W Alejandro Apaza-Perez

Antoine Girard
email: antoine.girard@l2s.centralesupelec.fr

Compositional Synthesis of Symbolic Controllers for Attractivity Specifications

Attractivity specifications consist in driving the state of a system to a target region and to keep it in that region afterwards. In this paper, we develop a compositional approach to symbolic controller synthesis for attractivity specifications. The approach consists in computing iteratively for each subsystem, refinements of the least-violating attractivity controller, and of the associated attractor. The controllers and attractors computed at a given iteration are used at the next iteration as control and external input constraints. The resulting fixed-point algorithm allows us to compute a decentralized attractivity controller for the interconnected system which minimizes the size of the attractor. To illustrate the effectiveness of our approach, we show an application for the temperature regulation of adjacent rooms of a building.

I. INTRODUCTION

Symbolic control is a computational approach to controller synthesis for nonlinear systems (see e.g. [START_REF] Tabuada | Verification and Control of Hybrid Systems[END_REF], [START_REF] Belta | Formal Methods for Discrete-Time Dynamical Systems[END_REF] and the references therein). The main concept of symbolic control is that of the symbolic model, also called discrete abstraction, which is a finite state/input approximation of a general continuous dynamical system. When a symbolic model is related to the original system by some formal behavioral relationship such as alternating simulation [START_REF] Tabuada | Verification and Control of Hybrid Systems[END_REF] or feedback refinement [START_REF] Reissig | Feedback refinement relations for the synthesis of symbolic controllers[END_REF], controllers designed for the symbolic model can be refined into controllers for the original system. This makes it possible to recourse to automatic controller synthesis techniques for finite state dynamical systems to synthesize controllers for continuous systems (see e.g. [START_REF] Tabuada | Verification and Control of Hybrid Systems[END_REF], [START_REF] Belta | Formal Methods for Discrete-Time Dynamical Systems[END_REF]). Symbolic control has several advantages. Firstly, it can be applied to very general classes of continuous systems with state and input constraints and subject to bounded disturbances. Secondly, it enables the automatic synthesis of controllers that are "correct-by-design" for various type of specifications such as safety, reachability, attractivity or more complex properties such as those described in Linear Temporal Logic; it also makes it possible to optimize certain performance criteria.

However, the symbolic control approach is subject to the curse of dimensionality and these techniques are effective mostly on low-dimensional dynamical systems. In the past decade, several compositional approaches have been proposed to tackle these scalability issues. The problem of computing a symbolic model for a system by abstracting its components individually has been investigated in [START_REF] Tazaki | Bisimilar finite abstractions of interconnected systems[END_REF], [START_REF] Pola | Symbolic models for networks of control systems[END_REF], [START_REF] Meyer | Compositional abstraction and safety synthesis using overlapping symbolic models[END_REF], [START_REF] Mallik | Compositional synthesis of finite-state abstractions[END_REF], [START_REF] Swikir | Compositional synthesis of finite abstractions for networks of systems: A small-gain approach[END_REF]. The main challenge though is arguably to design compositional approaches to controller synthesis. With that respect, assume-guarantee contracts offer an appealing theoretical framework [START_REF] Benveniste | Contracts for system design[END_REF] where a global system specification is decomposed in contracts specifying the obligations of a component under assumptions on the behavior of other components. Several approaches for compositional synthesis of symbolic controllers make use of such contracts [START_REF] Kim | Dynamic contracts for distributed temporal logic control of traffic networks[END_REF], [START_REF] Meyer | Compositional abstraction refinement for control synthesis[END_REF], [START_REF] Eqtami | A quantitative approach on assumeguarantee contracts for safety of interconnected systems[END_REF], [START_REF] Saoud | Contract-based design of symbolic controllers for safety in distributed multiperiodic sampleddata systems[END_REF]. Some other approaches are based on generalizations of the celebrated small-gain theorem [START_REF] Dallal | On compositional symbolic controller synthesis inspired by small-gain theorems[END_REF], [START_REF] Pola | Symbolic models for networks of control systems[END_REF], [START_REF] Apaza-Perez | Abstractionbased low complexity controller synthesis for interconnected nondeterministic systems[END_REF], [START_REF] Apaza-Perez | On distributed symbolic control of interconnected systems under persistency specifications[END_REF]. A third possibility is to start with partial synthesis of controllers on the components, to compute the composition and then to refine the composed controller to enforce the specification at the system level [START_REF] Saoud | Compositional abstraction-based synthesis for interconnected systems: An approximate composition approach[END_REF].

In this paper, we propose an approach for synthesizing compositionally attractivity controllers. Attractivity specifications consist in driving the state of a system to a target region and to keep it in that region afterwards. Compared to the approaches mentioned above, our approach is iterative and can be interpreted in the assume-guarantee framework as constructing a sequence of assume-guarantee contracts with increasingly strong assumptions and guarantees, captured here by the attractors of the system's components. The proposed approach extends the recent work of [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF], which allows to synthesize "least-violating" controllers, to the compositional framework of attractivity controller synthesis. Our approach thus consists in computing iteratively for each subsystem, refinements of the least-violating attractivity controller and of the associated attractor. The controllers and attractors computed at a given iteration are used at the next iteration as control and external input constraints. Then, the resulting fixed point algorithm allows us to compute a decentralized attractivity controller for the interconnected system. The closest work to the approach presented in this paper is [START_REF] Eqtami | A quantitative approach on assumeguarantee contracts for safety of interconnected systems[END_REF], where an iterative approach based on leastviolating safety controllers is proposed to design compositionally decentralized safety controllers. In the current work, we address a different type of specifications, namely attractivity. Another difference is that in the current work, we build a sequence of controllers that iteratively improves the behavior of the closed-loop system, while in [START_REF] Eqtami | A quantitative approach on assumeguarantee contracts for safety of interconnected systems[END_REF] a controller is only obtained after all iterations have finished.

The structure of this paper is as follows. In Section II, we provide the necessary theoretical background and provide a formal problem formulation. In Section III, we briefly recall the main results of [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF] on the synthesis of least-violating attractivity controllers. Section IV contains the main contribution of the paper, which is an algorithm to synthesize compositionally a decentralized attractivity controller for an interconnected system. Finally, in Section V, a numerical example of temperature regulation in a building is used to illustrate our approach.

Notation: R, R + 0 and N denote the sets of real, nonnegative real and natural numbers, respectively. For K ∈ N ∪ {+∞}, we define the following sets of integers N <K = {k ∈ N| k < K} and N ≤K = {k ∈ N| k ≤ K}. R denotes the set of extended real numbers, i.e. R = [-∞, +∞]. Given two sets A and B, a set-valued map f : A ⇒ B is a map from A to the set of subsets of B.

II. PROBLEM STATEMENT

In this section, we provide a formal statement of the problem under consideration.

A. Transition systems

We consider a class of transition systems formally defined as follows:

Definition 1: A transition system Σ is a quadruple Σ = (X, D,U, F) where X is the set of states; D is the set of external inputs; U is the set of control inputs; and F is the transition map F : X × D ×U ⇒ X.

