Numerical solution of the variational PDEs arising in optimal control theory
An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.
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Sociedade Brasileira de Matemática Aplicada e Computacional
2012
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oai:scielo:S1807-030220120001000032012-04-26Numerical solution of the variational PDEs arising in optimal control theoryCostanza,VicenteTroparevsky,Maria I.Rivadeneira,Pablo S. numerical methods first-order PDEs nonlinear systems optimal control Hamiltonian equations boundary-value problems An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.info:eu-repo/semantics/openAccessSociedade Brasileira de Matemática Aplicada e ComputacionalComputational & Applied Mathematics v.31 n.1 20122012-01-01info:eu-repo/semantics/articletext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003en10.1590/S1807-03022012000100003 |
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Costanza,Vicente Troparevsky,Maria I. Rivadeneira,Pablo S. |
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Costanza,Vicente Troparevsky,Maria I. Rivadeneira,Pablo S. Numerical solution of the variational PDEs arising in optimal control theory |
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Costanza,Vicente Troparevsky,Maria I. Rivadeneira,Pablo S. |
| author_sort |
Costanza,Vicente |
| title |
Numerical solution of the variational PDEs arising in optimal control theory |
| title_short |
Numerical solution of the variational PDEs arising in optimal control theory |
| title_full |
Numerical solution of the variational PDEs arising in optimal control theory |
| title_fullStr |
Numerical solution of the variational PDEs arising in optimal control theory |
| title_full_unstemmed |
Numerical solution of the variational PDEs arising in optimal control theory |
| title_sort |
numerical solution of the variational pdes arising in optimal control theory |
| description |
An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10. |
| publisher |
Sociedade Brasileira de Matemática Aplicada e Computacional |
| publishDate |
2012 |
| url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022012000100003 |
| work_keys_str_mv |
AT costanzavicente numericalsolutionofthevariationalpdesarisinginoptimalcontroltheory AT troparevskymariai numericalsolutionofthevariationalpdesarisinginoptimalcontroltheory AT rivadeneirapablos numericalsolutionofthevariationalpdesarisinginoptimalcontroltheory |
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1756431740471083008 |