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A Note on an Upper-Bound for the Sum of a Class K and an Extended Class K Function

Adrian Wiltz, Dimos V. Dimarogonas

Abstract

In this short note, we derive an upper-bound for the sum of two comparison functions, namely for the sum of a class K and an extended class K function. To the best of our knowledge, the relations derived in this note have not been previously derived in the literature.

A Note on an Upper-Bound for the Sum of a Class K and an Extended Class K Function

Abstract

In this short note, we derive an upper-bound for the sum of two comparison functions, namely for the sum of a class K and an extended class K function. To the best of our knowledge, the relations derived in this note have not been previously derived in the literature.
Paper Structure (2 theorems, 19 equations)

This paper contains 2 theorems, 19 equations.

Key Result

Lemma 1

Let $\alpha_{1}: \mathbb{R} \rightarrow \mathbb{R}$ be an extended class $\mathcal{K}_{e}$ function, and $\alpha_{2}: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ a convex class $\mathcal{K}$ function such that $\alpha_{1}(-x)\leq -\alpha_{2}(x)$ for all $x\in[0,A]$ and some finite $A>0$. Then, there

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • proof : Proof of Lemma \ref{['lemma:time-varying CBF without input constraints 1']}
  • proof : Proof of Lemma \ref{['lemma:time-varying CBF without input constraints 2']}