Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

TL;DR

This paper derives a two-term asymptotic expansion for the value function in large-time horizon nonlinear dissipative optimal control problems, revealing the dominant turnpike behavior and stabilization contributions.

Contribution

It introduces a novel two-term asymptotic expansion of the value function for dissipative nonlinear control problems as time approaches infinity.

Findings

The leading term is proportional to the time horizon T and corresponds to the static optimal value.

The second term captures stabilization effects related to the turnpike.

The expansion provides deeper insight into the structure of the value function for large T.

Abstract

Considering a general nonlinear dissipative finite dimensional optimal control problem in fixed time horizon T , we establish a two-term asymptotic expansion of the value function as $T\rightarrow+\infty$. The dominating term is T times the optimal value obtained from the optimal static problem within the classical turnpike theory. The second term, of order unity, is interpreted as the sum of two values associated with optimal stabilization problems related to the turnpike.