Yubo Bai

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Zhiqiang Wang

Exact controllability for a Rayleigh beam with piezoelectric actuator

Keywords: exact controllability, Rayleigh beam, piezoelectric actuator 2020 MSC: 93C20, 93B05, 35B65

In this paper, exact controllability problem for a Rayleigh beam with piezoelectric actuator is considered. Controllability results show that the space of controllable initial data depends on the regularity of the control function and the location of the actuator. Two different spaces of control function, L 2 (0, T ) and (H 1 (0, T )) ′ , correspond to two different controllability properties, L 2 -controllability and (H 1 ) ′ -controllability, respectively. The approach to prove controllability results is based on the Hilbert Uniqueness Method. Some non-controllability results are also obtained. In particular, non-controllability in short control time is studied by using Riesz basis property of exponential family in L 2 (0, T ). Finally, minimal control time for the exact controllability is obtained.

Introduction and main results

History and problem statement

In recent decades, there have been a large number of papers concerning the study of flexible structures. Three main directions of research can be considered: the modelling problem, the controllability problem and the stabilization problem. Modelling a flexible structure as a beam equation or a plate equation is an essential research field. In [START_REF] Han | Dynamics of transversely vibrating beams using four engineering theories[END_REF], four types of models for the transversely vibrating uniform beam, i.e., the Euler-Bernoulli beam, Rayleigh beam, shear beam and Timoshenko beam, were summarized and analysed. In the past few decades, the study of elastic structures with a piezoelectric actuator or sensor has gained a lot of attention. See [START_REF] Crawley | Detailed models of piezoceramic actuation of beams[END_REF][START_REF] Ph | Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction[END_REF] for a PDE modelling elastic structures with a piezoelectric actuator or sensor.

Concerning controllability for PDEs, boundary controllability for wave equation and plate equation was studied in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF] using the Hilbert Uniqueness Method (HUM). There were plenty of works on controllability for beam and plate based on HUM. In [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF][START_REF] Lagnese | Modelling analysis and control of thin plates[END_REF], boundary controllability for Kirchhoff plate equation was fully investigated. Exact controllability was obtained in sufficient large control time with a single boundary control (active on a sufficiently large portion of the boundary) in the case of clamped boundary conditions. As for beam equation, exact controllability for Euler-Bernoulli beam hinged at both ends with piezoelectric actuator was firstly considered in [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF]. Since the space dimension is one, Fourier series was used in [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF]. Then [START_REF] Crépeau | Control of a clamped-free beam by a piezoelectric actuator[END_REF] studied the exact controllability for the same beam equation with piezoelectric actuator in a different physical configuration: the clamped-free boundary conditions, i.e. a beam clamped at one end and free at the other end. In [START_REF] Özkan Özer | Exact controllability of a Rayleigh beam with a single boundary control[END_REF], Ingham inequality (see [START_REF] Baiocchi | Ingham type theorems and applications to control theory[END_REF][START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]) was used to obtain the exact controllability for Rayleigh beam equation with a single boundary control among four different boundary conditions.

In this paper, we consider the control problem modelling the transverse deflection of a Rayleigh beam which is subject to the action of an attached piezoelectric actuator. Assuming that the beam is hinged at both ends, the equation of Rayleigh beam can be written as (see, for instance, [START_REF] Crawley | Detailed models of piezoceramic actuation of beams[END_REF][START_REF] Ph | Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction[END_REF]), for (x, t) in (0, π) × (0, +∞),

w tt (x, t) -αw xxtt (x, t) + w xxxx (x, t) = u(t) d dx [δ η (x) -δ ξ (x)], (1.1a)
w(0, t) = w(π, t) = w xx (0, t) = w xx (π, t) = 0, (1.1b) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x).

(1.1c)

In the equations above w represents the transverse deflection of the beam, α > 0 is a physical constant, ξ and η stand for the ends of the actuator (0 < ξ < η < π), and δ y is the Dirac mass at the point y. The control is given by the function u : [0, T ] → R standing for the time variation of the voltage applied to the actuator. Our main purpose is to find the initial data that can be steered to rest by means of the control function u. To give the precise definitions of exact controllability, let us introduce for any ω in R the functional space Y ω as follows. Let Y 0 = L 2 (0, π). For ω > 0, let Y ω be the closure in H ω (0, π) of the set of y in C ∞ ([0, π]) satisfying the conditions y (2n) (0) = y (2n) (π) = 0 ∀n ≥ 0.

(1.2)

For ω < 0, let Y ω be the dual space of Y -ω with respect to the space Y 0 . Then we give the precise definitions.

Definition 1.1. The initial data (w 0 , w 1 ) in Y 2 × Y 1 is exactly L 2 -controllable in (ξ, η) at time T if there exists u in L 2 (0, T ) such that the solution w of (1.1) satisfies the condition w(•, T ) = w t (•, T ) = 0.

Definition 1.2. The initial data (w 0 , w 1 ) in Y 1 × Y 0 is exactly (H 1 ) ′ -controllable in (ξ, η) at time T if there exists u in (H 1 (0, T )) ′ such that the solution w of (1.1) satisfies the condition w(•, T ) = w t (•, T ) = 0.

Note that in Definitions 1.1 and 1.2, the spaces where the initial data (w 0 , w 1 ) can be taken depend on the wellposedness of (1.1) (see Section 3). In Definition 1.2, the space (H 1 (0, T )) ′ is the dual space of H 1 (0, T ) with respect to the space L 2 (0, T ). The study of controllability with less regular control function is inspired by [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF] which studied the controllability of changing the norm for wave equation and plate equation. Note that the system (1.1) is a time-reversible linear system, so the exact controllability is equivalent to the null controllability (see Theorem 2.41 of [11, p. 55]).

The paper is organized as follows. In the remaining part of this section, we present the main results of the paper, i.e. controllability, non-controllability and minimal time for the exact controllability. In Section 2, we provide some preliminaries on the theory of Diophantine approximation and the Riesz basis property of exponential family. The well-posedness results for the control problem (1.1) are given in Section 3. The main results are proved in Section 4, respectively. Appendix A provides the proof of a technical lemma which is used in the proof of noncontrollability in short control time.

Controllability results

In order to state the exact controllability results, let ε > 0 and let the sets A ⊂ (0, 1) and B ε ⊂ (0, 1) be the sets defined in Section 2. From Section 2, the set A is uncountable and has zero Lebesgue measure and the Lebesgue measure of set B ε is 1.

Our exact controllability results are the following. 1. Suppose that η+ξ 2π and η-ξ 2π belong to the set A. Then all initial data in Y 3 × Y 2 are exactly (H 1 ) ′ -controllable in (ξ, η) at time T . 2. Suppose that η+ξ 2π and η-ξ 2π belong to the set B ε . Then all initial data in Y 3+2ε × Y 2+2ε are exactly (H 1 ) ′controllable in (ξ, η) at time T .

In Theorem 1.4, (H 1 (0, T )) ′ control function brings new difficulties to the problem. The well-posedness of (1.1) with (H 1 (0, T )) ′ control function need to be proven while the well-posedness of (1.1) with L 2 (0, T ) control function is a known result (see Section 3).

Theorems 1.3 and 1.4 give some sufficient conditions for exact controllability. All the results show the dependence of the space of exactly controllable initial data on the location of the actuator. The differences between Theorem 1.4 and Theorem 1.3 are the different spaces of the control function and the different spaces of the controllable initial data. In Theorem 1.4, the control function belongs to (H 1 (0, T )) ′ rather than L 2 (0, T ) and the space of the controllable initial data is larger than the space in Theorem 1.3 with the same choice of ξ and η. Roughly speaking, the larger (less regular) the space of control function is, the larger (less regular) the space of controllable initial data is. To the best knowledge of the authors, such a result has not been developed yet for beam equation with piezoelectric actuator or interior control.

Non-controllability results

After the controllability results, we show some noncontrollability results. In Section 4, from Propositions 4.1 and 4.4 and the solution (3.5) of the adjoint problem, we can see that condition

η -ξ 2π , η + ξ 2π ∈ R \ Q (1.3)
is necessary to have any exact controllability result. The first non-controllability result concerns the insufficiency of condition (1.3). Under condition (1.3), there exist ξ and η and initial condition of problem (1.1) such that no control can steer this initial condition to the equilibrium.

Theorem 1.5. 1. For any β ≥ -1, there exist ξ and η satisfying (1.3) such that for any T > 0, the space Y β+3 × Y β+2 contains some initial data that are not exactly L 2 -controllable in (ξ, η) at time T .

2.

For any β ≥ -2, there exist ξ and η satisfying (1.3) such that for any T > 0, the space Y β+3 × Y β+2 contains some initial data that are not exactly (H 1 ) ′controllable in (ξ, η) at time T .

Theorem 1.3 (resp. Theorem 1.4) gives no information on the exact L 2 -controllability (resp. exact (H 1 ) ′controllability) of initial data in Y β+3 × Y β+2 for β < 1 (resp. for β < 0). A partial answer is given by the following result.

Theorem 1.6. Let ε > 0, T > 0 and ξ, η in (0, π) be arbitrary.

1. The set Y 3-ε × Y 2-ε contains some initial data that are not exactly L 2 -controllable in (ξ, η) at time T . 2. The set Y 2-ε × Y 1-ε contains some initial data that are not exactly (H 1 ) ′ -controllable in (ξ, η) at time T .
Notice that all the exact controllability results in Theorems 1.3 and 1.4 require T > 2π √ α, however, in [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF], the exact controllability results for Euler-Bernoulli beam have no requirement of control time. Consequently, a huge difference between Rayleigh beam and Euler-Bernoulli beam is revealed and the reason lies in various distributions of their eigenvalues. More precisely, under the same boundary condition (1.1b), the eigenvalues of Rayleigh beam equation are k 4 1+αk 2 for k in N * (see Section 3.1) while the eigenvalues of Euler-Bernoulli beam equation are k 4 for k in N * (see [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF]). Roughly speaking, this fact implies that Rayleigh beam equation possesses finite propagation speed and that Euler-Bernoulli beam equation possesses infinite propagation speed. For this reason, all the exact controllability results for Rayleigh beam require T > 2π √ α while the exact controllability results for Euler-Bernoulli beam hold for all T > 0 (see [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF]). Based on this fact, we give the non-controllability results for 0 < T < 2π √ α.

Theorem 1.7. Let 0 < T < 2π √ α and ξ, η in (0, π) be arbitrary. Notice that in Theorem 1.5, the lack of controllability holds for some special ξ and η which are related to the space of initial data. However, in Theorems 1.6 and 1.7, non-controllability holds for any ξ and η. From Theorem 1.7, we can see that T ≥ 2π √ α is necessary for exact controllability for Rayleigh beam equation. Therefore, minimal control time for the exact controllability is obtained. As far as we know, this is the first result stating a lack of controllability for Rayleigh beam in short control time.

Preliminaries

In this section, we provide some known results on the theory of Diophantine approximation (see [START_REF] Cassels | An introduction to Diophantine approximation[END_REF][START_REF] Lang | Introduction to Diophantine approximations, 2nd Edition[END_REF]) and the Riesz basis property of exponential family (see [START_REF] Avdonin | Families of exponentials[END_REF]).

For a real number ρ, we denote by ∥ρ∥ Z the difference, taken positively, between ρ and the nearest integer, i.e., ∥ρ∥ Z = min n∈Z |ρ -n|. Let us denote by A the set of all irrationals ρ in (0, 1) such that if [0, a 1 , . . . , a n . . . ] is the expansion of ρ as a continued fraction, then (a n ) is bounded. The set A is uncountable and its Lebesgue measure is equal to zero (see Theorem I of [13, p. 120]). The following property proven in Theorem 6 of [14, p. 23] is essential for this paper.

Proposition 2.1. A number ρ is in A if and only if there exists a constant C > 0 such that ∥qρ∥ Z ≥ C q (2.1)
for any strictly positive integer q.

The next proposition, which is proved in [13, p. 120], shows that an inequality slightly weaker than (2.1) holds for almost all points in (0, 1). Proposition 2.2. For any ε > 0, there exist a set B ε ⊂ (0, 1) having Lebesgue measure equal to 1 and a constant C > 0 such that for any ρ in B ε ,

∥qρ∥ Z ≥ C q 1+ε (2.2)
for any strictly positive integer q.

The following proposition on simultaneous approximation proven in Theorem VII of [13, p. 14] is useful to prove Theorem 1.6.

