation-iteration approach consists in replacing the original extremal problem with a sequence of grid problems that approximate it on a set of refining grids, and applying an iterative gradient-type method to each of the "approximate" extremal problems. In this case, we propose to construct only a few approximations to the solution for each of the "approximate" problems and to take the last of these approximations, using piecewise linear interpolation, as the initial approximation in the iterative process for the next "approximate" problem. The sequence of the corresponding piecewise linear interpolants is considered as a sequence of approximations to the solution of the original extremal problem. The paper discusses the theoretical foundations of this combined approach, as well as its advantages over traditional methods using the example of solving a model optimal control problem

Numerical solution of an optimal control problem for Oskolkov's system

2018 ◽  
Vol 41 (18) ◽  
pp. 9071-9080 ◽  
Author(s):  
Marina V. Plekhanova ◽  
Gusel D. Baybulatova ◽  
Pavel N. Davydov
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Solution

scholarly journals THE OPTIMAL CONTROL PROBLEM FOR ONE-DIMENSIONAL NONLINEAR SHRODINGER EQUATIONS WITH A SPECIAL GRADIENT TERM

2019 ◽  
pp. 49-66
Author(s):  
G. Yagub ◽  
N. S. Ibrahimov ◽  
M. Zengin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Quality Criterion ◽  
Necessary Condition ◽  
Gradient Term ◽  
Complex Coefficient ◽  
Existence And Uniqueness Theorem ◽  
One Dimensional ◽  
Nonlinear Part

In this paper we consider the optimal control problem for a one-dimensional nonlinear Schrodinger equation with a special gradient term and with a complex coefficient in the nonlinear part, when the quality criterion is a final functional and the controls are quadratically summable functions. In this case, the questions of the correctness of the formulation and the necessary condition for solving the optimal control problem under consideration are investigated. The existence and uniqueness theorem for the solution is proved and a necessary condition is established in the form of a variational inequality. Along with these, a formula is found for the gradient of the considered quality criterion.

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