Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

TL;DR

This paper establishes final state observability estimates for bi-continuous semigroups on Banach spaces, providing criteria based on uncertainty and dissipation, with applications to specific semigroups and a novel duality result.

Contribution

It introduces new observability estimates for bi-continuous semigroups and extends the duality between controllability and observability to locally convex spaces.

Findings

Final state observability estimates for Gauss-Weierstrass and Ornstein-Uhlenbeck semigroups.

A sufficient criterion based on uncertainty relation and dissipation estimate.

Extension of duality between controllability and observability to locally convex spaces.

Abstract

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time $T>0$ by taking into account the orbit of the initial value under the semigroup for $t\in [0,T]$, measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauss-Weierstrass semigroup and the Ornstein-Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the Banach spaces case.