Proposition 2.3. Let ρ 1 , . . . , ρ k be k irrationals in (0, 1). Then there exists a strictly increasing sequence of natural numbers q n such that for all n ≥ 1,

q 1 k n max i=1,...,k (∥q n ρ 1 ∥ Z , . . . , ∥q n ρ i ∥ Z , . . . , ∥q n ρ k ∥ Z ) ≤ k k + 1 .
The next proposition proven in Theorem II.4.18 of [15, p. 109] on the Riesz basis property of exponential family in L 2 (0, T ) is essential to prove Theorem 1.7.

Proposition 2.4. Let {λ n } n∈Z be a sequence of complex numbers such that sup n∈Z |Imλ n | < ∞ and inf n̸ =m |λ m - λ n | > 0. Let N (x, r) := ♯{λ n |x ≤ Reλ n < x + r} for x in R and r > 0,
where ♯A is the number of elements in the set A. Assume that for some T > 0,

lim r→∞ N (x, r) r = T 2π
holds uniformly relative to all x in R. Then for any T ′ in (0, T ), {e iλnt } n∈Z contains a subfamily {e iλq n t } n∈Z that forms a Riesz basis in L 2 (0, T ′ ). Moreover, if {λ n } n∈Z is a sequence of real numbers such that λ n = -λ -n , the subsequence {λ qn } n∈Z satisfies λ qn = -λ q-n .

Well-posedness of (1.1)

In Section 3.1, we show the well-posedness result of system (1.1) with L 2 (0, T ) control function which has been proved in [START_REF] Weiss | Well-posed linear systems-a survey with emphasis on conservative systems[END_REF]. In Section 3.2, we prove the well-posedness and regularity results of system (1.1) with (H 1 (0, T )) ′ control function. The approach is inspired by [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF].

3.1. Well-posedness of (1.1) with L 2 (0, T ) control function

We state the well-posedness result and show the proof here, because the process of the proof is also used in other sections.

Theorem 3.1. Suppose that (w 0 , w 1 ) belongs to Y 2 × Y 1 .
For any u in L 2 (0, T ) and for any ξ and η in (0, π), the initial and boundary value problem (1.1) admits a unique solution w having the regularity

w ∈ C([0, T ]; Y 2 ) ∩ C 1 ([0, T ]; Y 1 ). (3.1)
In order to prove Theorem 3.1, let us first consider the adjoint problem of (1.1) in (0, π) × (0, +∞),

ϕ tt (x, t) -αϕ xxtt (x, t) + ϕ xxxx (x, t) = 0, (3.2a) ϕ(0, t) = ϕ(π, t) = ϕ xx (0, t) = ϕ xx (π, t) = 0, (3.2b) ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x). (3.2c)
The next lemma proved in [START_REF] Weiss | Well-posed linear systems-a survey with emphasis on conservative systems[END_REF] provides the well-posedness of the adjoint problem (3.2) and some trace regularities needed in the proof of main results.

Lemma 3.2. For any initial data (ϕ 0 , ϕ 1 ) in Y 2 × Y 1 , there exists a unique weak solution ϕ of (3.2) in the class C([0, T ]; Y 2 )∩C 1 ([0, T ]; Y 1 ). Moreover, for any χ in (0, π), ϕ x (χ, •) belongs to H 1 (0, T ) and there exist C, C ′ > 0 such that

∥ϕ x (χ, •)∥ 2 H 1 (0,T ) ≤ C(∥ϕ 0 ∥ 2 H 2 (0,π) + ∥ϕ 1 ∥ 2 H 1 (0,π) ), (3.3) ∥ϕ x (χ, •)∥ 2 L 2 (0,T ) ≤ C ′ (∥ϕ 0 ∥ 2 H 1 (0,π) + ∥ϕ 1 ∥ 2 L 2 (0,π) ). (3.4)
Proof. It is easy to see, by the semigroup method, that the problem (3.2) is well-posed in the space Y 2 × Y 1 (see [17, p. 104]). Next we prove (3.3) and (3.4). Since the family of functions {x → sin(kx)} k∈N * is the orthogonal basis of Y 1 and Y 2 respectively, let ϕ 0 (x) = k≥1 a k sin(kx) and ϕ 1 (x) = k≥1 b k sin(kx) with (k 2 a k ) and (kb k ) in l 2 (R). By standard computation, we have

ϕ(x, t) = k≥1 a k cos k 2 √ 1 + αk 2 t + b k √ 1 + αk 2 k 2 sin k 2 √ 1 + αk 2 t sin(kx). (3.5)
Then for all T > 0, ϕ x (χ, •) belongs to H 1 (0, T ) and

T 0 |ϕ xt (χ, t)| 2 dx ≤ C k≥1 k 2 (a 2 k k 2 + b 2 k ),
which yields (3.3). And simultaneously we have

T 0 |ϕ x (χ, t)| 2 dx ≤ C ′ k≥1 (a 2 k k 2 + b 2 k ),
which clearly yields (3.4).

Proof of Theorem 3.1. Thanks to Lemma 3.2, the following backward adjoint problem in (0, π)×(0, τ

) is well-posed in Y 2 × Y 1 for every τ > 0 and g in Y 1 . v tt (x, t) -αv xxtt (x, t) + v xxxx (x, t) = 0, (3.6a) v(0, t) = v(π, t) = v xx (0, t) = v xx (π, t) = 0, (3.6b) v(x, τ ) = 0, v t (x, τ ) = g(x). (3.6c)
Moreover, for any χ in (0, π),

∥v x (χ, •)∥ L 2 (0,τ ) ≤ C∥g∥ Y0 . (3.7)
Since (1.1a) is linear and Lemma 3.2 holds, it is enough to consider the case w 0 = w 1 = 0. Suppose that g belongs to C ∞ 0 (0, π), and let v be the solution of (3.6). Define a linear operator L := I -α∂ xx . It is well-known that the operator L is an isomorphism from Y 2 to Y 0 and an isomorphism from Y 1 to Y -1 by Lax-Milgram Theorem. If we multiply (1.1a) by v and integrate by parts, we obtain

π 0 Lw(x, τ )g(x)dx = τ 0 u(t)(v x (η, t)-v x (ξ, t))dt. (3.8) Inequality (3.7) implies that τ 0 u(t)(v x (η, t) -v x (ξ, t))dt ≤ 2C∥u∥ L 2 (0,T ) ∥g∥ Y0 ,
so by (3.8), we obtain Lw(•, τ ) belongs to Y 0 , and hence w(•, τ ) belongs to Y 2 for all τ in [0, T ]. By replacing τ by τ + h in (3.8) we easily get that

w ∈ C([0, T ]; Y 2 ). (3.9) Denote R := (I -α∂ xx ) -1 . It follows from Lax-Milgram Theorem that the operator R is an isomorphism from Y -2
to Y 0 and an isomorphism from Y -1 to Y 1 . Applying R to both sides of (1.1a) yields

w tt (x, t)+Rw xxxx (x, t) = u(t)R d dx [δ η (x)-δ ξ (x)]. (3.10)
Regularity (3.9) implies that

Rw xxxx ∈ C([0, T ]; Y 0 ). (3.11)
As w satisfies (3.10) and dδ b dx belongs to Y -2 for all b in (0, π), we obtain from (3.11) that The conclusion (3.1) is now a consequence of (3.9) and (3.13) and of the general lifting result from [START_REF] Lasiecka | A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations[END_REF].

w tt ∈ L 2 (0, T ; Y 0 ). ( 3
3.2. Well-posedness of (1.1) with (H 1 (0, T )) ′ control function As the control function u belongs to (H 1 (0, T )) ′ , the dual space of H 1 (0, T ), we need to define the solution of (1.1) in the weak form. The next proposition gives a characterization of (H 1 (0, T )) ′ (see [20, p. 62]). Proposition 3.3. For every u in (H 1 (0, T )) ′ , there exist functions u 0 and u 1 in L 2 (0, T ) such that for all ϕ in H 1 (0, T ),

⟨u, ϕ⟩ (H 1 (0,T )) ′ ×H 1 (0,T ) = T 0 (u 0 ϕ + u 1 ϕ t )dt. (3.14) Moreover ∥u∥ (H 1 (0,T )) ′ = inf ∥(u 0 , u 1 )∥ (L 2 (0,T )) 2 = min ∥(u 0 , u 1 )∥ (L 2 (0,T )) 2 , (3.15)
the infimum being taken over, and attained on the set of all (u 0 , u 1 ) in (L 2 (0, T )) 2 for which (3.14) holds for every ϕ in H 1 (0, T ). And the element (u 0 , u 1 ) in (L 2 (0, T )) 2 satisfying (3.14) and (3.15) is unique.

Note that u 0 and u 1 also define a distribution u in D ′ (0, T ) as u = u 0 -u 1,t . We know from [20, p. 63] that the element u of (H 1 (0, T )) ′ is an extension to H 1 (0, T ) of the distribution u.

Inspired by [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], we define the weak solution of (1.1) by transposition and prove the well-posedness. We explain the results in three steps.

1. We prove the well-posedness and the trace regularity of a non-homogeneous problem (3.16). 2. We define the weak solution of (1.1) by transposition using the solution of non-homogeneous problem (3.16). 3. We prove the well-posedness of (1.1) with the control function u in (H 1 (0, T )) ′ .

Step

1. Let {f, θ 0 , θ 1 } belong to L 1 (0, T ; Y -1 ) × Y 2 × Y 1 .
Let us consider the following backward non-homogeneous problem in (0, π) × (0, T ),

θ tt (x, t) -αθ xxtt (x, t) + θ xxxx (x, t) = f (x, t), (3.16a) θ(0, t) = θ(π, t) = θ xx (0, t) = θ xx (π, t) = 0, (3.16b) θ(x, T ) = θ 0 , θ t (x, T ) = θ 1 . (3.16c)
The following proposition provides the well-posedness and the trace regularity of (3.16). Our approach to prove this proposition is inspired by ideas and methods used in [START_REF] Weiss | Well-posed linear systems-a survey with emphasis on conservative systems[END_REF][START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF].

Proposition 3.4. For any initial data (θ 0 , θ 1 ) in Y 2 × Y 1 and f in L 1 (0, T ; Y -1 ), there exists a unique weak solution θ of (3.16) in the class C([0, T ]; Y 2 )∩C 1 ([0, T ]; Y 1 ). Moreover, for any χ in (0, π), θ x (χ, •) belongs to H 1 (0, T ) and there exists C > 0 such that

∥θ x (χ, •)∥ H 1 (0,T ) ≤ C(∥θ 0 ∥ H 2 (0,π) + ∥θ 1 ∥ H 1 (0,π) + ∥f ∥ L 1 (0,T ;Y-1)
). (3.17)

Proof. Applying R to both sides of (3.16a), we obtain

θ tt (x, t) + Rθ xxxx (x, t) = Rf (x, t). ( 3
+ ∥θ 1 ∥ H 1 (0,π) + ∥f ∥ L 1 (0,T ;Y-1) ). (3.19)
Then we need to prove inequality (3.17). The following lemma proved in [START_REF] Weiss | Well-posed linear systems-a survey with emphasis on conservative systems[END_REF][START_REF] Ammari | Decay rates for a beam with pointwise force and moment feedback[END_REF] shows that the operator R∂ xxxx is "similar" to the elliptic operator

-1 α ∂ xx . Lemma 3.5. The linear operator L = -1 α ∂ xx -R∂ xxxx is bounded from Y 2 to Y 2 .
Using this lemma, we can reduce the proof of (3.17) to a regularity property for a string equation. We consider the initial value problem in (0, π) × (0, T ),

θ 1,tt (x, t) - 1 α θ 1,xx (x, t) = Rf (x, t), θ 1 (0, t) = θ 1 (π, t) = 0, θ 1 (x, T ) = θ 0 , θ 1,t (x, T ) = θ 1 .
The relations above imply that in (0, π)×(0, T ),

θ 2 = θ-θ 1 satisfies θ 2,tt (x, t) - 1 α θ 2,xx (x, t) = Lθ, θ 2 (0, t) = θ 2 (π, t) = 0, θ 2 (x, T ) = 0, θ 2,t (x, T ) = 0. Since θ belongs to C([0, T ]; Y 2 ) and L is bounded from Y 2 to Y 2 , Lθ belongs to C([0, T ]; Y 2 ).
Then by the classical theory for evolution equations of hyperbolic type (see [START_REF] Lasiecka | Nonhomogeneous boundary value problems for second order hyperbolic operators[END_REF]),

θ 2 belongs to C([0, T ]; H 3 (0, π)) ∩ C 1 ([0, T ]; H 2 (0, π)) and there exists a constant C 1 T > 0 such that ∥(θ 2 , θ 2,t )∥ C([0,T ];H 3 (0,π)×H 2 (0,π)) ≤ C 1 T ∥θ∥ C([0,T ];Y2
) . This inequality, combined with (3.19) and the standard trace theorem of hyperbolic equation, implies that there exists a constant C 2 T > 0 such that for any χ in (0, π),

∥θ 2,x (χ, •)∥ H 1 (0,T ) ≤ C 2 T (∥θ 0 ∥ H 2 (0,π) + ∥θ 1 ∥ H 1 (0,π) + ∥f ∥ L 1 (0,T ;Y-1) ). (3.20)
As for θ 1 , it is already proved by using multiplier methods in [START_REF] Fabre | Pointwise controllability as limit of internal controllability for the wave equation in one space dimension[END_REF] that there exists a constant C 3 T > 0 such that for any χ in (0, π),

∥θ 1,x (χ, •)∥ H 1 (0,T ) ≤ C 3 T (∥θ 0 ∥ H 2 (0,π) + ∥θ 1 ∥ H 1 (0,π) + ∥f ∥ L 1 (0,T ;Y-1) ). (3.21)
Then inequality (3.17) follows from (3.20) and (3.21).