A transition system with no external inputs can be written formally in the form of Definition 1 with a set of external inputs consisting of a unique dummy external input. In that case, with an abuse of notation, we denote Σ = (X,U, F).

A transition x ∈ F(x, d, u) means that Σ can evolve from state x to state x under external and control inputs d and u. For x ∈ X, u ∈ U, we denote F(x, D, u) = d∈D F(x, d, u). Define the set of enabled control inputs at a state x ∈ X as

enab F (x) = {u ∈ U | ∀ d ∈ D, F(x, d, u) = / 0}.
Let us remark that u is enabled at x if and only if there exist successors for all possible values of the external input. If enab F (x) = / 0 then x is said to be blocking. Within the framework of transition systems, we can define (memoryless state-feedback) controllers as follows:

Definition 2: A controller for system Σ is a set-valued map C : X ⇒ U such that C(x) ⊆ enab F (x), for all x ∈ X.
Closed-loop trajectories are then defined as follows:

Definition 3: A sequence (x t ) T t=0 , where T ∈ N ∪ {+∞}, x t ∈ X, for t ∈ N ≤T , is called a closed-loop trajectory of system Σ with controller C if and only if ∀t ∈ N <T , ∃d t ∈ D, x t+1 ∈ F(x t , d t ,C(x t )). A trajectory is called maximal if either T = +∞ or C(x T ) = / 0, it is complete if T = +∞.
The set of maximal closedloop trajectories starting from a given initial state x 0 ∈ X is denoted by T max (Σ,C, x 0 ).

In this paper, we consider uniform attractivity specifications, which are defined as follows:

Definition 4: X * ⊆ X is said to be uniformly attractive from initial state x 0 ∈ X for system Σ and controller C if there exists T 0 ∈ N, such that all maximal trajectories (x t ) T t=0 ∈ T max (Σ,C, x 0 ) are complete and satisfy x t ∈ X * , for all t ≥ T 0 .

The term uniform refers to the fact that the time bound T 0 after which the state remains in the target set X * is the same for all trajectories in T max (Σ,C, x 0 ). A discussion on the difference between uniform attractivity and (non-uniform) attractivity can be found in [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF]. In particular, it should be noted that uniform attractivity cannot be captured using specification formalisms such as linear temporal logic.

B. Interconnected systems

In this paper, we focus on interconnected transition systems. For the sake of simplicity, we consider systems consisting of two interconnected components but the approach can readily be extended to systems with an arbitrary number of interconnected components.

Hence, let us consider two components given by transition systems

Σ i = (X i , D i ,U i , F i ), i = 1, 2, with D 1 = X 2 and D 2 = X 1 .
The interconnected system is then described by a transition system with no external input Σ = (X,U, F),

where X = X 1 × X 2 , U = U 1 ×U 2 and for all x = (x 1 , x 2 ) ∈ X, u = (u 1 , u 2 ) ∈ U F(x, u) = F 1 (x 1 , x 2 , u 1 ) × F 2 (x 2 , x 1 , u 2 ). Let us consider controllers C i : X i ⇒ U i for components Σ i , i = 1, 2.
We define the associated decentralized controller for Σ, C : X ⇒ U given for all x = (x 1 , x 2 ) ∈ X by

C(x) = C 1 (x 1 ) ×C 2 (x 2 ). Let us remark that C(x) ⊆ enab F 1 (x 1 ) × enab F 2 (x 2 ) ⊆ enab F (x), so C satisfies Definition 2.
Let us consider a target set X * = X * 1 × X * 2 where X * i ⊆ X i , i = 1, 2. We now formally define the problem under consideration in this paper:

Problem 5: Synthesize controllers C i : X i ⇒ U i for com- ponents Σ i , i = 1, 2,
and a set of initial states B ⊆ X, such that X * is uniformly attractive for system Σ and controller C from all initial states x 0 ∈ B.

In the following, we present a compositional approach to solve Problem 5. Throughout the paper, we assume that the transition systems Σ i are symbolic, i.e. X i and U i are finite sets for i = 1, 2. However, let us remark that our approach can also be used to synthesize controllers for systems with infinite sets of states and inputs by resorting to abstraction techniques (see e.g. [START_REF] Tabuada | Verification and Control of Hybrid Systems[END_REF], [START_REF] Reissig | Feedback refinement relations for the synthesis of symbolic controllers[END_REF]), as shown in Section V.

III. LEAST-VIOLATING ATTRACTIVITY CONTROLLERS

In this section, we briefly recall results of [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF] on the synthesis of attractivity controllers. While the results in [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF] are stated for systems without external inputs, it is straightforward to extend them to this setting.

Consider a component Σ i = (X i , D i ,U i , F i ), i = 1, 2, and a target set X * i . We assume that X i is equipped with a metric

d i , let us consider function h i : X i → R ≥0 such that h i (x i ) = 0 if and only if x i ∈ X * i . An example of such function is h i (x i ) = min x i ∈X * i d(x i , x i ).
Let us consider the following sequence of dynamic programming fixed-point iterations:

W 0 i,S (x i ) = h i (x i ), (1)
W k+1 i,S (x i ) =        max h i (x i ), min u i ∈enab F i (x i ) max x + i ∈F i (x i ,D i ,u i ) W k i,S (x + i ) if enab F i (x i ) = / 0; +∞ if enab F i (x i ) = / 0; (2) for x i ∈ X i , k ∈ N.
We denote the fixed-point of ( 1)-( 2) by W * i,S . Then, let

W 0 i,A (x i ) = W * i,S (x i ), (3)
W k+1 i,A (x i ) =        min W * i,S (x i ), min u i ∈enab F i (x i ) max x + i ∈F i (x i ,D i ,u i ) W k i,A (x + i ) if enab F i (x i ) = / 0; W * i,S (x i ) if enab F i (x i ) = / 0; (4) for x i ∈ X i , k ∈ N.
We denote the fixed-point of ( 3)-( 4) by

W * i,A .
It has been shown in [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF] that for a symbolic system Σ i , there exists

K i ∈ N, such that for all x i ∈ X i , for all k ≥ K i , W k i,S (x i ) = W * i,S (x i ) and W k i,A (x i ) = W * i,A (x i ).
Hence, W * i,S and W * i,A can be computed in practice. Then, let the function k * i : X i → N be defined as follows for all

x i ∈ X i k * i (x i ) = min{k ∈ N |W k i,A (x i ) = W * i,A (x i )},
and for δ ∈ R, let us consider the following controller for

Σ i C δ i (x i ) =                      arg min u i ∈enab F i (x i ) max x + i ∈F i (x i ,D i ,u i ) W k * i (x i )-1 i,A (x + i ) if W * i,A (x i ) ≤ δ < W * i,S (x i ); u i ∈ enab F i (x i ) max x + i ∈F i (x i ,D i ,u i ) W * i,S (x + i ) ≤ δ if W * i,S (x i ) ≤ δ ; enab F i (x i ) if δ < W * i,A (x i ).
(5) We state the following claim that will be useful in further discussions and that follows directly from ( 5) and (2):

Claim 6: For all x i ∈ X i , C δ i (x i ) = / 0 if and only if enab F i (x i ) = / 0.
We also recall the following result of [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF]:

Theorem 7: There exists T i ∈ N, such that for all x i,0 ∈ X i with W * i,A (x i,0 ) ≤ δ , all maximal trajectories (x i,t ) T t=0 ∈ T max (Σ i ,C δ i , x i,0 ) are complete and satisfy ∀t ≥ T i , W * i,S (x i,t ) ≤ δ . Hence, the set

A i = {x i ∈ X i | W * i,S (x i ) ≤ δ } is uniformly attractive for system Σ i and controller C δ i from all initial states x i,0 ∈ B i = {x i ∈ X i | W * i,A (x i ) ≤ δ }.
We refer to A i as the attractor and to B i as the basin of attraction.

