SIAM Journal on Optimization, Volume 30, Issue 2, Page 1527-1554, January 2020.
In this paper, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time $k$, $x^*_k$, and $u^*_k$, with respect to the reference signal at time $i$, $d_i$, will have exponential decay in terms of $|k-i|$ with a decay rate $\rho$ independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al. can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.

中文翻译：

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非线性动态规划敏感性分析中的指数衰减

SIAM优化杂志，第30卷，第2期，第1527-1554页，2020年1月。
在本文中，我们研究了具有非线性动力学和目标的离散时间动态程序对初始条件和参考参数摄动的敏感性。在问题数据的统一可控性和有界性假设下，我们证明了相对于参考信号，在时间$ k $，$ x ^ * _ k $和$ u ^ * _ k $时最优状态和控制的方向导数在时间$ i $，$ d_i $，将具有以$ | ki | $表示的指数衰减，其衰减率$ \ rho $与时间范围无关。关键的技术步骤是证明Verschueren等人提出的凸化方法的一种版本。可以将其应用于KKT条件，并得出具有均匀边界数据的凸二次程序。反过来，可以进一步使用Riccati技术获得灵敏度结果，从观察中可以看出，方向导数是结构与KKT条件相似的二次程序的解。我们通过在小型非线性，非凸动态程序上的数值实验验证了我们的发现。