Step 2. Now we give the definition of the weak solution of (1.1). Denote by X a Banach space consisting of θ, the solution of (3.16). And give X a natural Banach structure such that {f, θ 0 , θ 1 } → θ is an isomorphism from

L 1 (0, T ; Y -1 ) × Y 2 × Y 1 to X . From Proposition 3.4, X is contained in C([0, T ]; Y 2 ) ∩ C 1 ([0, T ]; Y 1
) and (3.17) holds for all θ in X and χ in (0, π).

Assume that {f, θ 0 , θ 1 } belongs to L 1 (0, T ;

Y -1 ) × Y 2 × Y 1 and that {u, w 0 , w 1 } belongs to L 2 (0, T ) × Y 2 × Y 1 .
Denote by θ the solution of (3.16) and by w the solution of (1.1) given by Theorem 3.1. Multiplying (1.1a) by θ and integrating by parts, we obtain

T 0 π 0 wf dxdt + π 0 [Lw t (T )θ 0 -Lw(T )θ 1 ]dx = - T 0 u(t)(θ x (η, t) -θ x (ξ, t))dt + π 0 [Lw 1 θ(0) -Lw 0 θ t (0)]dx.
Now relaxing the assumption of {u, w 0 , w 1 } belonging to L 2 (0, T )×Y 2 ×Y 1 to {u, w 0 , w 1 } belonging to (H 1 (0, T )) ′ × Y 1 × Y 0 and considering the integral T 0 u(t)(θ x (η, t)θ x (ξ, t))dt as the linear form between u and θ x (η, •)θ x (ξ, •), we obtain from (3.14) that there exist functions u 0 and u 1 in L 2 (0, T ) such that

T 0 π 0 wf dxdt + π 0 [Lw t (T )θ 0 -Lw(T )θ 1 ]dx = - T 0 [u 0 (t)(θ x (η, t) -θ x (ξ, t)) + u 1 (t)(θ xt (η, t) -θ xt (ξ, t))]dt + π 0 [Lw 1 θ(0) -Lw 0 θ t (0)]dx. (3.22)
Let (3.22) be the definition of weak solution. Definition 3.6. Let T > 0, u in (H 1 (0, T )) ′ and (w 0 , w 1 ) in Y 1 × Y 0 be given. A solution of the problem (1.1) is a function w in C([0, T ]; Y 1 ) such that for every {f, θ 0 , θ 1

} in L 1 (0, T ; Y -1 ) × Y 2 × Y 1 , (3.22) holds and (w(T ), w t (T )) belongs to Y 1 × Y 0 .
Step 3. Then we are able to prove the well-posedness of (1.1) when u belongs to (H 1 (0, T )) ′ . Theorem 3.7. Suppose (w 0 , w 1 ) belongs to Y 1 × Y 0 . For any u in (H 1 (0, T )) ′ and for any ξ and η in (0, π), the initial and boundary value problem (1.1) admits a unique weak solution w in sense of Definition 3.6. And the map {w 0 , w 1 , u} → {w, w(T ), w t (T )} is continuous and linear with respect to the corresponding norm.

Proof. Since u belongs to (H 1 (0, T )) ′ , there exist u 0 and u 1 in L 2 (0, T ) such that (3.14) holds . Moreover, the map {f, θ 0 , θ 1 } → θ is an isomorphism from L 1 (0, T ; Y -1 ) × Y 2 × Y 1 to X . Therefore, we define a linear form Γ on X such that

Γ(θ) = - T 0 [u 0 (t)(θ x (η, t) -θ x (ξ, t)) + u 1 (t)(θ xt (η, t) -θ xt (ξ, t))]dt + π 0 [Lw 1 θ(0) -Lw 0 θ t (0)]dx. (3.23)
Since (w 0 , w 1 ) belongs to Y 1 × Y 0 , (Lw 0 , Lw 1 ) belongs to Y -1 × Y -2 . It follows from Proposition 3.4 that Γ is a continuous linear form on X . Denote by (f 1 , f 2 ) the linear form between Y -γ and Y γ for any γ ≥ 0 and denote by ⟨g 1 , g 2 ⟩ * the linear form between L ∞ (0, T ; Y 1 ) and L 1 (0, T ; Y -1 ). Therefore, for the linear form Γ in X ′ , there exists a unique element {w,

ζ * , ζ} in L ∞ (0, T ; Y 1 ) × Y -2 × Y -1 such that ⟨w, f ⟩ * + (ζ * , θ 0 ) + (-ζ, θ 1 ) = Γ(θ) ∀θ ∈ X . (3.24)
Next we claim that w above is actually the weak solution of (1.1). It is sufficient to prove that w satisfies (1.1) in weak sense, Lw(T ) = ζ and Lw t (T ) = ζ * . Notice that {x → sin(kx)} k∈N * is the family of eigenfunctions of R∂ xxxx . Let m(x) = sin(kx) for some k in N * and let h belong to L 1 (0, T ). Firstly we set Rf (t) = h(t)m, θ 0 = 0 and θ 1 = 0, then f (t) = h(t)Lm. Denote by λ 2 = k 4 /(1 + αk 2 ) the corresponding eigenvalue and take λ positively. Then we obtain from Proposition 3.4 that for t in (0, T ), θ(t) = q(t)m, where q satisfies q tt +λ 2 q = h and q(T ) = q t (T ) = 0. Clearly we have q(t) = -1 λ T t sin(λ(tσ))h(σ)dσ. Then we obtain from (3.23) and (3.24) that Now we have proved that there exists a unique element {w, w(T ), (3.22) holds and the map {w 0 , w 1 , u} → {w, w(T ), w t (T )} is continuous and linear with respect to the corresponding norm. In fact we have the property of w belonging to C([0, T ]; Y 1 ). Since when the known data {w 0 , w 1 , u} belongs to Y 2 × Y 1 × L 2 (0, T ), we have (3.1). Using a density argument, we conclude the proof of Theorem 3.7.

T 0 (w, Lm)(q tt + λ 2 q)dt = ⟨w, f ⟩ * = Γ(θ) = -(m x (η) -m x (ξ)) T 0 [u 0 (t)q(t) + u 1 (t)q t (t)]dt + q(0)(Lw 1 , m) -q t (0)(Lw 0 , m). (3.25) Notice that (w, Lm) = (Lw, m). Then (3.25) implies that (Lw, m) tt + λ 2 (Lw, m) = -(m x (η) -m x (ξ))u, (Lw, m)(0) =(Lw 0 , m), (Lw, m) t (0) =(Lw 1 , m).
(ζ, m) = Γ(θ) = (Lw 1 , m) sin(λT ) λ + (Lw 0 , m) cos(λT ) -(m x (η) -m x (ξ)) • T 0 u 0 (t) sin(λ(T -t)) λ -u 1 (t) cos(λ(T -t)) dt. (3.27) Moreover, it follows from (3.26) that (Lw, m)(T ) = (Lw 0 , m) cos(λT ) + (Lw 1 , m) sin(λT ) λ - m x (η) -m x (ξ) λ ⟨u, sin(λ(T -•))⟩ (H 1 (0,T )) ′ ×H 1 (0,T ) .
(ζ * , m) = Γ(θ) = (Lw 1 , m) cos(λT ) -(Lw 0 , m)λ sin(λT ) -(m x (η) -m x (ξ)) • T 0 [u 0 (t) cos(λ(T -t)) + u 1 (t)λ sin(λ(T -t))]dt.
w t (T )} in L ∞ (0, T ; Y 1 ) × Y 1 × Y 0 such that

Proofs of the main results

In this section, we prove the main results. On the one hand, for controllability results, namely Theorems 1.3 and 1.4, we first use the HUM to rewrite the control problem into observability inequalities of the adjoint equation.

Then we derive these observability inequalities by applying the Ingham inequality. The methods for proving Theorems 1.3 and 1.4 are inspired by the ideas and methods used in [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF] for Euler-Bernoulli beam with piezoelectric actuator. On the other hand, for non-controllability results, namely Theorems 1.5 to 1.7, we exhibit initial conditions so that the observability inequalities are false. The approaches in proofs of Theorems 1.5 and 1.6 are inspired by [START_REF] Tucsnak | Regularity and exact controllability for a beam with piezoelectric actuator[END_REF]. And the proof of the lack of the controllability in short control time, namely Theorem 1.7, is inspired by the methods used in [START_REF] Avdonin | Simultaneous controllability in sharp time for two elastic strings[END_REF]. 2 satisfying the compatibility conditions (1.2) and denote by ϕ(x, t) the solution of (3.2) with initial value (ϕ 0 , ϕ 1 ).

4.1. Exact L 2 -controllability (Proof of Theorem 1.3) Let (ϕ 0 , ϕ 1 ) in (C ∞ [0, π])
Consider a backward adjoint system in (0, π) × (0, T )

ψ tt (x, t) -αψ xxtt (x, t) + ψ xxxx (x, t) = u(t) d dx [δ η (x) -δ ξ (x)], (4.1a) ψ(0, t) = ψ(π, t) = ψ xx (0, t) = ψ xx (π, t) = 0, (4.1b) ψ(x, T ) = ψ t (x, T ) = 0, (4.1c)
where u in L 2 (0, T ) will be chosen later. Problem (4.1) is well-posed according to Theorem 3.1. Then, multiplying (4.1a) by ϕ and integrating by parts, we get

π 0 ϕ 0 (x)Lψ t (x, 0) -ϕ 1 (x)Lψ(x, 0)dx = T 0 u(t)(ϕ x (η, t) -ϕ x (ξ, t))dt. (4.2)
Let u(t) = ϕ x (η, t) -ϕ x (ξ, t). Since (3.4), u belongs to L 2 (0, T ). Define a linear operator Λ satisfying Λ(ϕ 0 , ϕ 1 ) = (Lψ t (•, 0), -Lψ(•, 0)).

Since (Lψ t (•, 0), -Lψ(•, 0)) belongs to Y -1 × Y 0 by Theorem 3.1, the operator Λ is well defined. In particular, we obtain from (4.2) that

⟨Λ(ϕ 0 , ϕ 1 ), (ϕ 0 , ϕ 1 )⟩ = T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt.
Therefore, we can define a seminorm

∥(ϕ 0 , ϕ 1 )∥ F := T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt 1 2
, for all (ϕ 0 , ϕ 1 ) in (C ∞ [0, π]) 2 satisfying the compatibility conditions (1.2).

A classical argument in HUM implies the following proposition.

Proposition 4.1. All initial data in Y β+3 × Y β+2 are exactly L 2 -controllable in (ξ, η) at time T if and only if there exists a constant c > 0 such that

T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt ≥ c(∥ϕ 0 ∥ 2 H -β + ∥ϕ 1 ∥ 2 H -β-1 ) (4.
3) for all (ϕ 0 , ϕ 1 ) in (C ∞ [0, π]) 2 satisfying the compatibility conditions (1.2). Equation (4.3) is called observability inequality. As in the proof of Lemma 3.2, the solution ϕ of the adjoint problem (3.2) has the form of (3.5), which implies that

T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt = 4 T 0 k≥1 k sin k(η + ξ) 2 • sin k(η -ξ) 2 a k cos k 2 t √ 1 + αk 2 + b k √ 1 + αk 2 k 2 sin k 2 t √ 1 + αk 2 2 dt. (4.4)
Remark 4.2. From (4.4), we can see that (1.3) is necessary for controllability.

