Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces
Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e.,...
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2025-07-01
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| author | Oscar Raúl Condori Mamani Bartolome Valero Larico María Luisa Torreblanca Wolfgang Kliemann |
| author_facet | Oscar Raúl Condori Mamani Bartolome Valero Larico María Luisa Torreblanca Wolfgang Kliemann |
| author_sort | Oscar Raúl Condori Mamani |
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| description | Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in <inline-formula data-eusoft-scrollable-element="1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" data-eusoft-scrollable-element="1"><semantics data-eusoft-scrollable-element="1"><msup data-eusoft-scrollable-element="1"><mi mathvariant="double-struck" data-eusoft-scrollable-element="1">R</mi><mi data-eusoft-scrollable-element="1">d</mi></msup></semantics></math></inline-formula>. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system. |
| format | Article |
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| language | English |
| publishDate | 2025-07-01 |
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| spelling | doaj-art-27e032e1efae4fa2903fe4d44d0899ba2025-08-20T03:08:06ZengMDPI AGMathematics2227-73902025-07-011314227310.3390/math13142273Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective SpacesOscar Raúl Condori Mamani0Bartolome Valero Larico1María Luisa Torreblanca2Wolfgang Kliemann3Facultad de Ciencias Naturales y Formales, Universidad Nacional de San Agustín de Arequipa, Arequipa 04001, PeruFacultad de Ciencias Naturales y Formales, Universidad Nacional de San Agustín de Arequipa, Arequipa 04001, PeruFacultad de Ciencias Naturales y Formales, Universidad Nacional de San Agustín de Arequipa, Arequipa 04001, PeruDepartment of Mathematics, Iowa State University, Ames, IA 50011, USABilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in <inline-formula data-eusoft-scrollable-element="1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" data-eusoft-scrollable-element="1"><semantics data-eusoft-scrollable-element="1"><msup data-eusoft-scrollable-element="1"><mi mathvariant="double-struck" data-eusoft-scrollable-element="1">R</mi><mi data-eusoft-scrollable-element="1">d</mi></msup></semantics></math></inline-formula>. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system.https://www.mdpi.com/2227-7390/13/14/2273bilinear control systemsLie theorycontrol setshomogeneous spacesLyapunov exponentsFloquet spectrum |
| spellingShingle | Oscar Raúl Condori Mamani Bartolome Valero Larico María Luisa Torreblanca Wolfgang Kliemann Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces Mathematics bilinear control systems Lie theory control sets homogeneous spaces Lyapunov exponents Floquet spectrum |
| title | Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces |
| title_full | Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces |
| title_fullStr | Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces |
| title_full_unstemmed | Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces |
| title_short | Controllability of Bilinear Systems: Lie Theory Approach and Control Sets on Projective Spaces |
| title_sort | controllability of bilinear systems lie theory approach and control sets on projective spaces |
| topic | bilinear control systems Lie theory control sets homogeneous spaces Lyapunov exponents Floquet spectrum |
| url | https://www.mdpi.com/2227-7390/13/14/2273 |
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