рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2022. Vol 41

Parametric Regularization of a Linear-quadratic Problem on a Set of Piecewise Linear Controls

Author(s)
Vladimir A. Srochko1, Elena V. Aksenyushkina2

1Irkutsk State University, Irkutsk, Russian Federation

2Baikal State University, Irkutsk, Russian Federation

Abstract
A linear-quadratic problem with arbitrary matrices in the functional and multidimensional control with convex constraint is considered. Acceptable controls are piecewise linear vector functions within an uneven grid of possible corner points. The reduction of the optimal control problem into a finite-dimensional format is carried out using vector formalization of the linear spline construction and block matrices together with the corresponding operations. The possibility of influencing the functional in the original problem is provided by using parameters with quadratic forms. The choice of these parameters is focused on the regularization of the functional in the sense of providing it with the properties of convexity or concavity at the level of a finite-dimensional model. The conditions for the choice of parameters are in the nature of inequalities with respect to the extreme eigenvalues of the block matrices forming the objective function. The corresponding convex or concave optimization problems can be solved in a finite number of iterations.
About the Authors

Vladimir A. Srochko, Dr. Sci. (Phys.–Math.), Prof., Irkutsk State University, Irkutsk, 664003, Russian Federation, srochko@math.isu.ru

Elena V. Aksenyushkina, Cand. Sci. (Phys.Math.), Assoc. Prof., Baikal State University, Irkutsk, 664003, Russian Federation, aks.ev@mail.ru

For citation
Srochko V. A., Aksenyushkina E. V. Parametric Regularization of a Linearquadratic Problem on a Set of Piecewise Linear Controls. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 41, pp. 57–68. (in Russian) https://doi.org/10.26516/1997-7670.2022.41.57
Keywords
linear-quadratic problem, multidimensional piecewise linear control, functional with parameters, regularization of the problem
UDC
517.977
MSC
49J15, 49M25
DOI
https://doi.org/10.26516/1997-7670.2022.41.57
References
  1. Arguchintsev A.V., Srochko V.A. Regularization procedure for bilinear optimal control problems based on a finite-dimensional model.Bulletin of St. Petersburg University. Applied mathematics. Computer science. Management processes, 2022, vol. 18, no. 1, pp. 179–187. https://doi.org/10.21638/11701/spbu10.2022.115 (in Russian)
  2. Vasiliev F.P. Optimization methods. Moscow, ICNMO Publ., 2011, 620 p. (in Russian)
  3. Gabasov R., Kirillova F.M., Pavlenok N.S. Constructing software and positional solutions to a linear-quadratic optimal control problem. Journal. calculation. matem. and checkmate. physics, 2008, vol. 48, no. 10, pp. 1748-1779. (in Russian)
  4. Izmailov A.F., Solodov M.V. Numerical optimization methods. Moscow, Fizmatlit Publ., 2005, 304 p. (in Russian)
  5. Parlett B. Symmetric eigenvalue problem. Numerical methods. Moscow, Mir Publ., 1983, 384 p. (in Russian)
  6. Srochko V.A. Iterative methods for solving optimal control problems. Moscow, Fizmatlit Publ., 2000. 160 p. (in Russian)
  7. Srochko V.A., Aksenyushkina E.V., Antonik V.G. Solution of linear-quadratic optimal control problem based on finite-dimensional models. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 33, pp. 3–16. https://doi.org/10.26516/1997-7670.2021.37.3 (in Russian)
  8. Strekalovsky A.S. Elements of nonconvex optimization. Novosibirsk, Nauka Publ., 2003, 356 p. (in Russian)
  9. Horn R., Johnson Ch. Matrix analysis. Moscow, Mir Publ., 1989, 653 p. (in Russian)

Full text (russian)