To prove the observability inequality (4.3) for some β, we apply the following Ingham inequality (see [START_REF] Baiocchi | Ingham type theorems and applications to control theory[END_REF][START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]) to our problem. Lemma 4.3. Let (ν k ) k∈Z be a strictly increasing sequence of real numbers and let γ ∞ be defined by γ ∞ = lim inf |k|→∞ |ν k+1 -ν k |. Assume that γ ∞ > 0. For any real T > 2π/γ ∞ , there exist two constants C 1 , C 2 > 0 such that for any sequence

(x k ) k∈Z in l 2 (C), C 1 k∈Z |x k | 2 ≤ T 0 k∈Z x k e iν k t 2 dt ≤ C 2 k∈Z |x k | 2 .
We apply Lemma 4.3 with

ν k = -ν -k = k 2 √ 1 + αk 2 , k ∈ N, 2x k = 2x -k = a k -i b k √ 1 + αk 2 k 2 • k sin k(η + ξ) 2 sin k(η -ξ) 2 , k ∈ N * , x 0 = 0. As lim |k|→∞ |ν k+1 -ν k | = 1/ √ α, then for any real T > 2π √ α, there exists a constant C 1 > 0 such that T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt ≥ C 1 k≥1 k 2 a 2 k + b 2 k (1 + αk 2 ) k 4 • sin k(η + ξ) 2 sin k(η -ξ) 2 2 . (4.5)
When η+ξ 2π and η-ξ 2π belong to A, it follows from (2.1) that there exists a constant C > 0 such that for all k ≥ 1,

sin k(η ± ξ) 2 = sin π k(η ± ξ) 2π -p ≥ sin πC k ≥ C k . (4.6)
Inequalities (4.5) and (4.6) imply that

T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt ≥ c k≥1 (a 2 k k -2 + b 2 k k -4 ),
which is exactly (4.3) for β = 1. This fact completes the proof of the first part of Theorem 1.3.

When η+ξ 2π and η-ξ 2π belong to B ε , it follows from (2.2) that there exists a constant C > 0 such that for all k ≥ 1,

sin k(η ± ξ) 2 ≥ C k 1+ε . (4.7)
Inequalities (4.5) and (4.7) imply that

T 0 |ϕ x (η, t) -ϕ x (ξ, t)| 2 dt ≥ c k≥1 (a 2 k k -2-4ε + b 2 k k -4-4ε ),
which is exactly (4.3) for β = 1 + 2ε. This fact completes the proof of the second part of Theorem 1.3.

4.2.

Exact (H 1 ) ′ -controllability (Proof of Theorem 1.4) Similar to in Section 4.1, we use the HUM to rewrite the controllability problem. Let (ϕ 0 , ϕ 1 ) in (C ∞ [0, π]) 2 satisfy the compatibility conditions (1.2). Denote by ϕ the solution of (3.2) with initial value (ϕ 0 , ϕ 1 ).

Then consider the backward adjoint system (4.1) where u in (H 1 (0, T )) ′ will be chosen later. Problem (4.1) is wellposed due to Theorem 3.7. Then taking the linear form between (4.1a) and ϕ and integrating by parts, we obtain

π 0 ϕ 0 (x)Lψ t (x, 0) -ϕ 1 (x)Lψ(x, 0)dx = ⟨u, ϕ x (η, •) -ϕ x (ξ, •)⟩ (H 1 (0,T )) ′ ×H 1 (0,T ) . (4.8) Recalling (3.14), this equation is equivalent to π 0 ϕ 0 (x)Lψ t (x, 0) -ϕ 1 (x)Lψ(x, 0)dx = T 0 [u 0 (t)(ϕ x (η, t) -ϕ x (ξ, t)) + u 1 (t)(ϕ xt (η, t) -ϕ xt (ξ, t))]dt, (4.9)
for some u 0 and u 1 in L 2 (0, T ).

Let u 0 (t) = 0 and u 1 (t) = ϕ xt (η, t)-ϕ xt (ξ, t). Function u 1 belongs to L 2 (0, T ) due to (3.3), and therefore, u is well defined. Define a linear operator Λ * satisfying Λ * (ϕ 0 , ϕ 1 ) = (Lψ t (•, 0), -Lψ(•, 0)).

Since (Lψ t (•, 0), -Lψ(•, 0)) belongs to Y -2 × Y -1 because of Theorem 3.7, operator Λ * is well defined. In particular, we have

⟨Λ * (ϕ 0 , ϕ 1 ), (ϕ 0 , ϕ 1 )⟩ = T 0 |ϕ xt (η, t) -ϕ xt (ξ, t)| 2 dt.
Therefore, we can define a seminorm

∥(ϕ 0 , ϕ 1 )∥ F * := T 0 |ϕ xt (η, t) -ϕ xt (ξ, t)| 2 dt 1 2
, for all (ϕ 0 , ϕ 1 ) in (C ∞ [0, π]) 2 satisfying the compatibility conditions (1.2).

A classical argument in HUM implies the following proposition.

Proposition 4.4. All initial data in Y β+3 × Y β+2 are exactly (H 1 ) ′ -controllable in (ξ, η) at time T if and only if there exists a constant c > 0 such that

T 0 |ϕ xt (η, t) -ϕ xt (ξ, t)| 2 dt ≥ c(∥ϕ 0 ∥ 2 H -β + ∥ϕ 1 ∥ 2 H -β-1 ) (4.
T 0 |ϕ xt (η, t) -ϕ xt (ξ, t)| 2 dt ≥ c k≥1 (a 2 k + b 2 k k -2 ),
which is exactly (4.10) when β = 0. This completes the proof of the first part of Theorem 1.4.

When η+ξ 2π and η-ξ 2π belong to B ε , there exists a constant c > 0 such that

T 0 |ϕ xt (η, t) -ϕ xt (ξ, t)| 2 dt ≥ c k≥1 (a 2 k k -4ε + b 2 k k -2-4ε ),
which is exactly (4.10) when β = 2ε. This completes the proof of the second part of Theorem 1.4. As mentioned in Remark 4.2, the condition (1.3) is necessary for controllability. We aim to prove the condition (1.3) is not sufficient in this section.

From Proposition 4.1, it is sufficient to show that for any β ≥ -1, there exist ξ and η satisfying (1.3) such that (4.3) is false for any c > 0. For any β ≥ -1, let

ν > max 3 2 β + 1, 2 . (4.11) We choose η + ξ 2π = ∞ n=1 a n 10 n! , (4.12)
where a n belongs to {0, 1, . . . , 9} for all n ≥ 1, and a n is not identically zero for great n. According to [START_REF] Valiron | Théorie des Fonctions[END_REF] the right-hand side of (4.12) is a Liouville number, i.e., it is transcendental and there exists a strictly increasing sequence of integers q n such that

sin q n η + ξ 2 ≤ π q ν n ∀n ≥ 1. (4.13)
Now we consider the sequence of initial data ϕ 0 n (x) = q µ n sin(q n x), ϕ 1 n (x) = 0 ∀x ∈ (0, π), (4.14) 2 and satisfies compatibility conditions (1.2) and

where µ = 3 2 β if β > 0 and µ = 1 if -1 ≤ β ≤ 0. Ob- viously, (ϕ 0 n , ϕ 1 n ) belongs to (C ∞ [0, π])
∥ϕ 0 n ∥ 2 H -β + ∥ϕ 1 n ∥ 2 H -β-1 → ∞. (4.15)
Moreover, we obtain from (4.4), (4.11), and (4.13) that Similarly, because of Proposition 4.4, it is sufficient to show that for any β ≥ -2, there exist ξ and η satisfying (1.3) such that (4.10) is false for any c > 0. The proof is quite similar to the proof above in this section. For any fixed β ≥ -2, we only need to change ν as

T 0 |ϕ n,x (η, t) -ϕ n,x (ξ, t)| 2 dt = 4 T 0 q n sin q n η + ξ 2 sin q n η -ξ 2 •q µ n cos q 2 n 1 + αq 2 n t 2 dt ≤ 4T q 2(µ+1) n sin q n η + ξ 2 2 ≤ 4πT q 2(µ+1-ν) n → 0, as n → ∞.
ν > max 3 2 β + 2, 3 , (4.17)
and to set µ = 3 2 β if β > 0 and µ = 1 if -2 ≤ β ≤ 0. By similar calculation, we obtain that (4.10) is false for any c > 0.

4.4. Non-controllability for initial data in less regular set (Proof of Theorem 1.6) Similarly to the previous section, we aim to prove (4.3) (resp. (4.10)) is false. According to Proposition 2.3, for any ξ and η in (0, π), there exists a strictly increasing sequence of positive integers {q n } n≥1 such that for all n ≥ 1,

sin q n η + ξ 2 ≤ π √ q n , sin q n η -ξ 2 ≤ π √ q n .
(4.18) First we consider the sequence of initial data

ϕ 0 n (x) = sin(q n x), ϕ 1 n (x) = 0 ∀x ∈ (0, π). (4.19)
We note that for any ε > 0,

∥ϕ 0 n ∥ 2 H ε + ∥ϕ 1 n ∥ 2 H ε-1 = Cq 2ε n → ∞, n → ∞, (4.20)
where C is a positive constant. By (4.4) and (4.18) we have that for all n ≥ 1,

T 0 |ϕ n,x (η, t) -ϕ n,x (ξ, t)| 2 dt = 4 T 0 q n sin q n η + ξ 2 sin q n η -ξ 2 • cos q 2 n 1 + αq 2 n t 2 dt ≤ K, (4.21)
where K is a positive constant. So (4.20) and (4.21) show that (4.3) is false for β = -ε and arbitrary c > 0.

Then we choose the sequence of initial data ϕ 0 n (x) = q -1 n sin(q n x), ϕ 1 n (x) = 0 ∀x ∈ (0, π). (4.22)

We note that for any ε > 0,

∥ϕ 0 n ∥ 2 H + ∥ϕ 1 n ∥ 2 H ε = Cq 2ε n → ∞, n → ∞, (4.23)
where C is a positive constant. By (3.5) and (4.18) we have that for all n ≥ 1,

T 0 |ϕ n,xt (η, t) -ϕ n,xt (ξ, t)| 2 dt = 4 T 0 sin q n η + ξ 2 sin q n η -ξ 2 • q 2 n 1 + αq 2 n sin q 2 n 1 + αq 2 n t 2 dt ≤ K, (4.24)
where K is a positive constant. So (4.23) and (4.24) show that (4.10) is false for β = -ε -1 and arbitrary c > 0.

4.5. The lack of controllability in short control time (Proof of Theorem 1.7)

We prove the lack of L 2 -controllability (resp. (H 1 ) ′controllability) when 0 < T < 2π √ α in this section. Let 0 < T < 2π √ α and ξ, η in (0, π) be arbitrary.

We first concern the lack of L 2 -controllability. For any β ≥ -1, we aim to find {(ϕ 0

m , ϕ 1 m )} m∈N * such that T 0 |ϕ m,x (η, t) -ϕ m,x (ξ, t)| 2 dt → 0, as m → ∞ and ∥ϕ 0 m ∥ 2 H -β + ∥ϕ 1 m ∥ 2 H -β-1 ≥ c > 0 for any m ≥ 1.
As in Section 4.1, we denote holds uniformly relative to x in R.

