Delay and Memory-Type Null Controllability for Heat Equations in Finite Dimensions

TL;DR

This paper studies null controllability for finite-dimensional linear heat-type systems that incorporate memory and time-delay effects. It introduces delay and memory-type null controllability and proves an exact duality with an augmented observability inequality for the adjoint system, capturing past, present, and terminal data. In the finite-dimensional setting, it derives sharp necessary and sufficient algebraic rank conditions that extend Kalman’s criterion to systems with memory and delay. The results explain why achieving $y(T)=0$ is insufficient without also extinguishing the memory integral and the state over the delay interval $[T-h,T)$, informing robust controller design for viscoelastic and delayed-feedback applications.

Abstract

We study null controllability for linear heat-type systems in finite dimensions that incorporate both memory and time-delay effects. A strengthened notion of controllability, referred to as delay and memory-type null controllability, is introduced, which requires the state, the memory functional, and the delayed history to vanish at the terminal time. Using a duality approach, we establish an augmented observability inequality for the adjoint system and show its equivalence to controllability. In the finite-dimensional setting, this leads to sharp necessary and sufficient algebraic rank conditions extending the classical Kalman criterion to systems with memory and delay.

Theorems & Definitions (1)

• Definition 1.1: Delay and Memory-Type Null Controllability