Controllability and Observability Imply Exponential Decay of Sensitivity in Dynamic Optimization

TL;DR

This paper demonstrates that uniform controllability and observability in dynamic optimization problems lead to exponential decay of sensitivity, which helps in understanding perturbation propagation and developing efficient solution schemes.

Contribution

It establishes that controllability and observability imply exponential decay of sensitivity by linking them to uniform second order conditions and constraint qualifications.

Findings

Exponential decay of sensitivity (EDS) holds under certain boundedness and regularity conditions.

Uniform controllability and observability imply the conditions for EDS.

Numerical examples illustrate the theoretical results.

Abstract

We study a property of dynamic optimization (DO) problems (as those encountered in model predictive control and moving horizon estimation) that is known as exponential decay of sensitivity (EDS). This property indicates that the sensitivity of the solution at stage $i$ against a data perturbation at stage $j$ decays exponentially with $|i-j|$. {Building upon our previous results, we show that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained under uniform controllability and observability. Hence, we have that uniform controllability and observability imply EDS.} These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.