It follows from

h i (x i ) ≤ W * i,S (x i ) that A i ⊆ {x i ∈ X i | h(x i ) ≤ δ }. Hence, if δ = 0, then A i ⊆ X *
i and X * i is also uniformly attractive for system Σ i and controller C δ i from all initial states x i,0 ∈ B i . It is shown in [START_REF] Girard | Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications[END_REF] that the controller C δ i is least-violating in the sense that W * i,A (x i,0 ) ≤ δ if and only if there exists a controller for Σ i making the set {x i ∈ X i | h(x i ) ≤ δ } uniformly attractive from x i,0 . Hence, the value of W * i,A (x i,0 ) gives us the smallest neighborhood of X * i which we can attract the trajectories of Σ i to.

We end the section by pointing out that the controller C δ i is set-valued and may provide more than one input, in particular for states belonging to the attractor.

IV. COMPOSITIONAL SYNTHESIS FOR ATTRACTIVITY

In this section, we present an approach for synthesizing compositionally controllers solving Problem 5. The main idea of our approach is to compute iteratively refinements of least violating attractivity controllers defined in the previous section, resulting in a nested sequence of attractors eventually contained in the target set. Let us consider two components

Σ i = (X i , D i ,U i , F i ), i = 1, 2 with D 1 = X 2 and D 2 = X 1 .

A. Initialization of controllers

For i = 1, 2, we start by computing W * ,0 i,S and W * ,0 i,A , the fixed-point of equations ( 1)-( 2) and ( 3)-(4), respectively, for system Σ i . Then, let us define

0 i = min x i ∈X i W * ,0 i,S (x i ) (6)
A 0 i = x i ∈ X i | W * ,0 i,S (x i ) ≤ 0 i (7)
B i = x i ∈ X i | W * ,0 i,A (x i ) ≤ 0 i (8)
and let C 0 i be the controller for Σ i defined as in [START_REF] Dallal | On compositional symbolic controller synthesis inspired by small-gain theorems[END_REF]

for δ = 0 i . Let A 0 = A 0 1 × A 0 2 , B = B 1 × B 2
and let the decentralized controller for Σ be given for all

x = (x 1 , x 2 ) ∈ X by C 0 (x) = C 0 1 (x 1 ) ×C 0 2 (x 2
). Proposition 8: There exists T 0 ∈ N, such that for all x 0 ∈ B, all maximal trajectories (x t ) T t=0 ∈ T max (Σ,C 0 , x 0 ) are complete and satisfy x t ∈ A 0 , for all t ≥ T 0 .

Proof: Let T i ∈ N be given as in Theorem 7 for δ = 0 i , i = 1, 2 and T 0 = max(T 1 , T 2 ). Let us consider x 0 ∈ B and a maximal trajectory (x t ) T t=0 ∈ T max (Σ,C 0 , x 0 ) where x t = (x 1,t , x 2,t ). We can verify that (x i,t ) T t=0 is a closedloop trajectory of Σ i with controller C 0 i , i = 1, 2. Moreover, maximality of (x t ) T t=0 implies that (x 1,t ) T t=0 or (x 2,t ) T t=0 is maximal. Let us assume without loss of generality that (x 1,t ) T t=0 is maximal (the other case is symmetric). Since W * ,0 1,A (x 1,0 ) ≤ 0 1 , it follows from Theorem 7 that (x 1,t ) T t=0 is complete and that W * ,0 1,S (x 1,t ) ≤ 0 1 , for all t ≥ T 1 . Completeness of (x 1,t ) T t=0 implies that T = +∞ and therefore (x 2,t ) T t=0 is also complete and thus maximal. Since W * ,0 2,A (x 2,0 ) ≤ 0 2 , it follows from Theorem 7 that W * ,0 2,S (x 2,t ) ≤ 0 2 , for all t ≥ T 2 . Hence, (x t ) T t=0 ∈ T max (Σ,C 0 , x 0 ) is complete and x t ∈ A 0 , for all t ≥ T 0 .

Let us remark that if A 0 ⊆ X * , then the controllers C 0 i , i = 1, 2, solve Problem 5. If this is not the case, we can try to refine these least-violating attractivity controllers. Indeed, the controllers C 0 i are set-valued and we can exploit this nondeterminism for further design to improve the closed-loop behavior of the system. In particular, knowing that after some time the states of the components are in the attractor A 0 , we can refine the controller C 0 i by synthesizing a least violating attractivity controller for component Σ i when its external input belongs to the attractor of the other component. This procedure can then be repeated iteratively, as shown in the next section.

B. Iterative refinement of controllers

In the following, we use the following notational convention 1 c = 2 and 2 c = 1.

For α ∈ N, starting from α = 0, for i = 1, 2, let us define the systems

Σ α+1 i = (X i , A α i c ,U i , F α+1 i )
where

F α+1 i (x i , d i , u i ) = F i (x i , d i , u i ) if u i ∈ C α i (x i ); / 0 if u i / ∈ C α i (x i ). (9)
Intuitively, Σ α+1 i describes the dynamics of component Σ i where external inputs are constrained by the attractor A α i c and the control inputs are constrained by the controller C α i . We then compute least violating attractivity controllers for these systems. Let W * ,α+1 i,S and W * ,α+1 i,A be the fixed-point of equations ( 1)-( 2) and ( 3)-( 4), respectively, for system Σ α+1 i . Then, let us define

α+1 i = max x i ∈A α i W * ,α+1 i,A (x i ) (10)
A α+1 i = x i ∈ A α i |W * ,α+1 i,S (x i ) ≤ α+1 i ( 11
)
and let C α+1 i be the controller for Σ α+1 i defined as in (5) for δ = α+1 i . Claim 9: For i = 1, 2, α ∈ N, the following hold:

• A α+1 i ⊆ A α i , • for all x i ∈ X i , C α+1 i (x i ) ⊆ C α i (x i ), • for all x i ∈ X i , C α+1 i (x i ) = / 0 if and only if C α i (x i ) = / 0.
Proof: The first item is a direct consequence of [START_REF] Meyer | Compositional abstraction and safety synthesis using overlapping symbolic models[END_REF]. From (9), we have that enab

F α+1 i (x i ) = C α i (x i ).
Then, the second item comes from (5) that gives

C α+1 i (x i ) ⊆ enab F α+1 i (x i ).
The third item comes from Claim 6 that gives C α+1 i

(x i ) = / 0 if and only if enab F α+1 i (x i ) = / 0. Let A α+1 = A α+1 1 × A α+1 2
, and let the decentralized controller for Σ be given for all

x = (x 1 , x 2 ) ∈ X by C α+1 (x) = C α+1 1 (x 1 ) ×C α+1 2 (x 2 )
. We can state the following result:

Theorem 10: For α ∈ N, there exists T α ∈ N, such that for all x 0 ∈ B, all maximal trajectories (x t ) T t=0 ∈ T max (Σ,C α , x 0 ) are complete and satisfy x t ∈ A α , for all t ≥ T α .

