# Stability proof for nonlinear MPC design using monotonically increasing weighting profiles without terminal constraints

## TL;DR

This paper introduces a new nonlinear MPC formulation that guarantees stability using monotonically increasing weighting profiles without terminal constraints, by adjusting the penalty rate.

## Contribution

It proposes a novel MPC approach with time-varying increasing stage costs and proves stability without terminal constraints under mild assumptions.

## Key findings

- Stability achieved with increasing stage cost profiles.
- Stability guaranteed by sufficiently high penalty increase rate.
- No terminal constraints needed for stability.

## Abstract

In this note, a new formulation of Model Predictive Control (MPC) framework with no stability-related terminal constraint is proposed and its stability is proved under mild standard assumptions. The novelty in the formulation lies in the use of time-varying monotonically increasing stage cost penalty. The main result is that the $0$-reachability prediction horizon can always be made stabilizing provided that the increasing rate of the penalty is made sufficiently high.