λ n = -λ -n = n 2 √ 1 + αn 2 , n ∈ N * . ( 4
We will prove this lemma in Appendix A. As 0

< T < 2π √ α, we can choose T ′ such that 0 < T < T ′ < 2π √ α. Let f in L 2 (0, 2π √
α) be a real valued function such that f (t) = 0 if 0 ≤ t ≤ T and ∥f ∥ L 2 (0,T ′ ) ̸ = 0. According to Lemma 4.5 and Proposition 2.4, the family {e iλnt } n∈Z * contains a subfamily {e iλq n t } n∈Z * which forms a Riesz basis in L 2 (0, T ′ ). Moreover, the subsequence {λ qn } n∈Z satisfies λ qn = -λ q-n . Then for the function f in L 2 (0, T ′ ) defined above, there exists a sequence {l n } n∈Z * in l 2 (C) such that f (t) = n∈Z * l n e iλq n t holds in L 2 (0, T ′ ) and 0 < n∈Z * |l n | 2 < ∞. Since f (t) is a real valued function, we have l n = l -n . Now we can define the sequence {(ϕ 0 m , ϕ 1 m )} m∈N * of initial data as the following,

ϕ 0 m (x) = 2 m n=1 Re(l n ) q n sin q n η + ξ 2 • sin q n η -ξ 2 -1 sin(q n x), ϕ 1 m (x) = -2 m n=1 Im(l n ) q n sin q n η + ξ 2 • sin q n η -ξ 2 -1 q 2 n 1 + αq 2 n sin(q n x). (4.27)
As mentioned in Remark 4.2, we assume (1.3) holds. Consequently, the sequence {(ϕ 0 m , ϕ 1 m )} m∈N * of initial data is well-defined.

Since 0 < n∈Z * |l n | 2 < ∞ and l n = l -n , there exists m 0 ≥ 1 such that l m0 ̸ = 0. So for m ≥ m 0 and for any As for the lack of exact (H 1 ) ′ -controllability for any β ≥ -2, we only need to change the initial data in (4.27) to

β ≥ -1, ∥ϕ 0 m ∥ 2 H -β + ∥ϕ 1 m ∥ 2 H -β-1 ≥ ∥ϕ 0 m0 ∥ 2 H -β + ∥ϕ 1 m0 ∥ 2 H -β-1 = c > 0. ( 4
ϕ 0 m (x) = 2 m n=1
Re(l n ) q n sin q n η + ξ 2

• sin q n η -ξ 2 Im(l n ) q n sin q n η + ξ 2

• sin q n η -ξ 2 -1 sin(q n x). Relations (4.28) and (4.32) finish the proof of the lack of exact (H 1 ) ′ -controllability for any β ≥ -2.

Conclusion

The exact controllability problem for Rayleigh beam equation with piezoelectric actuator has been fully considered. Two different types of controllability problem, namely L 2 -controllability and (H 1 ) ′ -controllability, are investigated. Moreover, several non-controllability results are proved. Especially, minimal control time for the exact controllability is deduced from the non-controllability result in short control time. As said in Remark 1. . Thus we define a real function g : (0, +∞) → (0, +∞) as g(r) = [(αr 2 + r √ α 2 r 2 + 4)/2] 1 2 and define {x} = q(x) -1, where q(x) = min q∈Z {q ≥ x}. Obviously, x -1 ≤ {x} < x. Then we have N (0, r) = {g(r)}. Notice that for x ≥ 0, N (x, r) = N (0, x + r) -N (0, x) = {g(x+r)}-{g(x)}. Therefore, we have g(x+r)-g(x)-1 ≤ N (x, r) ≤ g(x + r) -g(x) + 1. Now we need to estimate g(x + r) -g(x) for x, r > 0.

Let f (x) = x 2 √ 1+αx 2 for x > 0. Then simple calculation shows that 0 < x < 2/α implies f ′′ (x) > 0. Therefore, there exists N 0 in N satisfying N 0 > 2/α + 1 such that for all n ≥ N 0 , λ n+1 -λ n decreases and converges to 1/ √ α. Then for x ≥ N 0 , N (N 0 , r) ≤ N (x, r) ≤ lim x→+∞ N (x, r). Simple calculation shows that lim x→+∞ [g(x + r) -g(x)] = √ αr. Therefore, we obtain that g(N 0 + r)g(N 0 ) -1 ≤ N (x, r) ≤ √ αr + 1 holds for x ≥ N 0 . For 0 ≤ x ≤ N 0 , N (x, r) ≤ N (0, N 0 ) + N (N 0 , r) ≤ √ αr + 1 + g(N 0 ). Assuming that r > N 0 , we have Let θ belong to [0, 1] and |x| = θr. Note that N (0, |x|) + N (0, r -|x|) = {g(θr)} + {g((1 -θ)r)}. We obtain from the expression of g that lim r→∞ g(θr)/r = √ αθ. Consequently, (A.1) holds uniformly relative to -r ≤ x < 0.

N (x,
For x < -r, N (|x| -r, r) ≤ N (x, r) ≤ N (|x| -r, r) + 1. Let t = |x|-r ≥ 0. Then as same as in the situation x ≥ 0, we have that (A.1) holds uniformly relative to t ≥ 0, which means that (A.1) holds uniformly relative to x < -r.

Combining all the situations, we have that (A.1) holds uniformly relative to x in R. Lemma 4.5 is thus proved.

Theorem 1 . 3 . 1 .

 131 Let T > 2π √ α and ε > 0. Suppose that η+ξ 2π and η-ξ 2π belong to the set A. Then all initial data in Y 4 × Y 3 are exactly L 2 -controllable in (ξ, η) at time T . 2. Suppose that η+ξ 2π and η-ξ 2π belong to the set B ε . Then all initial data in Y 4+2ε × Y 3+2ε are exactly L 2 -controllable in (ξ, η) at time T . Theorem 1.3 gives us two exact L 2 -controllability results. The first result of Theorem 1.3 shows that, for the end of the piezoelectric actuator in an uncountable zero measure set, we have the exact L 2 -controllability in Y 4 ×Y 3 . The second result of Theorem 1.3 shows that, for almost all choices of the end of the piezoelectric actuator, we have the exact L 2 -controllability in more regular Sobolev spaces than Y 4 × Y 3 . The next theorem concerns the exact controllability in less regular spaces. Theorem 1.4. Let T > 2π √ α and ε > 0.

( 3 .

 3 26)Note that the first equation of (3.26) holds in (H 1 (0, T )) ′ . Since m(x) = sin(kx) and k ≥ 1 is an arbitrary natural number, then w satisfies (1.1) in weak sense. Now we set f = 0, θ 0 = 0 and θ 1 = -m. Then for t in (0, T ), θ(t) = 1 λ sin(λ(T -t))m. Therefore, (3.23) and (3.24) imply that

( 3 .

 3 28) Comparing (3.28) to (3.27), we obtain that (Lw, m)(T ) = (ζ, m), which proves Lw(T ) = ζ. Next we set f = 0, θ 0 = m and θ 1 = 0. Then for t in (0, T ), θ(t) = cos(λ(T -t))m. Thus (3.23) and (3.24) imply that

  3.26) implies that (Lw, m) t (T ) = (Lw 1 , m) cos(λT )-(Lw 0 , m)λ sin(λT ) -(m x (η) -m x (ξ)) • ⟨u, cos(λ(T -•))⟩ (H 1 (0,T )) ′ ×H 1 (0,T ) .(3.30) Comparing (3.30) to (3.29), we obtain that (Lw, m) t (T ) = (ζ * , m), which implies Lw t (T ) = ζ * .

  [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] for all (ϕ 0 , ϕ 1 ) in (C ∞ [0, π]) 2 satisfying the compatibility conditions (1.2).Assume T > 2π √ α. Using similar arguments as in Section 4.1, we have the following results. When η+ξ 2π and η-ξ 2π belong to A, there exists a constant c > 0 such that

4. 3 .

 3 The condition (1.3) is not sufficient (Proof of Theorem 1.5)

(4. 16 )

 16 Relations (4.15) and(4.16) show that (4.3) is false for any c > 0.

l n e λq n t 2 dt

 2 .28) Moreover, thanks to (4.4), we haveT 0 |ϕ m,x (η, t) -ϕ m,x (ξ, t)| f (t) = n∈Z * l n e λq n t in L 2 (0, T ), we obtain that T 0 |ϕ m,x (η, t) -ϕ m,x (ξ, t)| 2 dt → 0, as m → ∞. (4.30)Relations (4.28) and (4.30) finish the proof of the lack of exact L 2 -controllability for any β ≥ -1.

( 4 .l n e λq n t 2 dt

 42 31)Similarly, (4.28) holds for m ≥ m 0 and for any β ≥ -2. Moreover, we haveT 0 |ϕ m,xt (η, t) -ϕ m,xt (ξ, t)| → 0, as m → ∞.

5 k 2 √α 2 r 2 + 4 )

 5224 8, exact controllability in critical time is an open problem. Controllability problem for other types of beam equation with piezoelectric actuator, such as shear beam equation, also remains open. Appendix A. Proof of Lemma 4.Solving 1+αk 2 < r, we obtain that k < [(αr 2 + r √

  r) ≥ N (N 0 , r -N 0 + x) = N (0, x + r) -N (0, N 0 ) ≥ g(x + r) -g(N 0 ) -1 ≥ min x∈[0,N0] g(x + r) -g(N 0 ) -1.Then for all x ≥ 0 and r > N 0 , min x∈[0,N0] g(x + r)g(N 0 ) -1 ≤ N (x, r) ≤ √ αr + 1 + g(N 0 ) and hence, to x ≥ 0. For -r ≤ x < 0,N (0, |x|) + N (0, r -|x|) ≤ N (x, r) ≤ N (0, |x|) + N (0, r -|x|) + 1.

  .18) Notice that Rf belongs to L 1 (0, T ; Y 1 ). Then the problem (3.18) is well-posed in the state space Y 2 × Y 1 by the classical semigroup method (see [17, p. 106]), and then the problem (3.16) admits a unique solution θ in C([0, T ]; Y 2 ) ∩ C 1 ([0, T ]; Y 1 ). Moreover, there exists a constant C

T > 0 such that ∥θ∥ C([0,T ];Y2) ≤ C T (∥θ 0 ∥ H 2 (0,π)

  .25) Obviously, {λ n } n∈Z * is a strictly increasing sequence and lim |n|→∞ |λ n+1 -λ n | = 1/ √ α > 0. Adding or subtracting finite numbers in the sequence does not affect the result of Proposition 2.4, so we can apply Proposition 2.4 to the sequence {λ n } n∈Z * . Define N (x, r) as in Proposition 2.4 corresponding to {λ n } n∈Z * . We have the following lemma. Lemma 4.5. Let {λ n } n∈Z * and N (x, r) be defined above.

	We have that
	N (x, r) r	→	√	α, as r → ∞	(4.26)


Zeyi Shang

Sergey Verlan

Gexiang Zhang

Haina Rong

FPGA Implementation of Numerical P Systems

Keywords: Membrane Computing, numerical P system, enzymatic numerical P system, generalized numerical P system, Field Programmable Gate Array (FPGA), hardware implementation

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Membrane computing (MC) is a natural computation paradigm abstracted from structural and functional features of living cells [START_REF] Pȃun | Computing with membranes[END_REF][START_REF] Pan | Foreword: Starting jmc[END_REF]. Computing models in MC are called P Systems. There is an intense theoretic research in the area of MC focusing primarily on computability aspects, in particular on Turing-computability, see [START_REF] Narayanan | An overview of membrane computing[END_REF] for an overview. There is a huge number of Corresponding author: Gexiang Zhang, E-mail: zhgxdylan@126.com 1 different variants of P systems, we cite some important variants below: celllike P systems [START_REF] Pȃun | Computing with membranes[END_REF][START_REF] Narayanan | P systems with replicated rewriting[END_REF], tissue/population-like P systems [START_REF] Martín-Vide | Tissue P systems[END_REF][START_REF] Bernardini | Population P systems[END_REF][START_REF] Freund | Tissue P systems with channel states[END_REF], spiking neural P systems [START_REF] Ionescu | Computing with spiking neural P systems: Traces and small universal systems[END_REF][START_REF] García-Arnau | Spiking Neural P Systems: Stronger Normal Forms[END_REF][START_REF] Jiang | An improved universal spiking neural P system with generalized use of rules[END_REF] and P systems with active membranes [START_REF] Pȃun | P systems with active membranes: Attacking np-complete problems[END_REF][START_REF] Gazdag | A new method to simulate restricted variants of polarizationless P systems with active membranes[END_REF][START_REF] Vincenzo | From biopolymer duplication to membrane duplication and beyond[END_REF].