Proof: We proceed by induction. The case α = 0 corresponds to Proposition 8. Then, let us assume that the property is true for some α ∈ N, that is that there exists T α ∈ N, such that for all x 0 ∈ B, all maximal trajectories (x t ) T t=0 ∈ T max (Σ,C α , x 0 ) are complete and satisfy x t ∈ A α , for all t ≥ T α . From Claim 9, we have for all x ∈ X, C α+1 (x) ⊆ C α (x) and C α+1 (x) = / 0 if and only if C α (x) = / 0, then it follows that for all x 0 ∈ B, all maximal trajectories (x t ) T t=0 ∈ T max (Σ,C α+1 , x 0 ) are complete and satisfy

x t ∈ A α , for all t ≥ T α . Let us denote x t = (x 1,t , x 2,t ), then for all t ≥ T α , x i,t ∈ A α i , i = 1, 2. For i = 1, 2, we can then verify that (x i,t ) T t=T α is closed-loop trajectory of Σ α+1 i with controller C α+1 i . Moreover, x i,T α ∈ A α i implies by (10) that W * ,α+1 i,A (x i,T α ) ≤ α+1 i , therefore it follows that W * ,α+1 i,S (x i,t ) ≤ α+1 i
, for all t ≥ T α + T i , where T i is the uniform time bound given in Theorem 7. Letting T α+1 = T α + max(T 1 , T 2 ) we get from (11) that x t ∈ A α+1 , for all t ≥ T α+1 .

Our approach thus computes successive refinements of least-violating controllers C α i associated to a sequence of nested attractors A α . This allows us to build attractors that get closer to the target set X * , and that may eventually be contained in X * .

C. Algorithm

Our overall approach to synthesize decentralized attractivity controllers compositionally can be summarized in Algortihm 1.

We can now state the main result of the paper:

Theorem 11: Algorithm 1 terminates in a finite number of steps and the following properties hold: Algorithm 1: Compositional synthesis of attractivity controllers Input: Components Σ i = (X i , D i ,U i , F i ); target sets X * i ⊆ X i , i = 1, 2; Output: Controllers C i , i = 1, 2; attractor A; basin of attraction B; 1 A -1 := X 1 × X 2 ; α := 0; // Initialization 2 for i=1,2 do 3 Compute W * ,0 i,S and W * ,0 i,A , the fixed-points of (1)-( 2) and ( 3)-(4) for system Σ i ; 4 Compute 0 i , A 0 i , B i given by ( 6), ( 7), (8);

5
Compute C 0 i the controller for Σ i given by ( 5) with δ = 0 i ; 6 A 0 := A 0 1 × A 0 2 ; B := B 1 × B 2 ;

// Iterative refinement , the fixed-points of ( 1)-( 2) and ( 3)-(4) for system Σ α+1

7 while ¬(A α ⊆ X * 1 × X * 2 or A α = A α-1 ) do 8 for i=1,2 do 9 Compute system Σ α+1 i = (X i , A α i c ,U i , F α+1 i ) where F α+1
15 C i = C α i , i = 1, 2; 16 A := A α ;
• A is uniformly attractive for system Σ and controller C from all initial states x 0 ∈ B. • If A ⊆ X * , C 1 and C 2 are solutions to Problem 5.

Proof: From Claim 9, we get that A α ⊆ A α-1 for all α ≥ 0. Then since the transition systems are symbolic, X 1 × X 2 is a finite set. Hence, there exists α ≥ 0 such that A α = A α-1 . It follows that the while loop terminates after a finite number of iterations. Hence, Algorithm 1 terminates in a finite number of steps. Then, the first item is a direct consequence of Theorem 10 and the second item follows from the first item.

V. NUMERICAL

In this section, we show an illustrative application of Algorithm 1 to a model of temperature regulation in a four room building. We model the building as an interconnected systems of two components, each component consisting of two rooms. The continuous-time dynamics is given by: ẋ1

= αA(x 1 + x 2 ) -2αx 1 + β (T e 1 -x 1 ) + Γ(u 1 )(T h 1 -x 1 ) ẋ2 = αA(x 1 + x 2 ) -2αx 2 + β (T e 1 -x 2 ) + Γ(u 2 )(T h 2 -x 2 ) (12)
where The continuous-time dynamics of ( 12) is periodically sampled with period 5 minutes. We impose the state constraint T i ∈ [16, 24], i = 1, . . . , 4 and the control objective is to stabilize the temperature of all rooms in the intervals [19, 21]. We compute symbolic abstractions of both components using the approach described in [START_REF] Meyer | Compositional abstraction and safety synthesis using overlapping symbolic models[END_REF]. For that purpose, we use uniform partitions of the state intervals [START_REF] Swikir | Compositional synthesis of finite abstractions for networks of systems: A small-gain approach[END_REF]24] in 100 subintervals and uniform discretizations of the input intervals [0, 1] with 5 elements. Each component has then 10 4 states and 25 inputs.

x 1 = [ T 1 T 2 ]
We use Algorithm 1 to synthesize an attractivity controller for the system. The algorithm stops after 6 iterations of the main loop after the computed attractor is contained in the target set [19, 21] 4 . The total computation time is 45 seconds, with 13 seconds spent on computing abstractions of components, 11 seconds spent on initializing the controller and 22 seconds spent in refining the controller iteratively. We show on Figure 1 the sets that have been computed, where the black region represents uncontrollable states (i.e. states without guarantees of ever reaching the attractors), the blue region is the basin of attraction and the other regions are the sequence of attractors with the final attractor represented in yellow and contained in the target set in red.

Assuming initial temperatures x 1 (0) = [16 23] T and x 2 (0) = [16.5 22.5] T for the rooms, Figure 1 shows the trajectories of the room's temperatures in the state space with respect to the time, while applying the attractivity controllers obtained from Algorithm 1.

