Regional Analysis of Time-Fractional Diffusion Processes

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Introduction

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   In this chapter, we first introduce the concepts of CPSs, DPSs, and their new challenges. Then, based on the theory of CTRW and its approach to anomalous diffusion process, the advantages of using fractional order models to characterize the CTRW’s guide to anomalous diffusion are presented.

3. Chapter 2. Preliminary Results

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   This chapter aims to introduce some preliminary results to be used thereafter. To this end, some special functions and their properties are first worked out and we then state the basic knowledge of fractional calculus and semigroups At last, we present several useful lemmas, which are needed throughout this book.

4. Chapter 3. Regional Controllability

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   Generally speaking, to control a system is to make it behave according to our wishes at a least possible cost, in a way which is compatible with safety, regulations, and ethics. However, in many practical applications, it is not necessary to control all states due to the big cost and even not all the states could be reached; then the regional controllability concept is welcomed Afifi et al, Intern J Syst Sci 33(1):1–12, (2002), [1], El Jai, Pritchard, Sensors and controls in the analysis of distributed systems. Halsted Press, New York, (1988), [6], El Jai et al, Intern J Control 62(6):1351–1365, (1995), [7], i.e., the studied system is only exactly (or approximately) controllable on a subset of the whole space. For example, in an industrial furnace, people usually concern a control to guarantee that the temperature of the furnace is maintained at a certain subregion level. Then, the regional results here can be regarded as an extension of the existence contributions.

5. Chapter 4. Regional Observability

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   In the previous chapter, the regional controllability, regional gradient controllability and regional boundary controllability for Riemann–Liouville/Caputo type time fractional diffusion systems are displayed. Moreover, as we all know, the concept of observability of integer order diffusion systems is dual to that of controllability.

6. Chapter 5. Regional Detection of Unknown Sources

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   Simply because those natural disasters over recent years and their permanent risks, recently the environmental problems, and particularly the pollution problems have drawn increasing attention (Witherington and Martin in Understanding, assessing, and resolving light-pollution problems on sea turtle nesting beaches. Florida Marine Research Institute Technical Report, TR-2, 2000 [9], Ge et al. in 55th conference on decision and control, Las Vegas, USA, 2016 [7]). Moreover, we note that since the pollution problems are essentially due to the chemical or nuclear industries, etc., and one is worried about their expansion, the danger of pollution maybe increase when the pollution source remains unknown. As cited in El Jai and Afifi (Int J Appl Math Comput Sci 26(8):1447–1463, 1995) [4], according to its nature (pointwise, zone, boundary, fixed, or moving), intensity, and life duration, a source can be characterized by several parameters. So the source can be regarded as an unknown control which is observed and detected via sensors. In this chapter, we shall study the problem of regional detection of unknown sources for the Riemann–Liouville-type and Caputo-type time fractional diffusion systems, which are considered as extensions of previous work (see El Jai and Afifi in Int J Appl Math Comput Sci 26(8):1447–1463, 1995 [4], Afifi et al. in Int J Syst Sci 31(2):149-159, 2000 [1], Afifi et al. in Int J Appl Math Comput Sci 11(2):319-348, 2001 [2] for example).

7. Chapter 6. Spreadability

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   The aim of this chapter is to explore the spreadability of sub-diffusion process, in which the subdomains of the states to the system studied obeying a spatial property are nondecreasing. Since it is the first time for us to investigate the spreadability of sub-diffusion process. We focused on analytic results in this paper first. Simulation results will be presented in our forthcoming work.

8. Chapter 7. Regional Stability and Regional Stabilizability

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   Note that the concept of stability is one of the most important notions in system analysis. A steady state is said to be stable if the system evolves close to this state for small perturbations and for an unstable system, it is often asked if it is possible to stabilize it by a control input. However, in our real life, there exist systems which are unstable on the whole domain but do not perform in the same manner all over \(\varOmega \subseteq \mathbf {R}^n\), indeed they may be stable on some subregion \(\omega \) of \(\varOmega \).

9. Chapter 8. Conclusions and Future Work

   Fudong Ge, YangQuan Chen, Chunhai Kou

   Abstract

   The control theory of fractional diffusion systems constitutes one of the next big challenges of the engineering community. They will require cooperation of multidisciplines, such as mathematical modeling, engineering applications, and information sciences in order to be successful. Among those challenges, the notation of regional anaysis when the system under consideration is only studied on a subset of the whole space has to be one of the most significance to be explored, simply because that focusing on the regional anaysis would allow for a reduction in the number of physical actuators, offer the potential to reduce computational requirements in some cases, and also possible to discuss those systems which are not controllable on the whole domain, etc. So in this book, instead of analyzing a system by purely theoretical viewpoint, we study the time fractional diffusion system by using the notions of sensors and actuators, which allow us to understand the system better and consequently enable us to steer the real-world system in a better way.

10. Backmatter