Asymptotic controllability and optimal control

TL;DR

This paper investigates conditions under which a control system can be asymptotically stabilized to a target with bounded cost, introducing a novel inequality involving a control Lyapunov function to establish controllability and bounds on the value function.

Contribution

It introduces a new inequality involving a positive parameter and a control Lyapunov function that guarantees asymptotic controllability and bounds the value function without assumptions on the zero level set.

Findings

Control system is asymptotically controllable under the new condition.

The value function is bounded above by a scaled control Lyapunov function.

Provides a sufficient condition for controllability without zero set assumptions.

Abstract

We consider a control problem where the state must reach asymptotically a target while paying an integral payoff with a non-negative Lagrangian. The dynamics is just continuous, and no assumptions are made on the zero level set of the Lagrangian. Through an inequality involving a positive number $\bar p_0$ and a Minimum Restraint Function $U=U(x)$ --a special type of Control Lyapunov Function-- we provide a condition implying that (i) the control system is asymptotically controllable, and (ii) the value function is bounded above by $U/\bar p_0$.