VI. CONCLUSIONS

In this work, we presented a novel compositional approach to the synthesis of decentralized attractivity controllers. While the paper focuses on systems with two components, the approach can easily be extended to an arbitrary number of components. The approach is based on iterative refinements of least-violating attractivity controllers and of the associated attractors. We presented an algorithm for systems with finite state and input spaces and we have shown, using an example, how our approach can also be used to design controllers for systems with infinite state and input spaces using symbolic abstractions. Future work will focus on extending the approach to other types of specifications and to other types of interconnected systems such as systems with overlapping components as in [START_REF] Meyer | Compositional abstraction and safety synthesis using overlapping symbolic models[END_REF], [START_REF] Saoud | Contract-based design of symbolic controllers for safety in distributed multiperiodic sampleddata systems[END_REF].

i

  is given by (9); 10 Compute W * ,α+1 i,S and W * ,α+1 i,A

13 A 1 × A α+1 2 ; 14 α

 131214 α+1 := A α+1 := α + 1;

T and x 2 = [ T 3 T 4 ]

 234 T represent the temperatures in the rooms 1, 2, 3 and 4; T e is the temperature of the external environment of the building; T h 1 = [ Th 1 Th 2 ] T and T h 2 = [ Th 3 Th 4 ] T represent the maximal temperature of the heaters in the rooms; u 1 = [ ũ1 ũ2 ] T and u 2 = [ ũ3 ũ4 ] T are control inputs representing the heating power used in the rooms with 0 corresponding to no heating and 1 corresponding to full power. A = 0 1 1 0 represents the interconnections between rooms,1 = [ 1 1 ] T , Γ(v) = γ ṽ1 0 0 γ ṽ2 where v = [ ṽ1 ṽ2 ] T . α,β and γ are the conduction factors between the rooms, between the external environment and the rooms and between the heaters and the rooms, respectively. The values of α, β , γ are respectively 1/20, 1/200, 1/100; T e = 10 • C, Th 1 = 50 • C, Th 2 = 80 • C, Th 3 = 70 • C, Th 4 = 100 • C.

2 Fig. 1 .

 21 Fig. 1. (Left and middle) Sets computed by Algorithm 1, black region represents uncontrollable states (i.e. states without guarantees of ever reaching the attractors), the blue region is the basin of attraction and the other regions are the sequence of attractors with the final attractor represented in yellow and contained in the target set in red. (Right) Simulations (state and input evolutions) of the closed-loop system with initial condition x 1 (0) = [16 23] T and x 2 (0) = [16.5 22.5] T .

This work was supported in part by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 725144) and by the Agence Nationale de la Recherche (ANR) under Grant HANDY ANR-18-CE40-0010.


Salima Bouarfa

Philippe Jéhan

William Erb

Olivier Mongin

François-Hugues Porée

Thierry Roisnel

Ghenia Bentabed-Ababsa

Nicolas Le Yondre

Florence Mongin

Aza-aromatic polycycles based on triphenylene and acridine or acridone: synthesis and properties

Acridine-and acridone-based polycyclic aromatics were prepared by using as the key steps a copper-catalysed N-arylation of 2-aminobenzaldehyde, 2-aminophenones, or ethyl 2-aminobenzoate with 2-iodotriphenylene, and an acid-mediated cyclization. The regioselectivity of this intramolecular SEAr reaction was studied by performing Hückel theory calculations on the precursors. Due to their structural similarity with some MALDI-MS matrices, two acridine-based polycycles were evaluated for this purpose. Finally, in view of structure-properties relationships, preliminary studies of the photophysical properties of the synthesized acridine-and acridone-based polycycles were performed.

Introduction

Acridines and acridones are planar skeletons that play an important role in biologically active molecules, but also in organic materials for various applications. [1][2][3] For example, both acridine and acridone patterns have been found in fluorescent sensors 4,5 and in hole transporting materials. 6,7 Polyaromatic hydrocarbons (PAHs) are also structural units playing an important role in compounds endowed with biological and physicochemical (e.g. electronic and photophysical) properties. 8 Furthermore, the incorporation of one or more nitrogen atoms into PAHs has attracted the attention of chemists due to the impact of this heteroatom on the properties of these organic scaffolds. [9][START_REF] Miao | Polycyclic Arenes and Heteroarenes[END_REF][START_REF] Stępień | [END_REF][12] However, the development of materials based on nitrogencontaining PAHs depends on the availability of synthetic methods that can be used to access them. For example, few studies have been dedicated to the synthesis of acridine-and acridone-based PAHs with more than five rings. 13 In pioneered studies, Buu-Hoï and co-workers obtained benzo[c]and benzo[a]naphth[2,1-j]acridines from -and -naphthol, respectively, by regioselective Ullmann-Fetvadjian reaction with 2aminophenanthrene in the presence of paraformaldehyde (Figure 1, top). [START_REF] Cao | [END_REF] Similarly, benzo[c]-and benzo[a]naphth [2,3-h]acridines were respectively prepared from -and -naphthol by reaction with 1aminoanthracene while using 2-aminoanthracene furnished benzo[c]-and benzo[a]naphth [2,3-j]acridines, respectively (Figure 1, middle and bottom). 15 [2,3j]acridines already prepared. [START_REF] Cao | [END_REF]15 As evidenced by recent studies, accessing acridine-based PAHs is always a challenge. For example, Hashmi and co-workers reported in 2018 a -extending strategy towards dibenzo[a,c]acridines using gold-catalysis for C-H annulation of 2-ethynylbiaryls with anthranils. 16 We can also cite the modified iron-catalysed Povarov reaction involving 2-(2'-alkynylaryl)benzaldehydes and arylamines documented by Jana and co-workers in 2020 to access the same scaffolds 17 (Figure 2). 16,17 We recently showed that substrates such as 2aminobenzaldehyde 18 and 2-aminophenones 19 could be converted into either acridines, by N-arylation and subsequent cyclization, or acridones, by double N-arylation followed by cyclization. Inspired by these results, we here report how similar anilines can react with 2iodotriphenylene in order to deliver new PAHs based on acridine or acridone for which the photophysical properties were studied. In addition, due to their similarity with 9-aminoacridine (9AA) which is the reference matrix for negative mode MALDI, 20 two of the helicene-like compounds here prepared were evaluated for this purpose.

Figure 1 Benzo[c]-and benzo[a]naphth[2,1-j]acridines, benzo[c]-and benzo[a]naphth[2,3-h]acridines, and benzo[c]-and benzo[a]naphth

Results and discussion

Synthesis of acridine-based polycyclic aza-aromatics

In the course of a study dedicated to the synthesis of acridines, 18 we observed (as already noticed by others) 21,22 that the cyclization of both N,N'-(1,3-phenylene)bis(2-aminobenzaldehyde) and N,N'-(1,4phenylene)bis(2-aminobenzaldehyde) to dibenzophenanthrolines was regioselective, as displayed in Scheme 1. Inspired by a study on aromatic electrophilic substitution (SEAr) of condensed imidazole derivatives, 23 we applied the Fukui's concept (reaction of a compound with an electrophile at its carbon with the highest HOMO coefficient), 24 and rationalized these experimental results by performing Hückel theory calculations on A, B and C (see ESI). 25 This led us to consider the access to original helicene-like scaffolds 26 based on acridine by using a similar approach, i.e. (i) Narylation of 2-aminobenzaldehyde, or 2-aminophenones, with 2iodotriphenylene followed by (ii) acid-mediated cyclization to afford polycyclic aza-aromatics. Indeed, as shown in Scheme 2, even if the difference is less marked than before between both possible cyclization sites, an intramolecular SEAr reaction at the most hindered position is favoured under orbital or charge control. Scheme 1 Amplitudes of the highest occupied molecular orbital (HOMO) coefficients and charges (in brackets) obtained by using the HuLiS calculator 27 to rationalize the regioselective formation of dibenzophenanthrolines (R = H or Me).