Because of the biological background of P systems, they are used as modeling frameworks for biological and ecological systems, see [START_REF]Applications of Membrane Computing[END_REF][START_REF] Manca | Evolution and oscillation in p systems: Applications to biological phenomena[END_REF][START_REF] Miguel | Parallel simulation of population dynamics P systems: updates and roadmap[END_REF][START_REF] Nash | A P system model of swarming and aggregation in a Myxobacterial colony[END_REF] for more details. As for engineering application of P systems, it is a relatively new direction in its early stage comparing to biological/ecological modeling of P systems. Complex market interactions are modeled by population dynamic P systems in [START_REF] Sánchez-Karhunen | Modelling complex market interactions using pdp systems[END_REF]. Several variants of spiking neural P systems are used in power system fault diagnosis [START_REF] Wang | Fault diagnosis of electric power systems based on fuzzy reasoning spiking neural p systems[END_REF][START_REF] Rong | A novel approach for detecting fault lines in a small current grounding system using fuzzy reasoning spiking neural P systems[END_REF][START_REF] Peng | Fault Diagnosis of Power Systems Using Intuitionistic Fuzzy Spiking Neural P Systems[END_REF][START_REF] Zhang | Real-life applications with membrane computing[END_REF][START_REF] Rong | Automatic Implementation of Fuzzy Reasoning Spiking Neural P Systems for Diagnosing Faults in Complex Power Systems[END_REF]. Other applications can be found in [START_REF] Zhang | An Optimization Spiking Neural P System for Approximately Solving Combinatorial Optimization Problems[END_REF][START_REF] Zhang | A hybrid approach based on differential evolution and tissue membrane systems for solving constrained manufacturing parameter optimization problems[END_REF][START_REF] Zhang | Evolutionary membrane computing: A comprehensive survey and new results[END_REF].

Numerical P systems introduced in [START_REF] Pȃun | Membrane Computing and Economics: Numerical P Systems[END_REF] is a variant of P systems very different from the standard model. Instead of objects and rewriting rules it features real-valued variables which are updated in discrete time steps using a set of equations. This particularity, long time unexplored, is very interesting for applications in the area of control theory, as usually a control is described using a set of differential equations, which in many cases can be translated to the NPS equation set. In order to be applicable to practical case studies NPS model was extended to Enzymatic Numerical P systems (ENPS) [START_REF] Pavel | Enzymatic numerical P systems -a new class of membrane computing systems[END_REF] that allowed much more complex behaviors in the corresponding equations. This in turn lead to the concept of membrane controllers [START_REF] Buiu | Development of membrane controllers for mobile robots[END_REF] that are ENPS systems designed to act as controllers and running in some environment. As a test bed the control of differential wheeled robots [START_REF] Buiu | Development of membrane controllers for mobile robots[END_REF] (motion, obstacle avoidance, wall following, robot following) was developed. In [START_REF] Wang | Design and implementation of membrane controllers for trajectory tracking of nonholonomic wheeled mobile robots[END_REF] a kinematic controller and a proportional-integral-derivative controller are designed for wheeled mobile robots. Another interesting application is found in [START_REF] Wang | Multi-behaviors coordination controller design with enzymatic numerical P systems for robots[END_REF] where an environment classifier and a novel multi-behaviors control approach are proposed to enhance the reactive navigation performance of autonomous mobile robots. Membrane controllers mentioned above require several ingredients. The evolution of a robot in a real or simulated environment requires a program that reads/transmits the values of robot' sensors (usually distance and speed), runs the simulation of the controller (given in ENPS form) for one or several steps and then updates/transmits the values of actuators (usually robot wheel motors). Before running the simulation this program should assign initial values for the membrane controller and after the simulation it should retrieve corresponding output values from it. The controller itself is simulated using a custom simulator [START_REF] Buiu | Development of membrane controllers for mobile robots[END_REF][START_REF] George | Modelling multi-robot interactions using a generic controller based on numerical p systems and ros[END_REF] or by Matlab code [START_REF] Wang | Design and implementation of membrane controllers for trajectory tracking of nonholonomic wheeled mobile robots[END_REF][START_REF] Wang | Multi-behaviors coordination controller design with enzymatic numerical P systems for robots[END_REF]. In some cases [START_REF] George | Modelling multi-robot interactions using a generic controller based on numerical p systems and ros[END_REF][START_REF] Wang | Multi-behaviors coordination controller design with enzymatic numerical P systems for robots[END_REF] 2 the experiments were carried out in real robot environments. To speed-up the simulation in the case of complex controllers [START_REF] Wang | Multi-behaviors coordination controller design with enzymatic numerical P systems for autonomous mobile robots in unknown environments[END_REF] the use of graphical processing units (GPU) hardware architectures was proposed [START_REF] García-Quismondo | Implementing Enzymatic Numerical P Systems for AI Applications by Means of Graphic Processing Units[END_REF].

In this article we extend NPS to a new model that we call Generalized Numerical P systems (GNPS). The main idea behind this extension is to provide a theoretical background allowing us to build custom parallel hardware architectures using Field-Programmable Gate Array (FPGA) technology. We studied what operations can be efficiently performed in hardware and restricted the GNPS architecture to be a series of rules that imply that the dynamics of the system is described using equations written in Presburger arithmetic. This allows in turn a very efficient translation to a hardware description language (HDL) used for FPGA circuit design, allowing to run the model at the clock speed ( 10 8 steps/s) and using a low number of ressources. To assist this translation we developed a compiler that translates GNPS to Verilog HDL. This allows to simplify the design process and to rapidly develop real hardware prototypes. Moreover, it turns out that there is a tight link between GNPS, sequential circuits [START_REF] Shannon | The synthesis of two-terminal switching circuits[END_REF], Mealy/Moore automata [START_REF] Mealy | A method for synthesizing sequential circuits[END_REF][START_REF] Moore | Gedanken-experiments on sequential machines[END_REF]

DEFINITIONS

Numerical P Systems

Numerical P systems were introduced in [START_REF] Pȃun | Membrane Computing and Economics: Numerical P Systems[END_REF] as a model for the study of economical processes. They have a tree-like structure and each compartment contains a set of real-valued variables as well as evolution rules, called programs. They are formally defined as follows.

A numerical P system is the construct

Π = (m, H, µ, (V ar 1 , P r 1 , V ar 1 (0)), . . . , (V ar m , P r m , V ar m (0)))
where m > 0 is the degree of the system, H is a set of labels, µ is a membrane structure, V ar i , P r i and V ar i (0) are the set of variables, programs and initial values from compartment i, 1 ≤ i ≤ m. By convention, we will label variables with two indices such that V ar j = {x 1j , . . . , x kj j }.

A program (rule) P li ∈ P R i has the following form

P li : F li (x 1i , . . . , x kii ) → c l1 |v 1 + • • • + c lni |v ni
where variables v 1 , . . . , v ni belong to membrane i, or to the neighboring ones (the parent or the children of i).

The first part of the rule (function F ) is called the production function, while the second part (at the right-hand-side of the arrow) is called the repartition protocol.

A rule is applied as follows [START_REF] Pȃun | Membrane Computing and Economics: Numerical P Systems[END_REF]. First the value of the production function is computed, based on current values of the variables. Second, each variable v s , 1 ≤ s ≤ n i from the repartition protocol part receives the fraction c ls n i t=1 c lt of the computed production function value. If several rules update the same variable, then the corresponding amounts are added. Finally, the value of a variable at the beginning of each new step is reset to 0 if this variable was used in a computation of some production function.

It is not very difficult to observe that a computation in a NPS corresponds to a discrete time series, where the value of a variable at some time step is a function of the values of several variables at the previous time step. More precisely the evolution of the system can be described by the following equations:

x ji (t) = P lk has xji in rhs as vs

F lk (x 1k (t), . . . , x r k k (t)) c ls n k r=1 c lr + xji (t) (1)
where xji (t) = x ji (t) if x ji does not appear in any production function F lk , 0 otherwise.

Example 2.1. Consider the following NPS, also depicted in Figure 1, with two membranes nested as follows: It is not difficult to observe that the corresponding system can be rewritten as following time series with initial conditions a(0

[ 1 [ 2 ] 2 ] 1 . Let V ar 1 = {a, b, f }, V ar 2 = {x, y}, V ar 1 (0) = (0, 1, 3), V ar 2 (0) = (0, 1
) = 0, b(0) = 1, f (0) = 3, x(0) = 0, y(0) = 1.                  a(t + 1) = a(t) + b(t) b(t + 1) = x(t) + y(t) f (t + 1) = f (t) + a(t) + b(t)
x(t + 1) = x(t) + y(t) + 2 (a(t) + b(t))

y(t + 1) = x(t) + y(t) (2)
This can be also written in a matrix form as follows

       a b f x y        =        1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 2 2 0 1 1 0 0 0 1 1               a b f x y        (3)
In many cases systems of recurrences can be solved analytically using standard methods. The analytical solution for system defined by Equation ( 2) is given below:

a(t) = 2 × 3 t-2 , b(t) = 4 × 3 t-2 , f (t) = 3 t-1 + 3, x(t) = 8 × 3 t-2 , y(t) = 2 × 3 t-2 + 1, t > 1
We would like to notice that by definition it is allowed to have several rules in a membrane. In this case, at each step, one of them is chosen and applied non-deterministically. However, for practical considerations, most of the systems considered in the literature have only one rule per membrane.

Enzymatic Numerical P systems

Enzymatic numerical P systems (ENPS), introduced in [START_REF] Pavel | Enzymatic numerical P systems -a new class of membrane computing systems[END_REF] are an extension of NPS that adds a new type of rules (called enzymatic):

P li : F li (x 1i , . . . , x kii )(e →)c l1 |v 1 + • • • + c lni |v ni
The application of this rule is conditioned to the verification of the minimality condition between the values of e and x 1i , . . . , x kii . Unfortunately, there is no unique definition for this condition -several papers use different ones. Here is the list of most used conditions: e > min(c(x 1i ), . . . , c(x kii )), in [START_REF] Pavel | Enzymatic numerical P systems -a new class of membrane computing systems[END_REF] e > min(x 1i , . . . , x kii ), in [START_REF] Pavel | Enzymatic numerical P systems -a new class of membrane computing systems[END_REF] e > min(|x 1i |, . . . , |x kii |), in [START_REF] Brândusa | Robot localization implemented with enzymatic numerical P systems[END_REF].

The first definition above uses the function c(x), which is the concentration of x in the biological sense. For example, for a production function 2x + y we obtain c(x) = x/2, c(y) = y, see [START_REF] Pavel | Enzymatic numerical P systems -a new class of membrane computing systems[END_REF] for more details. As in the case of NPS, if there are several applicable rules, one of them is used nondeterministically. One can observe that ENPS also yield time series that use a conditional operator.

Example 2.2. Consider the following ENPS, also depicted in Figure 2, with three membranes nested as follows:

[ 1 [ 2 ] 2 [ 3 ] 3 ] 1 . Let V ar 1 = {a, b}, V ar 2 = {E}, V ar 3 = {c}, V ar 1 (0) = (0, 3), V ar 2 (0) = (2n), n > 0, V ar 3 (0) = 0.
The set of rules is defined as follows:

P r 1 : 2b + 1(E →)1|a, P r 2 : E -1 → 1|E, P r 3 : 2(c + 1)(E →)1|c + 1|b.
It is not difficult to observe that the corresponding system can be rewritten as following time series.

a(t + 1) = if E(t) > b(t) then 2b(t) + 1 else a(t) b(t + 1) = c(t) + 1 c(t + 1) = c(t) + 1 E(t + 1) = E(t) -1 a(0) = 0, b(0) = 0, c(0) = 0, E(0) = 2n
Clearly, we obtain that a(t)

= t i=0 2i + 1 = t 2 (for t ≤ n).

Binary normal forms for (E)NPS

Traditionally, NPS evolve in min 1 mode (from each membrane a single program is selected and applied), see [START_REF] Freund | A formal framework for static (tissue) P systems[END_REF]. In [START_REF] Leporati | Improved Universality Results for Parallel Enzymatic Numerical P Systems[END_REF] so called all-parallel mode is introduced for (E)NPS, when all applicable rules from a membrane are applied at the same step (we remark that contrary to traditional P systems there is no competition for symbols/variable values). The advantage of such a mode is that the computation becomes deterministic. Another advantage is that the system allows a very nice binary normal form.

Definition 2.1. A (E)NPS is said to be in the binary normal form if all rules are of form

P : F (x 1 , . . . , x n ) → c|x k + L|λ, or P : F (x 1 , . . . , x n )(E →)c|x k + L|λ,
for some n, k > 0; c, L ≥ 0 and where λ is a special dummy variable.

