Pareto optimality in the infinite horizon cooperative difference game
This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary co...
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| Published in | IET control theory & applications Vol. 14; no. 3; pp. 386 - 395 |
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| Format | Journal Article |
| Language | English |
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12.02.2020
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| Abstract | This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary conditions for the existence of the Pareto solutions are put forward. Furthermore, two conditions are presented to guarantee that zero does not belong to the Lagrange multiplier set. In addition, it is shown that the necessary conditions are also sufficient under certain convexity assumptions and a transversality condition. Next, the indefinite linear quadratic case is discussed. For a fixed initial state, under the condition of controllability, the necessary conditions are put forward. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional as well as a transversality condition provide the sufficient conditions for a control to be Pareto optimal. For an arbitrary initial state, if the system is stabilisable, then the solvability of the related algebraic Riccati equation provides a sufficient condition under which all Pareto optimal strategies are obtained by the weighted sum minimisation method. |
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| AbstractList | This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary conditions for the existence of the Pareto solutions are put forward. Furthermore, two conditions are presented to guarantee that zero does not belong to the Lagrange multiplier set. In addition, it is shown that the necessary conditions are also sufficient under certain convexity assumptions and a transversality condition. Next, the indefinite linear quadratic case is discussed. For a fixed initial state, under the condition of controllability, the necessary conditions are put forward. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional as well as a transversality condition provide the sufficient conditions for a control to be Pareto optimal. For an arbitrary initial state, if the system is stabilisable, then the solvability of the related algebraic Riccati equation provides a sufficient condition under which all Pareto optimal strategies are obtained by the weighted sum minimisation method. |
| Author | Lin, Yaning |
| Author_xml | – sequence: 1 givenname: Yaning surname: Lin fullname: Lin, Yaning email: yaning_lin@163.com organization: School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, People's Republic of China |
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| CitedBy_id | crossref_primary_10_1016_j_jfranklin_2023_08_006 crossref_primary_10_1016_j_jfranklin_2021_05_013 crossref_primary_10_1049_cth2_12437 |
| Cites_doi | 10.1002/oca.2395 10.1016/j.automatica.2013.03.004 10.1109/TAC.2015.2509958 10.1007/978-1-4614-9038-8 10.1016/j.fss.2015.06.014 10.1016/j.jfranklin.2016.11.021 10.1017/CBO9780511805127 10.1016/j.automatica.2018.04.044 10.1002/asjc.686 10.1109/TNNLS.2017.2652478 10.1109/TAC.2009.2026854 10.1016/j.automatica.2018.03.017 10.1109/TAC.2014.2305933 10.2307/2224098 10.1016/j.automatica.2008.01.022 10.1137/080726227 10.1109/TFUZZ.2014.2312985 10.1016/j.jfranklin.2013.12.019 10.1016/j.jfranklin.2018.04.025 |
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| Copyright | The Institution of Engineering and Technology 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology |
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| Keywords | Pareto optimisation transversality condition infinite horizon cooperative difference game matrix algebra sufficient conditions convexity condition optimisation necessary conditions Riccati equations Pareto optimality Pareto solutions Pareto optimal strategies Lagrange multiplier set minimisation sufficient condition linear quadratic control |
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| SubjectTerms | convexity condition infinite horizon cooperative difference game Lagrange multiplier set linear quadratic control matrix algebra minimisation necessary conditions optimisation Pareto optimal strategies Pareto optimality Pareto optimisation Pareto solutions Research Article Riccati equations sufficient condition sufficient conditions transversality condition |
| Title | Pareto optimality in the infinite horizon cooperative difference game |
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