Under acidic conditions, 28,29 the cyclization of the 2-(triphenylenyl)aminobenzaldehyde 1 (prepared by N-arylation 18,19,30 of 2-aminobenzaldehyde using 2-iodotriphenylene) took place regioselectively to provide the expected phenanthro[9,10-a]acridine (2) in high yield. Its structure, already established by NMR (NOESY correlation observed between H1 and H16), was confirmed by X-ray diffraction (Scheme 3). Note that our attempt to perform the cyclization under smoother conditions, by using catalytic trifluoromethanesulfonic acid (10 mol%) in 1,1,1,3,3,3-hexafluoro-2propanol 31 at room temperature overnight, also led to the polycyclic aza-aromatic 2 but in a lower 47% yield.

ARTICLE

Please do not adjust margins Please do not adjust margins In contrast, by starting from 2-aminophenones, the N-arylationcyclization sequence rather led to the phenanthro[9,10-b]acridines 4 in good yields. This was established unambiguously from the NMR spectra of 4b, owing to the NOESY correlations observed between H1 and H16, and between H8 and H9 (Scheme 4). A more important steric hindrance in the case of ketones when compared with aldehyde, hampering an orbital control of the reaction, could be advanced to rationalize the regioselectivity switch observed.

Various attempts to functionalize compound 2 at its 16-position proved unsuccessful. For example, our efforts to introduce an amino group, as reported for the conversion of acridine into its 9-amino derivative (urea and sodium hydride in dimethylsulfoxide), 32 only led to the recovery of the starting material. Similarly, our attempt to oxidize 2 into the corresponding acridone, as reported in the case of acridines (sodium hydride in dimethylsulfoxide), 33 failed (starting material recovered, even after heating at 80 °C overnight).

As an alternative way to generate 16-substituted phenanthro[9,10-a]acridines, we considered the 16-chloro derivative as starting point. This would in turn be accessible from the corresponding phenanthro[9,10-a]acridones. Indeed, compared with that of a ketone function, steric hindrance of the acyl chloride required for its synthesis should be reduced. However, our attempts to oxidize the aldehyde 1 into the corresponding carboxylic acid failed, for example by using iodine and potassium hydroxide in acetonitrile 34 (very low conversion), or with sodium chlorite in the presence of 2-methyl-2-butene and sodium dihydrogenophosphate in tert-butanol-water 35 (no reaction). Similarly, the conversion of the aldehyde into the ethyl ester 5, by using iodine and potassium hydroxide in ethanol, 34 failed (very low conversion). Thus, we selected ethyl anthranilate to perform its N-arylation 30,36 with 2-iodotriphenylene towards the diarylamine 5, isolated in 88% yield. After hydrolysis of the ester function, 37 the carboxylic acid 6 was treated by phosphorus(V) oxychloride under reflux 37 for 3 days. Under these conditions, the expected chloride 7 was obtained in 84% yield (Scheme 5).

Scheme 5 Synthesis of 16-chlorophenanthro[9,10-a]acridine (7).

The replacement of chlorine of 9-chloroacridine by an amino group is documented in the literature, and notably by employing ammonium carbonate in phenol at 120 °C. 38 However, when applied to the chloride 7, the procedure did not furnish any amino derivative, but provided instead the phenyl ether 8 (Scheme 6, top).

Then, alternative ways to access the amino derivative were tested. The treatment of 7 with an excess of ammonia in dimethylsulfoxide at 120 °C (sealed tube) did not give the amino derivative satisfactorily (starting material was mainly recovered while degradation was noticed). Reaction of 7 with allylamine (used at solvent) at 60 °C (sealed tube) afforded the allylamino derivative to some extent, but the subsequent cleavage of the allyl group Please do not adjust margins Please do not adjust margins (methanesulfonic acid and palladium on charcoal in ethanol under reflux) did not allowed the expected amino derivative to be isolated. Finally, because the aryl ether 8 was easily obtained, we attempted the reaction of 4-aminophenol with the chloride 7 in the presence of a base. By performing the reaction in dimethylsulfoxide at 100 °C, 39 the aminoether 9 was formed. However, due to difficult purification, it was isolated in a low 8% yield (Scheme 6, bottom).

To prepare polycyclic aza-aromatics based on triphenylene and NH-free acridone, we first examined the mono-N-arylation of anthranilic acid with 2-iodotriphenylene, and the subsequent cyclization using polyphosphoric acid as described in similar cyclizations, 40 but we failed in isolating a triphenylene-based acridone by following this approach. Because Zhou and co-workers succeeded in cyclizing 2-(phenylamino)acetophenone into NH-free acridone by using copper(I) iodide (0.2 equiv) in dimethylsulfoxide at 140 °C under air atmosphere, 41 we attempted a similar reaction from the ketone 3c. However, after 36 h, a complex mixture was obtained from which a NH-free acridone could not be detected.

These failures led us to rather turn to the synthesis of Nsubstituted triphenylene-based acridones. To this goal, we first attempted the cyclization of the ketones 3a and 3b in the presence of copper(I) iodide (0.2 equiv) in dimethylsulfoxide at 120 °C under air since these conditions were employed by Zhou and Deng to convert 2-(phenylamino)benzophenone into N-phenylacridone. 42 However, these conditions failed in delivering the expected products.

Consequently, we capitalized on the possible double coppercatalysed N-arylation of 2-aminobenzaldehyde we recently reported. 18 By reaction with 2-iodotriphenylene (2 equiv) in the presence of an excess of base (3 equiv) in refluxing diethyl ether for 24 h, the aldehyde 10 was obtained in 62% yield (Scheme 7, top). Next, we studied its cyclization, and first tested the acidic conditions employed to convert the compounds 3 into 4; however, degradation took place, and no acridone could be isolated.

In 2013, Yang and co-workers cyclized N-methyl-N-phenyl-2aminobenzaldehydes into N-methylacridones by recourse to catalytic ytterbium(III) triflate or scandium(III) triflate in the presence of sodium sulfate in dimethylformamide at 100 °C. 43 Inspired by this work, we applied the protocol using ytterbium(III) triflate, and observed the formation of the N-triphenylated triphenylene-based acridone 11. However, it proved difficult to purify this compound which was only isolated in a low 10% yield by recrystallization from ethyl acetate (Scheme 7, bottom). To rationalize the regioselectivity of the reaction, which probably takes place by intramolecular SEAr reaction followed by oxidation, 43 we calculated as above both amplitudes of the HOMO coefficients and charges on the aldehyde 10 (see ESI and Scheme 8, left). Albeit in favour of the most hindered position, the difference between both cyclization sites on the triphenylene ring is not very marked and steric effects might become predominant. This is probably why a helicene-like acridone was not formed in this case.