It is relatively easy to see that any (E)NPS working in all-parallel mode can be transformed to an equivalent one in the binary normal form. To achieve this, any rule/program

F (x 1j , . . . , x kj ) → c 1 |v 1 + • • • + c l |v l
can be replaced by l rules that assign the same proportion of F to each vari-

able x l F (x 1j , . . . , x kj ) → c s |v s + (K -c s )|λ, 1 ≤ s ≤ l, K = l p=1 c p .
We remark that when l = 1, then L = K -c s = 0, so there is no variable λ introduced in the rule.

A similar transformation can be done with enzymatic programs.

Moreover, if we relax the condition that variables of the production function should be from the same membrane, then it is possible to obtain a stronger result. This allows to combine all rules related to a single variable into one rule by choosing appropriate coefficients. In this case several programs of type

F n (x 1j , . . . , x kj ) → c n |x + z n |λ can be combined as n p=1 c p c p + z p F p (x 1j , . . . , x kj ) → 1|x. (4)
(E)NPS having only rules of the above type is said to be in unary normal form. It is easy to observe that Equation (4) corresponds to the following time series

x(t + 1) = n p=1 c p c p + z p F p (x 1j (t), . . . , x kj (t)) + x(t). (5)
which is exactly the Equation (1).

In this section we introduce a generalization of (E)NPS, called Generalized Numerical P System (GNPS), which has some interesting properties helpful for the hardware implementation of the model. As a starting point we take the notion of the membrane controller [START_REF] Buiu | Development of membrane controllers for mobile robots[END_REF], hence from the beginning we assume to be interested not in the final result of the computation, but in the dynamics of the model. This naturally leads to the inclusion of the concept of dedicated input and output variables. The functioning of the system supposes that input variables are read-only and can be updated by an external entity at each step.

The output variables are write-only and an external entity may use their values at each step. Such a definition allows to effectively build controllers based on GNPS, without using any additional tools or mechanisms to pass the values and start/stop the computation.

From the structural point of view we use a structural abstraction intermediate between a tree-based structure and a flattened system (more precisely a hypergraph structure), being the equivalent of the network of cells [START_REF] Freund | A formal framework for static (tissue) P systems[END_REF] in NPS. This allows to have the notion of the locality (useful for hardware implementation as it can trigger the use of neighbor cells), but does not impose the strong restriction of a tree structure -some examples of membrane controllers spend an enormous amount of time for the data propagation because of the imposed tree structure. Concretely, this allows production functions to contain variables defined in a different membrane and also the repartition protocol may involve variables from any combination of membranes.

The main difference of GNPS with respect to previous models is a new type of rules that generalize all previous ones. This comes from the observation that rules of ENPS are rather limited (and also have a poorly defined semantics). The ENPS simulator PeP [START_REF] George | GitHub-PeP: (Enzymatic) Numerical P System simulator[END_REF] already proposed to use some simple arithmetic predicates to control the applicability of the rules. With GNPS we propose to go further and to use conditional rules of form (we separate by a semicolon local variables from the other ones):

P (x 1 , . . . , x k ; E 1 , . . . , E m ); F (x 1 , . . . , x k ) → c 1 |v 1 , . . . , c n |v n , ( 6
)
where P is predicate in Presburger arithmetic (we recall that this is the firstorder theory of the natural numbers with addition, i.e. one can use comparisons, Boolean operations, additions, subtractions and constant multiplications in expressions). Moreover, in the basic variant of the definition we will restrict production functions F to be Presburger as well. However, in order to accommodate real-case scenarios we will allow the usage of a finite algebraic signature (a set of functions that can be used in addition to the operations in Presburger arithmetic) for both production functions and predicates.

Example 3.1. We can consider the following predicate for a rule

P (x, y, z; E, F ) = E > x ∧ (F > y * 2 + 3 * z).
If we consider an algebraic signature containing the ordinary multiplication operation (σ = {×}), then it would be possible to write the following predicate

P (x, y, z; E, F ) = E > x ∧ ((F > y * 2 + 3 * z) ∨ (E + F > x × y + z)).
Finally, in order to obtain a deterministic evolution of the system, we assume that GNPS works in all-parallel mode, i.e. all applicable rules are applied at each step. This allows to greatly simplify the design of the hardware implementations.

Formally, we define a GNPS as the following tuple

Π = (m, I, O, (V ar 1 , V ar 1 (0)), . . . , (V ar m , V ar m (0)), P r, σ),
where m, V ar i and V ar i (0) have the same meaning as in NPS (the number of cells/membranes, the vectors of internal variables and their initial values).

The rules are no more specific to some membrane, so they are all collected in the set P r. Each rule is of form [START_REF] George | GitHub-PeP: (Enzymatic) Numerical P System simulator[END_REF]. In the case of an always true predicate, it can be omitted. Used variables in each rule induce a dependency hypergraph.

When this hypergraph is a tree, we may use a Venn diagram notation and place rules in corresponding cells/membranes. The input (resp. output) variables are given by the set I (resp. O). The algebraic signature σ contains the list of additional functions used (with respect to the addition/subtraction and constant multiplication). If σ = ∅ then it may be omitted from the definition.

We recall that the evolution of the system is performed in all-parallel mode,

i.e. all applicable rules are applied in parallel at each step.

FPGA IMPLEMENTATION

According to the discussion in Section 2.2 and the fact that Presburger arithmetic is recursive, any GNPS system can be rewritten as the following time series (where X(t), Y (t) and Q(t) are the vectors of input, output and internal variables, respectively, at time t):

Q(t + 1) = F (Q(t), X(t)) (7)
Y (t + 1) = G(Q(t), X(t)) (8)
These equations are the generalization (using real numbers instead of Boolean values) of equations used in switching algebra [START_REF] Shannon | The synthesis of two-terminal switching circuits[END_REF] for the definition of the concept of Mealy automaton [START_REF] Mealy | A method for synthesizing sequential circuits[END_REF], which together with Moore automaton [START_REF] Moore | Gedanken-experiments on sequential machines[END_REF] form the basis of modern synchronous circuit design. Since from the implementation point of view real numbers should be encoded using a fixed bit size, it appears that GNPS are very similar to vector Moore/Mealy machines. This in turn allows a straight implementation using hardware FPGA technology.

For implementation efficiency we considered following restrictions for GNPS:

• Real values are replaced by their approximation using a fixed-point binary representation.

• The production functions are linear.

• The predicates are Presburger-definable.

The above restrictions allow to relatively easy obtain the Mealy/Moore machine in form of Equation [START_REF] George | Modelling multi-robot interactions using a generic controller based on numerical p systems and ros[END_REF]. The corresponding functions F and G are linear enriched with conditional statements as it is shown in Example 4.1.

Remark 4.1. In the basic case we consider an empty signature σ as this allows a straight translation to Verilog. For more complex computations, corresponding functions should be implemented additionally as Verilog modules.

This can induce a delay, as in many cases it is not possible to compute corresponding functions in one clock step.

Remark 4.2. In the case of fixed-point encoding, it is possible to easily implement the multiplication operation working in one time step. This can be done either directly (by using multiplication code dependent on the width of the encoding), or using a special component of FPGA called DSP slice that allows to perform multiplication operations in one step (up to 48-bit width).

Remark 4.3. Contrary to multiplication, it is not easy to implement the division operation in one time step. However, the division by a constant c can be seen as the multiplication by c -1 .

Example 4.1. Consider the system depicted on Fig. 3. It has two input, two output and two internal variables. The system computes the average value of its inputs and also indicates if this value changed by more than 0.1% on the previous step. We recall that all rules are executed in parallel.

The set of equations corresponding to this system is the following (a(0) = b(0) = out 1 (0) = out 2 (0) = 0).

a(t + 1) = in 1 (t) + in 2 (t) 2 b(t + 1) = a(t) out 1 (t + 1) = a(t) -b(t) out 2 (t + 1) = if |a(t) -b(t)| < 0.001 then 0 else 1
It can be directly transformed to Verilog as follows (we assume a fixed point encoding of real numbers over 32 bits and using 12 bits for the fractional part).

In the below listing the fixed-point (constant) multiplication is performed by the function _mult (recall that 2048 is 0.5 in the chosen fixed-point encoding). The synthesis of this circuit using Vivado tools uses 87 logic cells.

It can be seen that the translation is rather straightforward. A compiler FPNtoVerilog was developed in order to assist in this translation. As input it takes the GNPS model in form of Equations ( 7) and ( 5) and produces as output behavioral Verilog code implementing the corresponding Mealy/-Moore automaton.

The compiler performs the following steps:

1. Parse the input file.

2. Identify input and output symbols.

3. Flatten the obtained system.

4. Perform constant propagation.

5. Convert all constants to fixed-point real number representation.

6. Write Verilog output.

These steps are performed using standard compiling techniques. The last step is straightforward as a sequential Boolean (switching) function/circuit can be directly translated to Verilog. As a result a file containing the synthesizable (in FPGA) Verilog module whose code simulates each step of the GNPS at each clock tick is generated. Two case studies are designed as target models and their FPGA implementation process is detailed to elucidate how a GNPS can be implemented in a FPGA. Since the algorithm we implement is making use of the square root function, we consider that the signature of the system is σ = { √ •}.

Our

θ[0.2] b 1 [0] b 2 [0] b 3 [0] b 31 [0] b 4 [0] b 41 [0]
        Skin FIGURE 4
GNPS model for case study 1. It implements the core computations of Sobel image edge detection algorithm. GNPS1 has a skin membrane containing 4 programs and 19 real-value variables. The predicate for each program is taken to a separate line before it. All the programs execute in parallel if their conditional rules are satisfied. Input variables are a1, a2, a3, a4 (highlighted in red). Output variables are b1, b2, b3, b4 (highlighted in blue). Others are intermediate variables.

After inputting GNPS1 to our FPNtoVerilog compiler, the output is a behavioral model that specifies the behavior of GNPS1. We also provided the implementations of the functions from the signature σ (square root and the square function). Next, this model is translated to register transfer level (RTL) using Vivado tools. The upper-most level of GNPS1 schematic generated by Vivado is depicted in Apparently, behavioral simulation suggests that four programs of GNPS1 execute simultaneously and output variables emit results 20 ns later. As stated above, behavioral simulation neglects any delays so the waveform cannot reflect the real timing situation. Post implementation timing simulation which can only be conducted after implementation operation can provide more reliable timing waveform.

As the platform for design & validation of prototype circuits [START_REF] Pong | FPGA Prototyping by Verilog Examples[END_REF], the essential task of FPGA implementation is to obtain circuits. Synthesis procedure bridges the gap between RTL models and circuits. Models can be synthesized after behavioral simulation if it behaves as expected behaviors. There are two important tasks should be done after synthesizing target model: setting constraints and debug cores. Constraints include timing constraints and physical

FIGURE 6

Behavioral simulation of GNPS1. The first rising edge of clock is at 5 ns. b2 gets its result at 15 ns and other three variables get their results at 5 ns since all the delays are neglected. This is not the reality but a simplification for assessing the behaviors of target model. constraints. In timing constraints, the period of the clock and input/output delays are set, while physical constraints specify I/O configurations, mapping ports of model to pins of FPGA. The clock period is set 10 ns in the two case studies. To save pins, only b 3 and b 4 are set as output ports for GNPS1. All the constraints are written in constraint file (.xdc) in the format of industry standard Synopsys Design Constraints (SDC) [START_REF]Xilinx. Vivado design suite user guide: Using constraints[END_REF]. The variables to be debugged in hardware debug procedure can be marked and set after synthesis.

The subsequent procedure of Synthesis is Implementation, which performs plan & route of the synthesized circuits and other vital operations such as power and hardware resource consumption analysis, real timing analysis. As mentioned above, behavior simulation ignores any delays so the timing situation cannot be evaluated from its waveform. Nevertheless, gate delays and data path delays of a model are taken into account after implementation so the timing of a design can be revealed by post implementation timing simulation, as shown in Figure 7. According to design timing summary provide by Vivado, the worst negative slack (WNS) is 3.97 ns, worst hold slack (WHS) 0.058 ns and worst pulse width slack (WPWS) 3.75 ns.

