Enlarged Controllability of Riemann–Liouville Fractional Differential Equations

Abstract

IntroductionThe purpose of fractional calculus is to generalize standard derivatives into non-integer order operators. As well acknowledged in the literature, many dynamical systems are best characterized by dynamic models of fractional order, based on the notion of non-integer order differentiation or integration. The study of fractional order systems is, however, more delicate. Indeed, fractional systems are, on one hand, memory systems, notably for taking into account the initial conditions, and on the other hand they present much more complex dynamics [1,2,3].Since traditional computation is based on differentiation and integer order integration, the concept of fractional computation has an enormous potential to change the way we see, model, and control the "nature" around us [1,4,5]. Several theoretical and experimental studies show that some electrochemical [6,7], thermal [8] and viscoelastic [9,10] systems are governed by non-integer order differential equations. The use of classical models, based on a complete differentiation, is therefore not appropriate. As a result, models based on non-integer differential equations have been strongly developed in the past few decades [11,12,13].To backtrack the root of fractional calculus, one needs to go back to the XVII century (more accurately to 1695), when L'Hôpital was questioning Leibniz about the possible meaning of half order differentiation. This question has attracted the

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

“…For distributed parameter systems, several works deal with the problem of regional observability, which we study here in the fractional context, investigating the possibility to reconstruct the initial state or gradient only on a subregion ω of the evolution domain Ω [31][32][33][34][35]. For results on controllability, see [36][37][38][39]. The interest to study the concept of observability for fractional differential equations is not new: see [17,[40][41][42].…”

Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces

“…Our work can be extended in several directions: (i) to a case of enlarged controllability using different fractional derivatives; (ii) by developing methods to determine the control predicted by our existence theorem, e.g., by using RHUM and penalization approaches [10,40,41]; (iii) or by giving applications of neutral systems to epidemiological problems [42,43]. Many other questions remain open, as is the case of regional controllability and regional discrete controllability for problems of the type considered here.…”

Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

“…erefore, the research studies of fractional-order calculus attract lots of attention for these kinds of systems with several fractional derivatives (for more details, see [7][8][9][10] and the references therein). Zhou and Jiao [11] introduced a concept of a mild solution based on Laplace transform and probability density functions; several authors presented a tremendous amount of valuable results on controllability and observability, stability analysis, and so on [12][13][14][15].…”