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

TL;DR

This paper investigates a new form of controllability for finite-dimensional heat systems with memory and delay effects, establishing algebraic conditions extending classical criteria.

Contribution

It introduces delay and memory-type null controllability and derives sharp algebraic rank conditions extending Kalman's criterion for such systems.

Findings

Established an augmented observability inequality.

Proved equivalence between observability and controllability.

Extended Kalman criterion to systems with memory and delay.

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.