Lack of controllability of the heat equation with memory

We consider a model for the heat equation with memory, which has infinite propagation speed, like the standard heat equation. We prove that, in spite of this, for every T > 0 there exist square integrable initial data which cannot be steered to hit zero at time T , using square integrable controls. We show that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamic

Approximate controllability and lack of controllability to zero of the heat equation with memory

In this paper we consider the heat equation with memory in a bounded region $\Omega \subset\mathbb{R}^d$, $d\geq 1$, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class $C^1$. We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of $\Omega$. We prove approximate controllability of the system and, in contrast with this, we prove the existence of initial conditions which cannot be steered to hit the target $0$ in a certain time $T$, of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the heat equation

Lack of controllability of thermal systems with memory

Heat equations with memory of Gurtin-Pipkin type have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when the laplacian appears also out of the memory term, the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are $L^2$-initial conditions which cannot be controlled to zero. The proof of this fact (presented in previous papers) consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every systems with smooth memory kernel

On a three-dimensional and two four-dimensional oncolytic viro-therapy models

We revisit here and carry out further works on tumor-virotherapy compartmental models of [Tian, 2011, Wang et al., 2013, Phan and Tian, 2017, Guo et al., 2019]. The results of these papers are only slightly pushed further. However, what is new is the fact that we make public our electronic notebooks, since we believe that easy electronic reproducibility is crucial in an era in which the role of the software becomes very important.Comment: 41 pages, 15 figure

Control of functional differential equations with function space boundary conditions

Problems involving functional differential equations with terminal conditions in function space are considered. Their application to mechanical and electrical systems is discussed. Investigations of controllability, existence of optimal controls, and necessary and sufficient conditions for optimality are reported