Exponential Decay in the Sensitivity Analysis of Nonlinear Dynamic Programming

TL;DR

This paper demonstrates that the sensitivity of solutions in nonlinear dynamic programming decays exponentially with respect to the temporal distance between initial perturbations and the point of evaluation, under certain controllability assumptions.

Contribution

It establishes an exponential decay rate for the sensitivity of optimal states and controls in nonlinear dynamic programs, extending previous linear results to nonlinear settings.

Findings

Sensitivity decays exponentially with |k-i|

Decay rate is independent of horizon length

Validation through numerical experiments

Abstract

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 $k$, $\boldsymbol{x}^*_{k}$ and $\boldsymbol{u}^*_{k}$, with respect to the reference signal at time $i$, $\boldsymbol{d}_{i}$, 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.