When does stabilizability imply the existence of infinite horizon optimal control in nonlinear systems?

TL;DR

This paper investigates conditions under which stabilizability guarantees the existence of infinite horizon optimal controls in nonlinear systems, linking stability properties with control existence and manifold estimates.

Contribution

It identifies classes of nonlinear systems where infinite horizon optimal controls exist based on stability, stabilizability, detectability, and growth conditions.

Findings

Existence regions for stable manifolds are estimated.

Conditions linking stabilizability to infinite horizon control are established.

Applications include analysis of turnpike properties in nonlinear control.

Abstract

The paper addresses an existence problem for infinite horizon optimal control when the system under control is exponentially stabilizable or stable. Classes of nonlinear control systems for which infinite horizon optimal controls exist are identified in terms of stability, stabilizability, detectability and growth conditions. The result then applies to estimate the existence region of stable manifolds in the associated Hamiltonian systems. Applications of the results also include the analysis for turnpike property in nonlinear finite horizon optimal control problems by a geometric approach.