One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution
We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop o...
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MDPI AG
2024-11-01
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| author | Valery Y. Glizer Vladimir Turetsky |
| author_facet | Valery Y. Glizer Vladimir Turetsky |
| author_sort | Valery Y. Glizer |
| collection | DOAJ |
| description | We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players’ controls of this game. Using the game’s solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader’s control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior of the open-loop optimal players’ controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players’ controls are designed. An illustrative example is presented. |
| format | Article |
| id | doaj-art-1a350a1190eb4176aed3e15e2b1631d3 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-1a350a1190eb4176aed3e15e2b1631d32024-11-26T17:50:58ZengMDPI AGAxioms2075-16802024-11-01131180110.3390/axioms13110801One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop SolutionValery Y. Glizer0Vladimir Turetsky1The Galilee Research Center for Applied Mathematics, Braude College of Engineering, Karmiel 2161002, IsraelDepartment of Mathematics, Braude College of Engineering, Karmiel 2161002, IsraelWe consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players’ controls of this game. Using the game’s solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader’s control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior of the open-loop optimal players’ controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players’ controls are designed. An illustrative example is presented.https://www.mdpi.com/2075-1680/13/11/801Stackelberg linear–quadratic differential gamecheap control gamesingularly perturbed boundary-value problemasymptotic analysisopen-loop suboptimal players’ controls |
| spellingShingle | Valery Y. Glizer Vladimir Turetsky One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution Axioms Stackelberg linear–quadratic differential game cheap control game singularly perturbed boundary-value problem asymptotic analysis open-loop suboptimal players’ controls |
| title | One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution |
| title_full | One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution |
| title_fullStr | One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution |
| title_full_unstemmed | One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution |
| title_short | One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution |
| title_sort | one class of stackelberg linear quadratic differential games with cheap control of a leader asymptotic analysis of an open loop solution |
| topic | Stackelberg linear–quadratic differential game cheap control game singularly perturbed boundary-value problem asymptotic analysis open-loop suboptimal players’ controls |
| url | https://www.mdpi.com/2075-1680/13/11/801 |
| work_keys_str_mv | AT valeryyglizer oneclassofstackelberglinearquadraticdifferentialgameswithcheapcontrolofaleaderasymptoticanalysisofanopenloopsolution AT vladimirturetsky oneclassofstackelberglinearquadraticdifferentialgameswithcheapcontrolofaleaderasymptoticanalysisofanopenloopsolution |