Control the Coefficient of a Differential Equation as an Inverse Problem in Time
There are many problems based on solving nonautonomous differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo&g...
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MDPI AG
2024-01-01
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/329 |
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| author | Vladimir Ternovski Victor Ilyutko |
| author_facet | Vladimir Ternovski Victor Ilyutko |
| author_sort | Vladimir Ternovski |
| collection | DOAJ |
| description | There are many problems based on solving nonautonomous differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> represents the coordinate of a material point and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> is the angular frequency. The inverse problem involves finding the bounded coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>. Continuity of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is not required. The trajectory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is also unknown, but the initial and final values of the phase variables are given. The variation principle of the minimum time for the entire dynamic process allows for the determination of the optimal solution. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made. |
| first_indexed | 2024-03-08T10:41:53Z |
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| id | doaj.art-fe59d17b33c54043b06d9ff2528fb989 |
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| last_indexed | 2024-03-08T10:41:53Z |
| publishDate | 2024-01-01 |
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| spelling | doaj.art-fe59d17b33c54043b06d9ff2528fb9892024-01-26T17:33:47ZengMDPI AGMathematics2227-73902024-01-0112232910.3390/math12020329Control the Coefficient of a Differential Equation as an Inverse Problem in TimeVladimir Ternovski0Victor Ilyutko1Department of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, International University Park Road 1, Shenzhen 518172, ChinaDepartment of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, International University Park Road 1, Shenzhen 518172, ChinaThere are many problems based on solving nonautonomous differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> represents the coordinate of a material point and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> is the angular frequency. The inverse problem involves finding the bounded coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>. Continuity of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is not required. The trajectory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is also unknown, but the initial and final values of the phase variables are given. The variation principle of the minimum time for the entire dynamic process allows for the determination of the optimal solution. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made.https://www.mdpi.com/2227-7390/12/2/329optimal controlreachability setinverse problem |
| spellingShingle | Vladimir Ternovski Victor Ilyutko Control the Coefficient of a Differential Equation as an Inverse Problem in Time Mathematics optimal control reachability set inverse problem |
| title | Control the Coefficient of a Differential Equation as an Inverse Problem in Time |
| title_full | Control the Coefficient of a Differential Equation as an Inverse Problem in Time |
| title_fullStr | Control the Coefficient of a Differential Equation as an Inverse Problem in Time |
| title_full_unstemmed | Control the Coefficient of a Differential Equation as an Inverse Problem in Time |
| title_short | Control the Coefficient of a Differential Equation as an Inverse Problem in Time |
| title_sort | control the coefficient of a differential equation as an inverse problem in time |
| topic | optimal control reachability set inverse problem |
| url | https://www.mdpi.com/2227-7390/12/2/329 |
| work_keys_str_mv | AT vladimirternovski controlthecoefficientofadifferentialequationasaninverseproblemintime AT victorilyutko controlthecoefficientofadifferentialequationasaninverseproblemintime |