For the sake of comparing the computing speed of FPGA hardened GNPS and software simulation of GNPS, speedup is defined as the ratio of elapsed time of two methods. A software called PeP which dose not have a GUI dedi- Here we can see that there is no one-to-one correspondence between a clock cycle and a GNPS step. For complex arithmetic computations, one step of GNPS requires more than one clock cycle. cates to emulate (E)NPS [START_REF] George | GitHub-PeP: (Enzymatic) Numerical P System simulator[END_REF]. For the sake of simulating other types P systems, one can resort to P-Lingua [START_REF] Pérez-Hurtado | P-lingua in two steps: flexibility and efficiency[END_REF]. GNPS1 is transformed to its ENPS counterpart and emulated by PeP, which outputs results and elapsed time to compute the results, as shown in Figure 8. Then the speedup of FPGA hardened GNPS1 is calculated in Equation 9. The maximum error of output variables is computed in Equation 10.

The estimated power consumption is reported after implementation, shown in Figure 9 (a). Because the function of GNPS1 and GNPS2 is not complicate, the dynamic power merely shares 13% to 14% of total power and clock power makes up more than 70% of dynamic power.

(a) Power consumption of GNPS1 is 0.078 w.

(b) Power consumption of GNPS2 is 0.08 w.

FIGURE 9

Total power consumption is the sum of device static power and dynamic power. Power consumption of the two cases are nearly the same, although GNPS1 works in all parallel and GNPS2 works in sequential.

After implementation of a model, its Place & Route planning is also performed. Vivado provides powerful view check feature so the place and route can be zoomed in to see each path clearly. Hardware resource cost for GNPS1 is listed in Table 1.

If design objectives are satisfied after implementation, one can proceed to generate bitstream which contains design specifications and program device that download bitstream to FPGA to carry out physical plan & route.

The real computing results of FPGA cannot be observed straightforwardly, but requires a particular procedure called hardware debug. The variables to be debugged should be marked in Verilog codes or marked in the net list.

We

Case study 2

In practical applications such as image processing and robot path planning, computation process comprises several sequential procedures. for GNPS2 conforms to these delays and the behavioral simulation is shown in Figure 13, in which variable cout1 is the edge detection signal that can detect the rising edge of cont. Be ware that b i (i = 1, 2, 3, 4) gets value instantly for the neglect of delays in behavioral simulation.

Synthesize GNPS2 and open the synthesized model, complete constraints design and debug core set as that of GNPS1, then implement GNPS2. Run

Hardware consumption for GNPS2 is summarized in Table 2. It can be seen that GNPS2 utilizes a little bit more resources than GNPS1 for its more complicate logic. At last, perform hardware debug to verify that FPGA hardened GNPS2 obtained correct outcomes, shown in Figure 16.

Finally, the last tests were performed using an autonomous execution of the system without output and using distributed read-only memory data storage for the input. Under this setup the speed of 100Mhz (corresponding to the system clock) was achieved. This means that a GNPS model can be simulated at a speed of 10 8 steps per second. We would like to remark that in real-use cases the reaction speed will be dependent on the input/output delay. It is pointed out that the input/output circuits are not system-specific and can be reused for different simulations. However, at the present state they need to be integrated manually in the final hardware design. At this moment, the development of FPNtoVerilog continues in order to integrate the automatic generation of input/output modules. This will allow a generation of a hardware circuit directly from the GNPS specification, without any user intervention.

CONCLUSIONS

In this work we provide an efficient custom FPGA-based hardware architecture design for the implementation of generalized numerical P systems. This allows a high-speed simulation of the corresponding system as well as a direct on-chip handling of input and output data. This architecture has a solid theoretical basis based on GNPS systems and consumes a low number of ressources. However, at the moment a manual intervention during the design process is still necessary and we concentrate our future effort on the complete authorization of the design. In perspective, this will allow to create custom chips performing a control function and handling their input/output in an automated manner, without having any hardware programming knowledge. The method for FPGA implementation of GNPS is proposed and verified. Results show that it is feasible to implement GNPS on FPGA to exploit their parallelism to speedup computations. Comparing to software simulation of GNPS, the speedup achieved is an order of 10 4 . Obviously, the hardware architecture of GNPS is a parallel one. However, the fact that programs and membranes worked as processing units to handle tasks inspires us that a FPGA hardened GNPS seems to be a heterogeneous multicore processor for different programs and membranes working in parallel have different functions. As a consequence, this heterogeneous architecture defined by P systems differing from current architectures can be applied in computation intensive

  and synchronous programming languages like Esterel and Lustre. We considered several test cases and in this paper we present two of them, describing implementation results of Sobel image edge detection algorithm on Diligent Basis 3 FPGA board, based on Xilinx Artix 7 architecture. This paper is organized as follows: Section 2 introduces the definition of NPS and ENPS, analyzing their relations with systems of difference equations, together with the definition of a normal form of (E)NPS. Section 3 extends the original definition of NPS to GNPS. Section4 discusses the FPGA implementation of two case studies in detail. Conclusions are drawn in Section 5.

6 aFIGURE 2

 62 FIGURE 2 Enzymatic numerical P system from Example 2.2.

FIGURE 3 P

 3 FIGURE 3 P system from Example 4.1.

  module A # ( parameter WIDTH = 3 2 , parameter BPPOS = 1 2 ) ( o u t p u t [WIDTH : 0 ] o u t 1 , o u t p u t [WIDTH : 0 ] o u t 2 , i n p u t [WIDTH : 0 ] i n 1 , i n p u t [WIDTH : 0 ] i n 2 , i n p u t c l k ) ; r e g [WIDTH : 0 ] a = 0 ; r e g [WIDTH : 0 ] b = 0 ; r e g [WIDTH : 0 ] c = 0 ; r e g [WIDTH : 0 ] d = 0 ; a l w a y s @( p o s e d g e c l k ) b e g i n a <= m u l t ( i n 1 + i n 2 , 2 0 4 8 ) ; b <= a ; o u t 1 <= a-b ; o u t 2 <= a-b < 4 && a-b > -4 ? 0 : 4 0 9 6 ; end endmodule

Figure 5 .

 5 Before proceeding to the next processing step, RTL model should be evaluated to verify that it dose possess required functions/behaviors. A test file named testbench is designed to perform such verification work. In the testbench, a model should be instantiated at first. Then input data/impulse to its input ports of the model and analyze outcomes represented in the form of timing waveforms. It is notable that outcomes are simulated by software (Vivado for our case, this simulation process is called Behavioral Simulation in Vivado.) instead of FPGA. In this research, two case study models are constructed as sequential circuits, namely clock is involved as a metronome to synchronize operations. Rising edge of clock is the trigger signal, i.e., operations can only be carried out after a rising edge and variables hold their values until next rising edge. The period of clock is set to 10 ns in the test bench. Behavioral simulation omits any gate delays and data path delays, which means that results are output instantaneously, at the same time of trigger edge for sequential circuits [3] and the change instant of signals for combinational circuits. GNPS1's behavioral simulation timing waveform is shown in Figure 6, from which we can see that b 1 , b 3 , b 4 get their outcomes after the first rising edge of clock. While the result of b 2 arises after the second rising edge, for the sum of two operands should be computed in the first period of clock and during that time the initial value for variable to be radicated is zero. Here we resort to corresponding Vivado IP core to compute the square root [42].

FIGURE 5

 5 FIGURE 5 Block diagram of GNPS1 RTL model. After the design of a model, the corresponding schematic can be drawn automatically by Vivado. The schematic characterizes the same functions/behaviors as RTL model representing by HDL.

FIGURE 7

 7 FIGURE 7Post implementation timing simulation of GNPS1. Port b1, b3 and b4 obtain their steady output value after the eleventh rising edge of clock, indicating it costs 110 ns to get results. For b1, its steady output value emerges after sixteenth rising edge, costing 160 ns to compute outcome.

FIGURE 8

 8 FIGURE 8Software simulation of GNPS1. It is assumed that GNPS1 evolves one step to stop.Here we can see that there is no one-to-one correspondence between a clock cycle and a GNPS step. For complex arithmetic computations, one step of GNPS requires more than one clock cycle.

  × 100% ≈ 1.225765 × 10 -3

mark b 3

 3 and b 4 in Verilog codes as debug signals. After programing device, the integrated logic analyzer window open automatically. Debug signals should be added into the window manually, then run debug to exhibit values computed by FPGA, shown in Figure 10.

  post implementation timing simulation to check the timing situation, as shown in Figure 14. PeP simulation of GNPS2 shown in Figure 15, so the speedup of FPGA implementation is computed in Equation 11. Power consumption of GPNS2 is given in Figure 9 (b).9.306 × 10 6 480 ≈ 1.9388 × 10 4

FIGURE 11 GNPS

 11 FIGURE 11 GNPS model for case study 2 is numbered as GNPS2. The equations inside are the core computations of Sobel image edge detection algorithm. GNPS2 has 5 membranes and evolves 4 steps to reach halt condition. Programs in each membrane compute concurrently while each membrane execute serially.

FIGURE 12

 12 FIGURE 12 Block diagram of GNPS2 RTL model. Each membrane is modeled in Verilog basic functional unit, module. The bug icons indicate variables to be debugged in Hardware Debug procedure.

FIGURE 13

 13 FIGURE 13 Behavioral simulation of GNPS2. At a rising edge of clock, if couti = 1, execute programs in the next membrane. Three edge detection signal appears at 105 ns, 265 ns and 375 ns respectively and lasts 10 ns.

(a) Variable b 1

 1 gets its value in the 11th cycle, while b 2 obtains its value in the 25th cycle. (b) Variable b 3 gets its value in the 38th cycle, while b 4 obtains its value in the 49th cycle.

FIGURE 14

 14 FIGURE 14Post implementation timing simulation of GNPS2. The real timing of b2 and b3is a little different than expected.

FIGURE 15

 15 FIGURE 15Software simulation of GNPS2. It is assumed that GNPS1 evolves four steps to stop. CPU of the host computer costs 0.009306 s to get results.

  (a) Hardware debug of b 1 (connect to b 11 ) and b 2 (connect to b 12 ). (b) Hardware debug of b 3 and b 4 .

FIGURE 16

 16 FIGURE 16 Hardware debug of GNPS2. Values are represented in hexadecimal, 070cd is 28877 in decimal. 28877 ÷ 2 11 = 14.10009765625, which is the value of b1. 17c5 is 6085 in decimal, 6085 ÷ 2 11 = 2.97119140625, which is the value of b2.

  Another important point is to test the obtained chips by using them to directly perform the robot control. Input/output from serial ports, leds and switches on the BASYS 3 board were tested to build data transmission channels. We have made several preliminary investigations showing the possibility to acquire sensor data and to send control signals to the Pioneer 3 DX wheeled robot from the GNPS controller. A further development would need to design the corresponding interface and accompanying functions in the compiler. We have also shown that the underlying model is extremely close to well known circuit design abstractions based on Mealy/Moore automata. This can give a new research direction investigating the links between GNPS and synchronous programming languages.

TABLE 1

 1 Hardware resources utilization of GNPS1 FIGURE 10 Hardware debug of GNPS1. Input variables cannot be debugged so there is no clock signal. Values are represented in hexadecimal, 01f66 is 8038 in decimal. 8038÷2 11 = 3.9248046875, which is the value of b4. 005eb is 1515 in decimal, 1515 ÷ 2 11 = 0.73974609375, which is the value of b3. to sequential mode, resulting GNPS2, depicted in Figure 11. Conditional rule of membrane 1 is met at beginning, so program P r 11 and P r 21 take place simultaneously. Other conditional rules are not met for the initial value of conditional variables are zeros. After P r 21 modifying the value of e 2 from 0 to 3, programs in membrane 2 are triggered to execute. So does membrane 3 and 4. It is worth to note that P r 21 and P r 22 consume e 1 and e 2 so each program can only execute once. In short, a train-type ignition method is used to control the execution sequence of programs. Input GNPS2 to FPNtoVerilog obtaining the RTL model. The block diagram of GNPS2 RTL model is illustrated in Figure 12. Edge detection is used as the trigger signal to impel the next membrane to execute. By this way, the train-type ignition is realized. According to Figure 7 reflecting actual timing condition of each program, program P r 11 , P r 13 , P r 14 cost 11 clock cycles to compute while P r 12 needs 16 clock cycles. The testbench designed

TABLE 2 Hardware

 2

resources utilization of GNPS2

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (61972324, 61672437, 61702428), by Beijing Advanced Innovation Center for Intelligent Robots and Systems (2019IRS14), Artificial Intelligence Key Laboratory of Sichuan Province (2019RYJ06) and the Sichuan Science and Technology Program (2018GZDZX0043, 2018GZ0185, 2018GZ0086).