Abstract:
In the framework of control parameterization methods a number of optimization problems of linear phase systems with quadratic
and bilinear functionals is considered. Approximation of the control is carried out in the class of piecewise linear functions
and is formed as a linear combination of a special set of support functions with
coefficients that are variables of the finite-dimensional problem. At the same time the interval
control constraint in the variational problem automatically passes into similar constraints on the variables of the finite-dimensional problem. To characterize and effectively solve these problems, explicit expressions of the selected functionals are obtained with respect to
parameters of approximations. As a result, a series of quadratic mathematical programming problems with the simplest
restrictions on variables is formulated. The quadratic forms of the objective functions are determined by the Gram and Hessenberg matrices. It should be emphasized that the parametrization preserves the convexity property of the original optimal control problem.
In addition, the simplest optimal control problem with a linear terminal functional after parameterization is solved without iterations. The connection between the coordinated problems at the level of optimality conditions is established. It consists in the fact that the differential condition of
the extremum in a finite-dimensional problem is locally equivalent to the maximum principle for the variational problem at the points of set.
Keywords:
linear control system, quadratic and bilinear functionals, piecewise-linear approximation, finite dimensional problems.
Citation:
V. A. Srochko, E. V. Aksenyushkina, “Parameterization of some control problems by linear systems”, Bulletin of Irkutsk State University. Series Mathematics, 30 (2019), 83–98
\Bibitem{SroAks19}
\by V.~A.~Srochko, E.~V.~Aksenyushkina
\paper Parameterization of some control problems by linear systems
\jour Bulletin of Irkutsk State University. Series Mathematics
\yr 2019
\vol 30
\pages 83--98
\mathnet{http://mi.mathnet.ru/iigum397}
\crossref{https://doi.org/10.26516/1997-7670.2019.30.83}
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A. V. Arguchintsev, V. P. Poplevko, “Variatsionnoe uslovie optimalnosti granichnogo upravleniya v sostavnoi modeli lineinykh differentsialnykh uravnenii raznykh tipov”, Differentsialnye uravneniya i optimalnoe upravlenie, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 224, VINITI RAN, M., 2023, 3–9
Elena D. Kotina, Ekaterina B. Leonova, Viktor A. Ploskikh, “Displacement field construction based on a discrete model in image processing problems”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 39 (2022), 3–16
D. A. Ovsyannikov, L. V. Vladimirova, I. D. Rubtsova, A. V. Rubanik, V. A. Ponomarev, “Modifitsirovannyi geneticheskii algoritm poiska globalnogo ekstremuma v sochetanii s napravlennymi metodami”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 39 (2022), 17–33
A. V. Arguchintsev, V. A. Srochko, “Protsedura regulyarizatsii bilineinykh zadach optimalnogo upravleniya na osnove konechnomernoi modeli”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 18:1 (2022), 179–187
A. V. Arguchintsev, V. P. Poplevko, “Variatsionnoe uslovie optimalnosti v zadache upravleniya lineinoi giperbolicheskoi sistemoi pervogo poryadka s zapazdyvaniem na granitse”, Geometriya, mekhanika i differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 212, VINITI RAN, M., 2022, 3–9
D. O. Trunin, “Ob odnom podkhode k optimizatsii lineinykh po sostoyaniyu upravlyaemykh sistem s terminalnymi ogranicheniyami”, Geometriya, mekhanika i differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 213, VINITI RAN, M., 2022, 89–95
V. A. Srochko, E. V. Aksenyushkina, V. G. Antonik, “Reshenie lineino-kvadratichnoi zadachi optimalnogo upravleniya na osnove konechnomernykh modelei”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 37 (2021), 3–16
Alexander Arguchintsev, Vasilisa Poplevko, “An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations”, Games, 12:1 (2021), 23
Alexander Buldaev, Communications in Computer and Information Science, 1476, Mathematical Optimization Theory and Operations Research: Recent Trends, 2021, 463
V. A. Srochko, E. V. Aksenyushkina, “On resolution of an extremum norm problem for the terminal state of a linear system”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 34 (2020), 3–17
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