Abstract
ABSTRACT
The 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
Introduction
The 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
The Time Fractional Diffusion System
Let Ω be an open bounded subset of R n , n = 1, 2, 3, with a regular boundary ∂Ω.
We consider the following time fractional order diffusion system:
where 0 D α t and 0 I 1−α t denote, respectively, the Riemann-Liouville fractional order derivative and integral with respect to time t. For details on these operators, see e.g. [30, 31] . Here we just recall their definition:
where 0 < α < 1. The second order operator A in (1) is linear with dense domain such that the coefficients do not depend on t. It generates a C 0 -semi-group (S(t)) t≥0 on the Hilbert space Y := L 2 (Ω). We refer the reader to Engel and Nagel [32] and Renardy and Rogers [33] for more properties on operator A. The initial datum y 0 is in Y . The operator B : R m → Y is the control operator, which depends on the number m of actuators and u ∈ L 2 (0, T ; R m ).
Preliminaries
In this section, we recall some notions and facts needed in the sequel. We begin with the concept of mild solution, that has been used in fractional calculus in several different contexts [34, 35, 36] .
Definition 1 (See, e.g., [37, 38, 39] ). For any given function f ∈ L 2 (0, T ;Y ) and α ∈ (0, 1), we say that function g ∈ L 2 (0, T ;Y ) is a mild solution of the system
where
and ϕ α is the probability density function defined by
Remark 1. The probability density function satisfies
Moreover,
For results on existence and uniqueness of mild solutions for a class of fractional neutral evolution equations with nonlocal conditions, we refer to Zhou and Jiao [40] . Here we note that a mild solution of system (1) can be written as
Remark 2. Throughout the paper, y(x,t) is the variable of system (1) . It depends on time t and space x. We use y(x,t; u) as the solution of system (1) when it is excited with a control u. We also denote it formally by y(u).
for all u ∈ L 2 (0, T ; R m ) and assume that (S * (t)) t≥0 is a strongly continuous semi-group generated by the adjoint operator of A on the state space Y . Let ·, · be the duality pairing of space Y . It is easy to see that
For any v ∈ Y , it follows from (4) that
where B * is the adjoint operator of B, and
Let ω ⊂ Ω be a given region of positive Lebesgue measure. We define the restriction operator χ ω and its adjoint χ
In order to prove our results, the following lemma is needed.
Lemma 1 (See [41]). Let the reflection operator Q on the interval [0, T ] be defined by Q h(t) := h(T − t), for some function h which is differentiable and integrable in the Riemann-Liouville sense. Then the following relations hold:
Q 0 I α t h(t) = t I α T Q h(t), Q 0 D α t h(t) = t D α T Q h(t) and 0 I α t Q h(t) = Q t I α T h(t), 0 D α t Q h(t) = Q t D α T h(t).
Enlarged Controllability and Characterization
We define exact enlarged controllability (EEC) as follows. In our case, we take G as a sub-vectorial closed space of Y := L 2 (Ω). The following result holds.
Theorem 2. Our system is exact enlarged controllable (i.e., system (1) is exactly G-controllable in ω in the sense of Definition 2) if and only if
Proof. Suppose (6) holds. Then, there exists
Therefore, y(u) ∈ G and we have EEC in ω. Conversely, assume that one has EEC of (1) in ω, which means that χ ω y(u) ∈ G. Using (2) and (3), we have
Let us denote w = χ ω y(u) − χ ω T α−1 K α (T )y 0 = χ ω Hu. Then, one has w ∈ Im χ ω H and
We just proved (6).
Fractional HUM Approach
We now extend the Hilbert uniqueness method (HUM), introduced by Lions in [29] , to the fractional setting (1) and try to compute the (optimal) control that steers system (1) into G. First of all, we prove under what conditions we can find enlarged controllability. Then we compute the (optimal) control that steers our system into G. The reader interested in the HUM approach, in the context of fractional calculus, is referred to [42, 43, 44, 45] .
The Case of a Zone Actuator
Let us consider system (1) excited with a zone actuator (D, f ), where D ⊆ Ω is the support of the actuator and f its spatial distribution. For details about actuators, we refer to [20, 46] . System (1) can be written as follows:
in Ω.
