Lie algebra structure of fitness and replicator control
Lie algebra structure of fitness and replicator control
复制标题
适应度和复制控制的李代数结构
DOI：
--
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
P. Krishnaprasad
中科院分区：
文献类型：
--
作者：
Vidya Raju;P. Krishnaprasad
50多年来，通过博弈论的透镜，动力系统的观点在进化生物学和经济学中发挥了突出的作用。特别是，研究复制微分方程的标准（概率）单纯形，指定的健身地图或支付功能，产生了深刻的见解，这种系统的时间行为。然而，行为受到背景和环境因素的影响，具有改变游戏规则的性质（即，操纵适应度图）。本文开发了一个原则性的几何方法来建模和理解这样的影响，将复制动力学到一个更广泛的控制理论框架。我们的方法的核心是建设一个李代数结构的健身地图的空间，映射同态李代数的复制向量场。这类似于经典力学，其中函数的泊松括号李代数映射到相关的哈密顿向量场。我们表明，延长Svirezhev在1972年的工作，复制向量场的轨迹是一个解决方案的基础积分曲线定义的余切丛的单纯形的哈密顿系统。此外，我们利用李代数结构的适应度映射，以确定一类复制系统的可控性。
For over fifty years, the dynamical systems perspective has had a prominent role in evolutionary biology and economics, through the lens of game theory. In particular, the study of replicator differential equations on the standard (probability) simplex, specified by fitness maps or payoff functions, has yielded insights into the temporal behavior of such systems. However behavior is influenced by context and environmental factors with a game-changing quality (i.e., fitness maps are manipulated). This paper develops a principled geometric approach to model and understand such influences by incorporating replicator dynamics into a broader control-theoretic framework. Central to our approach is the construction of a Lie algebra structure on the space of fitness maps, mapping homomorphically to the Lie algebra of replicator vector fields. This is akin to classical mechanics, where the Poisson bracket Lie algebra of functions maps to associated Hamiltonian vector fields. We show, extending the work of Svirezhev in 1972, that a trajectory of a replicator vector field is the base integral curve of a solution to a Hamiltonian system defined on the cotangent bundle of the simplex. Further, we exploit the Lie algebraic structure of fitness maps to determine controllability properties of a class of replicator systems.
DOI：
10.1109/cdc.2018.8619147
发表时间：
2018
期刊：
2018 IEEE Conference on Decision and Control (CDC
影响因子：
--
作者：
Raju, Vidya;Krishnaprasad, P. S.
通讯作者：
Krishnaprasad, P. S.