Volume 11, no. 4Pages 55 - 66

About One Approach to Numerical Solution of Nonlinear Optimal Speed Problems

A.S. Buldaev, I.D. Burlakov
Optimal speed problems are among the most important problems of the theory of controlled systems. In the qualitative theory of nonlinear speed problems one of the main results is the Pontryagin maximum principle. For the numerical solution of nonlinear speed problems, along with methods based on the maximum principle, methods of reducing to auxiliary problems of optimal control using linearization, parameterization, discretization, and other techniques are widely used. The complexity of numerical methods is determined by the number of iterations to find a solution to the speed problem with a given accuracy. A universal computational procedure that is effective for calculating a variety of speed problems does not currently exist. Therefore, it is actual to develop special approaches to reduce the amount of calculations and reduce the number of iterations. The paper proposes a new approach based on the reduction of a nonlinear speed problem to an auxiliary optimization problem with mixed control functions and parameters. To search for a solution to the emerging auxiliary problem, a specially developed form of conditions for nonlocal improvement of admissible control in the form of a fixed-point problem of the control operator, and a constructed iterative algorithm for successive improvement of admissible controls are used. Approbation and comparative analysis of the computational efficiency of the proposed fixed point approach is carried out on known models of optimal speed problems.
Full text
optimal speed problem; conditions for improving control; fixed point problem.
1. Tyatyushkin A.I. Mnogometodnaya tehnologiya optimizacii upravljaemyh sistem [Multi-Method Optimization of Controllable Systems]. Novosibirsk, Nauka, 2006. (in Russian)
2. Gornov A.Yu. Vychislitel'nye tehnologii resheniya zadach optimal'nogo upravleniya [Computational Technologies for Solving Optimal Control Problems]. Novosibirsk, Nauka, 2009. (in Russian)
3. Srochko V.A. Iteracionnye metody resheniya zadach optimalnogo upravleniya [Iterative Methods for Solving Optimal Control Problems]. Moscow, Fizmatlit, 2000. (in Russian)
4. Buldaev A.S., Khishektueva I.-Kh. The Fixed Point Method in Parametric Optimization Problems for Systems. Automation and Remote Control, 2013, vol. 74, no. 12, pp. 1927-1934.
5. Buldaev A.S., Daneev A.V. New Approaches to Optimization of Parameters of Dynamic Systems on the Basis of Problems about Fixed Points. Far East Journal of Mathematical Sciences, 2016, vol. 99, no. 3, pp. 439-454.
6. Buldaev A.S., Khishketueva I.-Kh. Fixed Point Methods in Problems of Optimization of Nonlinear Systems by Control Functions and Parameters. The Bulletin of Irkutsk State University, Series: Mathematics, 2017, vol. 19, pp. 89-104. (in Russian) DOI: 10.26516/1997-7670.2017.19.89
7. Buldaev A.S. Methods of Fixed Points on the Basis of Design Operations in Optimization Problems of Control Functions and Parameters of Dynamical Systems. The Buryat State University Bulletin, Series: Mathematics, Informatics, 2017, vol. 1, pp. 38-54. (in Russian) DOI: 10.18101/2304-5728-2017-1-38-54