Please do not adjust margins

Please do not adjust margins Similar calculations performed on the aldehyde 12 show a not very marked difference between the phenyl and the less hindered 2triphenylenyl possible cyclization sites (see ESI and Scheme 8, right). Since there is also no peculiar difference between both sides regarding steric hindrance, a cyclization involving the phenyl ring might be expected. We therefore decided to synthesize the aldehyde 12.

To attain compound 12, it is possible to successively N-arylate 2aminobenzaldehyde by using iodobenzene and 2-iodotriphenylene. In our hands, the first N-arylation under the conditions used in Scheme 3 to prepare 1 afforded 2-(phenylamino)benzaldehyde 18 in 86% yield; however, the second N-arylation using 2-iodotriphenylene proceeded in a low 22% yield (not shown).

When we reversed the step order, and thus involved 1 and iodobenzene, we could obtain 12 more satisfactorily (Scheme 9, top). Then, the cyclization of 12 was performed as before, by using either ytterbium(III) triflate or scandium(III) triflate, and the predicted acridone 13 was isolated in 83% yield (Scheme 9, bottom) and 64% yield (not shown), respectively. It is worth noting that N-arylation of acridone by using 2-iodotriphenylene (1.5 equiv) in the presence of potassium carbonate (2 equiv), copper(I) iodide (0.20 equiv) and 2,2,6,6-tetramethyl-3,5-heptanedione (0.40 equiv) in dimethylformamide at reflux for 24 h, as reported to access nitrogenembedded quinoidal pentacenes, 44 only allowed us to isolate 13 in 8% yield (not shown).

Qualitative evaluation of acridine-based polycyclic azaaromatics as matrices for MALDI-MS

With the structural similarity of 8 and 9 with 9-aminoacridine (9AA), the gold standard matrix for negative mode in MALDI-MS, 20 and azahelicenes which were described as potential candidates for MALDI or MAILD matrices, 45 we compared the ability of 8 and 9 to ionize a group of nine target compounds (mainly drugs, see ESI). Therefore, we designed a quick MALDI-MS method with the idea to focus on qualitative detection. We performed the experiments in both positive and negative mode, with the five different matrices DCTB, HCCA, 9-AA, 8 and 9, with two solvents for a two-layer spotting 46 (90 spots and 180 experiments). Each experiment aimed to reach 500 counts on the base peak by increasing gradually the laser power. A better quality of MALDI mass spectra in term of resolution and exact mass precision is achieved within the little power as possible, this phenomenon is called irradiance or fluence. 47 We scaled this factor [Laser(+) or Laser(-)] from zero (upper than 66% of laser power) up to six (less than 15%). The goal was to obtain target compounds as base peaks, but if not possible, the relative intensity was exported. We scaled this factor [Target.in(+) or Target.in(-)] from zero (not seen) up to five (seen as base peak). A product factor [Product.LT] representing both the laser power applied and the presence of target compounds was calculated. We scaled specifically a factor [Target.BP(+), Target.BP(-) or Target.BP] representing the number of target compounds obtained as a base peak, in positive, negative or in both, without redundancy. The solvent influence was not relevant.

At the first sight, the Kiviat's diagram of average values of the factors for the five matrices shows that DCTB was surprisingly the best matrix in both ionization modes. 9-AA has slightly higher result in term of overall ionization of target compounds in negative mode but the ionization efficiency is bad (Figure 3).

Figure 3

Qualitative analysis of the five matrices.

For more details, HCCA is a matrix mainly applied for positive mode in MALDI-MS. The Kiviat's diagram of average values for DCTB vs. HCCA matrices shows that HCCA ionized one target compound only in positive mode as expected, in comparison DCTB is the best in positive mode with three target compounds as base peak (see Figure 4). In a matter of fact, in this experiment DCTB is better than HCCA in positive ionization. 9-AA is the gold standard matrix for negative mode in MALDI-MS. 20 The Kiviat's diagram of average values for DCTB vs. 9-AA matrices shows that 9-AA was suitable to ionize three target compounds only in negative mode as expected, in comparison DCTB has also three target compounds as base peak (see ESI and Figure 5). 9-AA has slightly higher result in term of overall ionization of target compounds in negative mode (Target.in(-): 9-AA (42.2%) and DCTB (38.8%)). But DCTB ionization efficiency is really better (Laser(-): DCTB (76.8%) and 9-AA (29.5%)). In this work, DCTB is a better matrix than 9-AA in negative ionization. For the comparison between 9-AA, 8 and 9, the Kiviat's diagram shows that the three matrices were suitable to ionize three target compounds in negative mode as expected (see ESI and Figure 6). Although 9-AA gave the best result in term of overall ionization of target compounds in negative mode (Target.in(-) 9-AA (42.2%), 8 (35.6%) and 9 (32.2%)), the ionization efficiency of 9 is promising (Laser(-) 9 (36.2%) vs. 8 (29.7%) and 9-AA (29.5%)).

Photophysical properties of the synthesized acridine-and acridone-based aza-polycycles

Both acridine and acridone derivatives can display interesting fluorescence properties, in particular when substituted by electrondonating groups. 18,48 Thus, to establish structure-properties relationships, preliminary studies of the photophysical properties of most of the nitrogen-containing polycycles were performed. Their UV-visible absorption and emission properties were investigated in toluene, and the results are gathered in Table 1. The three phenanthrol[9,10-a]acridines 2, 7 and 8 exhibit similar absorption and emission properties, i.e. a lowest energy absorption band in the 350-450 nm range, with molar extinction coefficients of ~ 10 4 M -1 cm -1 , and an unresolved emission band in the violet-blue part of the visible (Figure 7). Substitution at 16-position leads to bathochromic shifts in both absorption and emission, with larger shifts with chloro (7) than with phenoxy (8). Their fluorescence quantum yields are rather low (2-6%), probably in relation with their non-planar helicene-like shape. The three phenanthrol[9,10-b]acridines 4a-c exhibit a structured absorption band with a maximum at around 415 nm and a structured emission band in the blue-green part of the visible (Figure 8). Introduction of a chlorine atom at the 13-position leads to small red-Please do not adjust margins Please do not adjust margins shifts of the absorption (5 nm) and emission (4 nm) bands, whereas replacement of a phenyl ring by a weaker donor methyl group at the 15-position has almost no effect on the absorption but leads to a 10 nm blue-shift of the emission band. The fluorescence quantum yields of 4a-c range between 14 and 30%. These compounds are therefore clearly more emissive than the helicene-like compounds 2, 7 and 8, in relation with their better planarity. Phenanthrol [9,10-b]acridone 11 and acridone 13 exhibit similar fine-structured lowest energy absorption and emission bands, but those of compound 11 are strongly red-shifted (45 nm in absorption and 47 nm in emission), in relation with the extended -electron system of 11 in comparison with that of 13 (Figure 9). This extension also leads to a significant increase of the fluorescence quantum yield from 3% for 13 to 8% for 11.

