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Volume 28, Issue 4
An Efficient Method for Multiobjective Optimal Control and Optimal Control Subject to Integral Constraints

Ajeet Kumar & Alexander Vladimirsky

DOI: 10.4208/jcm.1003-m0015

J. Comp. Math., 28 (2010), pp. 517-551.

Published online: 2010-08

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  • Abstract

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem [25]. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.

  • Keywords

Optimal control, Multiobjective optimization, Pareto front, Vector dynamic programming, Hamilton-Jacobi equation, Discontinuous viscosity solution, Semi-Lagrangian discretization.

  • AMS Subject Headings

90C29, 49L20, 49L25, 58E17, 65N22, 35B37, 65K05.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-517, author = {Ajeet Kumar and Alexander Vladimirsky}, title = {An Efficient Method for Multiobjective Optimal Control and Optimal Control Subject to Integral Constraints}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {4}, pages = {517--551}, abstract = {

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem [25]. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1003-m0015}, url = {http://global-sci.org/intro/article_detail/jcm/8535.html} }
TY - JOUR T1 - An Efficient Method for Multiobjective Optimal Control and Optimal Control Subject to Integral Constraints AU - Ajeet Kumar & Alexander Vladimirsky JO - Journal of Computational Mathematics VL - 4 SP - 517 EP - 551 PY - 2010 DA - 2010/08 SN - 28 DO - http://doi.org/10.4208/jcm.1003-m0015 UR - https://global-sci.org/intro/article_detail/jcm/8535.html KW - Optimal control, Multiobjective optimization, Pareto front, Vector dynamic programming, Hamilton-Jacobi equation, Discontinuous viscosity solution, Semi-Lagrangian discretization. AB -

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem [25]. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.

Ajeet Kumar and Alexander Vladimirsky. (2010). An Efficient Method for Multiobjective Optimal Control and Optimal Control Subject to Integral Constraints. Journal of Computational Mathematics. 28 (4). 517-551. doi:10.4208/jcm.1003-m0015
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