Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality
In the paper we consider nonlinear systems depending linearly on control with one-dimensional output. As is well known, under the analyticity requirement, the output can be expressed as a series of iterated integrals of controls with scalar coefficients. Since iterated integrals generate a free asso...
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| Format: | Article |
| Language: | English |
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Oles Honchar Dnipro National University
2023-11-01
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| Series: | Journal of Optimization, Differential Equations and Their Applications |
| Subjects: | |
| Online Access: | https://model-dnu.dp.ua/index.php/SM/article/view/185 |
| _version_ | 1827397629804281856 |
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| author | Daria M. Andreieva Svetlana Yu. Ignatovich |
| author_facet | Daria M. Andreieva Svetlana Yu. Ignatovich |
| author_sort | Daria M. Andreieva |
| collection | DOAJ |
| description | In the paper we consider nonlinear systems depending linearly on control with one-dimensional output. As is well known, under the analyticity requirement, the output can be expressed as a series of iterated integrals of controls with scalar coefficients. Since iterated integrals generate a free associative algebra, algebraic and combinatorial tools can be applied. Developing ideas of the algebraic approach to homogeneous approximation of nonlinear control systems, we propose the definition of a homogeneous approximation for series of iterated integrals with one-dimensional coefficients and study its algebraic properties. In addition, we describe relations between the homogeneous approximation of one-dimensional series of iterated integrals and an approximation in the sense of time optimality. Namely, we give conditions under which the optimal time and optimal controls for the minimal realization of the initial series are approximated by the optimal time and optimal controls for the minimal realization of its homogeneous approximation in a neighborhood of the origin. |
| first_indexed | 2024-03-08T19:12:19Z |
| format | Article |
| id | doaj.art-1dd922f257724431a32b431583ec4afb |
| institution | Directory Open Access Journal |
| issn | 2617-0108 2663-6824 |
| language | English |
| last_indexed | 2024-03-08T19:12:19Z |
| publishDate | 2023-11-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Journal of Optimization, Differential Equations and Their Applications |
| spelling | doaj.art-1dd922f257724431a32b431583ec4afb2023-12-27T09:19:58ZengOles Honchar Dnipro National UniversityJournal of Optimization, Differential Equations and Their Applications2617-01082663-68242023-11-0131212310.15421/142308177Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time OptimalityDaria M. Andreieva0Svetlana Yu. Ignatovich1Department of Applied Mathematics, V. N. Karazin Kharkiv National UniversityDepartment of Applied Mathematics, V. N. Karazin Kharkiv National UniversityIn the paper we consider nonlinear systems depending linearly on control with one-dimensional output. As is well known, under the analyticity requirement, the output can be expressed as a series of iterated integrals of controls with scalar coefficients. Since iterated integrals generate a free associative algebra, algebraic and combinatorial tools can be applied. Developing ideas of the algebraic approach to homogeneous approximation of nonlinear control systems, we propose the definition of a homogeneous approximation for series of iterated integrals with one-dimensional coefficients and study its algebraic properties. In addition, we describe relations between the homogeneous approximation of one-dimensional series of iterated integrals and an approximation in the sense of time optimality. Namely, we give conditions under which the optimal time and optimal controls for the minimal realization of the initial series are approximated by the optimal time and optimal controls for the minimal realization of its homogeneous approximation in a neighborhood of the origin.https://model-dnu.dp.ua/index.php/SM/article/view/185nonlinear control systemseries of iterated integralsfree associative algebracore lie subalgebrahomogeneous approximationtime-optimal control problem |
| spellingShingle | Daria M. Andreieva Svetlana Yu. Ignatovich Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality Journal of Optimization, Differential Equations and Their Applications nonlinear control system series of iterated integrals free associative algebra core lie subalgebra homogeneous approximation time-optimal control problem |
| title | Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality |
| title_full | Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality |
| title_fullStr | Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality |
| title_full_unstemmed | Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality |
| title_short | Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality |
| title_sort | homogeneous approximation of one dimensional series of iterated integrals and time optimality |
| topic | nonlinear control system series of iterated integrals free associative algebra core lie subalgebra homogeneous approximation time-optimal control problem |
| url | https://model-dnu.dp.ua/index.php/SM/article/view/185 |
| work_keys_str_mv | AT dariamandreieva homogeneousapproximationofonedimensionalseriesofiteratedintegralsandtimeoptimality AT svetlanayuignatovich homogeneousapproximationofonedimensionalseriesofiteratedintegralsandtimeoptimality |