# Prescribed-time convergence with input constraints: A control Lyapunov function based approach

## TL;DR

This paper introduces a control Lyapunov function-based framework for ensuring prescribed-time convergence of control-affine nonlinear systems under input and spatiotemporal constraints, using a time transformation technique and a novel parameter to manage convergence rate.

## Contribution

The paper proposes a new control approach that guarantees prescribed-time convergence with input constraints by transforming the system and characterizing initial condition sets.

## Key findings

- Successfully achieves prescribed-time convergence under input constraints.
- Characterizes initial conditions for guaranteed convergence.
- Demonstrates effectiveness through a numerical example.

## Abstract

In this paper, we present a control framework for a general class of control-affine nonlinear systems under spatiotemporal and input constraints. Specifically, the proposed control architecture addresses the problem of reaching a given final set $S$ in a prescribed (user-defined) time with bounded control inputs. To this end, a time transformation technique is utilized to transform the system subject to temporal constraints into an equivalent form without temporal constraints. The transformation is defined so that asymptotic convergence in the transformed time scale results into prescribed-time convergence in the original time scale. To incorporate input constraints, we characterize a set of initial conditions $D_M$ such that starting from this set, the closed-loop trajectories reach the set $S$ within the prescribed time. We further show that starting from outside the set $D_M$, the system trajectories reach the set $D_M$ in a finite time that depends upon the initial conditions and the control input bounds. We use a novel parameter $\mu$ in the controller, that controls the convergence-rate of the closed-loop trajectories and dictates the size of the set $D_M$. Finally, we present a numerical example to showcase the efficacy of our proposed method.