临界附近对称破缺动力学的最优控制
Optimal control of symmetry-breaking dynamics near criticality
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中文总结 AI 辅助
研究叉形分岔附近动力系统最优控制问题，借助庞特里亚金极大值原理渐近展开，在三种动力状态下研究最优控制律，推导广义振幅方程，经数值解验证，分析分岔结构并构建渐近解。
中文摘要 AI 辅助
我们研究叉形分岔附近动力系统的最优控制问题，其受外部线索在细胞命运选择和其他自然过程中引导对称破缺转变的作用所驱动。利用从庞特里亚金极大值原理获得的最优性条件的渐近展开，在由控制强度相对于临界距离的缩放所区分的三种动力状态下，研究了一般n维系统的主导阶最优控制律。在强控制极限下结果简化为线性二次控制的已知近似，而在弱控制和中间控制状态下，我们推导了描述优化轨迹的状态和协态变量共同演化的广义振幅方程。这些控制范式通过双稳生化开关的典型模型的完整最优控制问题的数值解得到验证。在弱控制状态下分析了最优控制问题的分岔结构。最后，我们展示了在该状态下使用边界层方法在长时间极限中构建渐近解。
英文摘要
We study the problem of optimal control for dynamical systems near a pitchfork bifurcation, motivated by the role of external cues in guiding symmetry-breaking transitions in cell-fate selection and other natural processes. Using an asymptotic expansion of the optimality conditions obtained from the Pontryagin maximum principle, the leading-order optimal control law for a general n-dimensional system is examined across three dynamical regimes distinguished by scaling of control strength with respect to the distance from criticality. While in the strong control limit the results reduce to known approximations from linear-quadratic control, we derive generalized amplitude equations for the co-evolution of state and costate variables describing the optimized trajectory in the weak and intermediate control regimes. These control normal forms are validated against numerical solutions of the full optimal control problem for a canonical model of a bistable biochemical switch. The bifurcation structure of the optimal control problem is analyzed in the weak control regime. Finally, we demonstrate the construction of asymptotic solutions in the long time limit in this regime using boundary-layer methods.