The main question we answer is the following one. Does there exist a (
Let us introduce G • as the polar space of G. Then, ϕ 0 ∈ G • , that is, ϕ 0 , φ = 0 for all φ ∈ G, where ·, · denotes the scalar product in Y . For ϕ 0 ∈ G • , we consider the following backward system:
where Q is the reflection operator on the interval [0, T ] defined in Lemma 1. Hence, system (8) can be rewritten as
System (9) has a unique mild solution given by
Moreover, we define the following semi-norm on G:
Lemma 3. Equation (10) defines a norm.
Proof. To prove that (10) is a norm, we show that ϕ 0
, is equivalent to f , ϕ(·,t) = 0. It follows from the uniqueness theorem in [47] that ϕ ≡ 0.
We also consider the problem:
in Ω.
We obtain ψ such that ψ :
The control (12) ensures EEC and we have ψ = y(u), where y(u) is a formal notation of the solution y(x, T ; u). In order to easily explain (11) with an equation, we define
where M is an affine operator from G • to the orthogonal G ⊥ of G. Thus, all return to solve the equation Mϕ 0 = 0. Let us decompose M into a linear part and a constant one:
in Ω.
Then,
Let Λ be the operator given by
With these notations, the exact enlarged controllability problem leads to the solution of equation
The following result holds.
Theorem 4. System (7) is exact enlarged controllable relatively to G. Moreover, the control
steers the system into G.
Proof. From Lemma 3, we have that (10) is a norm. Now, we show that (15) admits a unique solution in G. For any ϕ 0 ∈ G • , by (14) it follows that
Existence of a unique solution follows from [23, Theorem 1.1].
The Case of a Pointwise Actuator
Consider now system (1) with a pointwise internal actuator, which can be written in the form
where b is the position of the actuator. For any ϕ 0 ∈ G • , we consider system (8) and we define the following semi-norm:
dt.
Using similar arguments as in Section 5.1, we can easily prove that this semi-norm is indeed a norm. Let us consider u(t) = ϕ(b,t) and the following two systems:
in Ω.
Then, the exact enlarged controllability problem is equivalent to solve equation
Similarly to the proof of Theorem 4, for the case of a zone actuator, we prove exact enlarged controllability for a pointwise actuator.
Theorem 5. System (16) is exact enlarged controllable relatively to G. Moreover, the control u
steers the system into G.
Minimum Energy Control
The study of fractional optimal control problems is a subject under strong development: see [48, 49, 50, 51] and references therein. In this section, inspired by the results of [52, 53] , we show that the steering controls found in Section 5 are minimizers of a suitable optimal control problem. For that, let us consider the following minimization problem:
where
The proof of our Theorem 6 is based on a penalization method [54, 55] . Proof. Let ε > 0. Suppose that there is EEC relatively to G and consider the following problem:
The set S of pairs (z, u) verifying (18) is nonempty. Consider the penalized problem of (17) given by
where
Let {u ε , z ε } be the solution of (19) and let us define
Because we did assume that U ad is nonempty, we have
where C represents various positive constants independent of ε. It follows from (20) 
Hence, when ε → 0, we have that u ε is bounded and we can extract a sequence such that
Using the semi-continuity of J , one has
Then,
and u * =ũ. With respect to problem (19), we have
For u ∈ U ad and η such that
we deduce that p ε verifies
in Ω with η(T ), p ε (T ) = 0 for all η such that η(T ) ∈ G. Then, p ε (T ) ∈ G • . If we suppose that
, then we can switch to the limit when ε → 0. Moreover, if we have exact enlarged controllability relatively to G, then
in Q z(ξ,t) = 0 on Σ lim
Thus, we take p(T ) ∈ G • and we introduce the solution ϕ of (9). Then, ψ = z if ψ(T ) ∈ G, which proves that (13) has a unique solution for ϕ 0 ∈ G • .
Conclusions
This paper deals with the notion of regional exact enlarged controllability for Riemann-Liouville time fractional diffusion systems. Our results extend the ones in [42, 56, 57] . They can be extended to complex fractional-order distributed parameter dynamic systems. Other difficult questions are still open and deserving further investigations, e.g., the problem of boundary enlarged controllability for fractional systems; and the problem of gradient enlarged controllability/observability for fractional order distributed parameter systems. These and other questions, as to give numerical results and a real application to support our theoretical analysis, are being considered and will be addressed elsewhere.

{'paper_id': '32686369', 'title': 'Enlarged Controllability of Riemann-Liouville Fractional Differential Equations', 's2_url': 'https://api.semanticscholar.org/CorpusID:32686369', 'source': 'S2ORC'}