论文标题
非线性动力学的不确定性传播：多项式优化方法
Uncertainty propagation for nonlinear dynamics: A polynomial optimization approach
论文作者
论文摘要
我们使用类似Lyapunov的函数和凸优化来传播不确定性的非线性系统的初始条件，由普通微分方程控制。我们考虑了没有近似的完整非线性动力学，即使知道只有有限的初始条件（例如平均值和方差）的限制统计数据，也会在预期的未来价值上产生严格的界限。对于在紧凑型集合中演变的动力系统，最佳的上部（下部）结合与所有初始状态分布中最大（最小）的期望相吻合，这与已知的统计数据一致。对于由多项式方程和多项式关注量控制的系统，可以使用多项式优化和半决赛编程中的工具来计算最佳界限的单侧估计。此外，这些数值界限可证明在紧凑型情况下会收敛到最佳范围。我们说明了范德尔振荡器和混乱制度中的洛伦兹系统的方法。
We use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations. We consider the full nonlinear dynamics without approximation, producing rigorous bounds on the expected future value of a quantity of interest even when only limited statistics of the initial condition (e.g., mean and variance) are known. For dynamical systems evolving in compact sets, the best upper (lower) bound coincides with the largest (smallest) expectation among all initial state distributions consistent with the known statistics. For systems governed by polynomial equations and polynomial quantities of interest, one-sided estimates on the optimal bounds can be computed using tools from polynomial optimization and semidefinite programming. Moreover, these numerical bounds provably converge to the optimal ones in the compact case. We illustrate the approach on a van der Pol oscillator and on the Lorenz system in the chaotic regime.
