Abstract
We investigate the exact enlarged controllability and optimal control of a fractional diffusion equation in Caputo sense. This is done through a new definition of enlarged controllability that allows us to extend available contributions. Moreover, the problem is studied using two approaches: a reverse Hilbert uniqueness method, generalizing the approach introduced by Lions in 1988, and a penalization method, which allow us to characterize the minimum energy control.
Highlights
The calculus of fractional order began more than three centuries ago
The subject has been developed by several mathematicians such as Euler, Fourier, Liouville, Grunwald, Letnikov and Riemann, among many others, up to the present day where many authors study such kind of operators and propose new fractional derivatives [1,2,3,4,5]
Fractional calculus is used in the field of automatic control to obtain more accurate models, to develop new control strategies and to improve the characteristics of control systems [14,15]
Summary
The calculus of fractional order began more than three centuries ago. It was first mentioned by the celebrated Leibniz, in a letter replying to l’Hopital, addressing the question whether the derivative remains valid for a non-integer order. Optimal control can be regarded as a branch of Mathematics whose goal is to improve the state variables of a control system in order to maximize a benefit or minimize a given cost This is applicable to practical situations, where state variables can be temperature, a velocity field, a measure of information, etc. We deal with the controllability problem of Caputo fractional diffusion equations when in presence of constraints on the state variables This is related with the notion of enlarged controllability, which was first investigated by Lions in 1988 for hyperbolic systems [27] and later developed for linear and semilinear parabolic systems [28,29,30, 31].
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