# American Institute of Mathematical Sciences

April  2022, 9(2): 165-189. doi: 10.3934/jdg.2022002

## Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games

 1 Applied Research Laboratory, Penn State University, University Park, PA 16802, USA 2 Naval Postgraduate School, Monterey, CA 93940, USA

* Corresponding author: James Fan

Received  February 2019 Revised  November 2020 Published  April 2022 Early access  February 2022

Fund Project: CG is supported by NSF grant CMMI-1463482 and CMMI-1932991

We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than 1 and in which the payoff to the winner is controlled by a control variable $\gamma$. Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function.

Citation: Christopher Griffin, James Fan. Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games. Journal of Dynamics and Games, 2022, 9 (2) : 165-189. doi: 10.3934/jdg.2022002
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The phase portrait of the first order system representing the limiting behavior of the open loop control $\gamma$ and it's first derivative
The control function, its approximations and the $\left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2$, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 3 strategy cyclic game (rock-paper-scissors)
The control function, its approximations and the $\left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2$, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 5 strategy cyclic game (rock-paper-scissors-Spock-lizard)
The control function, its approximations and the $\left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2$, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 7 strategy cyclic game
The control function, its approximations and the $\left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2$, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 9 strategy cyclic game
A solution that violates the sufficient condition for optimality $r > {\bf{H}}^T\mathit{\boldsymbol{\lambda}}$ but can be shown to be optimal by appealing to the matrix Riccatti equation, which has bounded solutions for $t \in [0,t_f]$. We compare the solution to the ordinary controller derived from the ordinary Euler-Lagrange equations and the controller that is directly computed using Lemma 4.2. Note they are identical as expected
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