Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization
Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization
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DOI：
10.1137/15m1053347
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发表时间：
2015-12
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通讯作者：
Giovanni Fantuzzi;D. Goluskin;Deqing Huang;Sergei I. Chernyshenko
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文献类型：
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作者：
Giovanni Fantuzzi;D. Goluskin;Deqing Huang;Sergei I. Chernyshenko
我们描述了证明确定性动力系统中无限时间平均值和随机系统中平稳期望的上限和下限的方法。动力学和有界量被假定为状态变量的多项式函数。这些方法是计算机辅助的，使用平方和多项式来制定可以通过半定规划检查的充分条件。在确定性情况下，我们寻求适用于特定局部吸引子的严格界限。证明这种界限的一个障碍是它们在全球范围内不成立。它们通常会被从当地吸引力盆地之外开始的轨迹所违反。我们描述了两种密切相关的方法来克服这个障碍：一种方法需要知道吸引力盆地的子集，另一种方法考虑相应随机系统的零噪声极限。使用范德波尔振荡器来说明边界方法。我们将确定性平均值限制在吸引极限上......
We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit c...