Conclusions

Thus, in the present study, we evidenced the value of coppercatalysed N-arylation of anilines bearing an adjacent carbonylated function to access nitrogen-containing polyaromatic hydrocarbons (PAHs). Interestingly, the cyclization to PAHs occurred regioselectively; while formation of helicene-like scaffolds was observed in some cases (which corresponds to both orbital and charge controls), more linear polycycles prevailed from bulkier precursors.

Regarding the unexpected results obtained with DCTB as a MALDI-MS matrix, the introduction of a trans-2-[2-methyl-2propenyliden-3-yl]malononitrile group (R-CH=C(CH3)-CH=C(CN)2) on our original PAHs could be a promising strategy to reach more efficient matrices. Besides, other applications for the prepared aza-polycycles could be expected. Indeed, the introduction of heteroatoms, and in particular nitrogen, in nanographenes (which behave as graphene segments) can have a great impact on their properties (e.g. resistance to oxidation and dimerization, ability to molecular packing, optical and electrochemical behaviours, etc.), and can be used to tune them. 49 Therefore, as our strategy is able to easily deliver azapolycycles, further development could be expected. Efforts are actually ongoing and will be reported in due course.

Experimental

The general information, synthetic procedures and analyses of the intermediates can be found in the ESI. †

Synthesis of the compounds 2 and 4 (General procedure).

They were prepared by adapting a literature procedure. 28 To 2-(2-triphenylenyl)aminobenzaldehyde (1; 0.37 g, 1.0 mmol), or the required 2-(2-triphenylenyl)aminophenone (1.0 mmol), in CH3CO2H (3 mL) was added 96% H2SO4 (2.8 mmol, 0.15 mL). The mixture was stirred at 110 °C for 2 h. After cooling to room temperature, water (5 mL) was added and the mixture was basified using 25% NH4OH before extraction using AcOEt (3x20 mL), drying over MgSO4, and removal of the solvent. The crude was purified as specified in the product description. It was obtained by adapting a literature procedure. 38 A mixture of 16chlorophenanthro[9,10-a]acridine (7; 0.36 g, 1.0 mmol) and phenol (0.11 g, 1.2 mmol) was stirred and heated at 70 °C before addition of ammonium carbonate (0.13 g, 1.65 mmol). The internal temperature was raised to 120 °C and the reaction mixture was stirred at this temperature for 45 min. It was next cooled to 30 °C and poured into acetone (3 mL) cooled at 0 °C. A precipitate took form which was collected and washed with acetone. This solid was treated by sodium hydroxide (0.26 g) in water (6 mL), and the mixture extracted by CH2Cl2 (3 x 20 mL). The organic phase was dried over MgSO4, filtered, and concentrated under reduced pressure. Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt Please do not adjust margins Please do not adjust margins N-(2-Triphenylenyl)-9-acridone (13). It was prepared under an argon atmosphere by adapting a procedure reported previously. 43 A mixture of N-phenyl-N-(2-triphenylenyl)-2aminobenzaldehyde (12; 0.44 g, 1.0 mmol), ytterbium triflate (31 mg, 50 mol) and sodium sulfate (71 mg, 0.50 mmol) in DMF (5 mL) was heated at 100 °C for 20 h. After cooling to room temperature, diethyl ether (0.40 L) was added, and the organic phase was washed with water (100 mL). After drying the organic phase over sodium sulfate, the solvent was evaporated under vacuum. Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt 80:20; Rf = 0.325) gave 13 in 83% yield (0.36 g) as a beige solid: mp > 260 °C; IR (ATR): 722, 751, 801, 818, 864, 912, 938, 1027, 1076, 1160, 1258, 1306, 1361, 1449, 1458, 1486, 1507, 1597, 1631, 2962, 3068 cm -1 ; 1 H NMR (CDCl3)  6.89 (d, 2H, J = 8.6 Hz, H4 and H5), 7.31 (t, 2H, J = 7.4 Hz, H2 and H7), 7.49 (t, 2H, J = 7.6 Hz, H3 and H6), 7.62-7.66, 7.73 and 7.77-7.79 (m, 2H, t, 1H, J = 7.6 Hz and m, 2H; H3', H6', H7', H10' and H11'), 8.52 (d, 1H, J = 8.1 Hz, H12'), 8.65 (d, 2H, J = 7.9 Hz, H1 and H8), 8.68 (s, 1H, H1'), 8.71-8.79 (m, 3H), 9.00 (d, 1H, J = 8.5 Hz); 13

Figure 2

 2 Figure 2Dibenzo[a,c]acridines already prepared.16,17

Scheme 2

 2 Scheme 2 Retrosynthetic plan to helicene-like scaffolds based on acridine and amplitudes of the HOMO coefficients and charges (in brackets) obtained (same values for R = H, Me, Ph, OH or Cl) by using the HuLiS calculator 27 (see ESI for more details).

Scheme 4

 4 Scheme 4 Synthesis of the phenanthro[9,10-b]acridines 4.

Scheme 6

 6 Scheme 6 Conversion of 16-chlorophenanthro[9,10-a]acridine (7) and ORTEP diagram (30% probability) of 16-phenoxyphenanthro[9,10-a]acridine (9).

Scheme 7

 7 Scheme 7 Synthesis of N-(2-triphenylenyl)phenanthro[9,10-b]-15-acridone (11).

Scheme 9

 9 Scheme 9 Synthesis of N-(2-triphenylenyl)-9-acridone(13).

Figure 4

 4 Figure 4 Qualitative analysis of DCTB vs. HCCA.

Figure 5

 5 Figure 5 Qualitative analysis of DCTB vs. 9-AA.

Figure 6

 6 Figure 6 Qualitative analysis of 9-AA, 8 and 9.

Figure 7

 7 Figure 7 Absorption (solid line) and emission (dotted line) of phenanthrol[9,10-a]acridines 2, 7 and 8 in toluene.

Figure 8

 8 Figure 8 Absorption (solid line) and emission (dotted line) of phenanthrol[9,10-b]acridines 4a-c in toluene.

Figure 9

 9 Figure 9 Absorption (solid line) and emission (dotted line) of acridones 11 and 13 in toluene.

Phenanthro[ 9 ,

 9 10-a]acridine (2). It was prepared according to the general procedure by starting from 2-(2triphenylenyl)aminobenzaldehyde (1). Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt 80:20; Rf = 0.375) gave 2 in 97% yield (0.34 g) as a yellow solid: mp 224 °C; IR (ATR): 723, 747, 766, 828, 856, 929, 965, 1144, 1244, 1373, 1418, 1433, 1477, 1495, 1523, 1539, 1610, 1717, 1965, 2850, 2919, 3059 cm -1 ; 1 H NMR (CDCl3)  7.59 (t, 1H, J = 7.45 Hz, H14), 7.69 and 7.71-7.75 (t, 1H, J = 7.0 Hz and m, 3H, H2, H3, H6 and H7), 7.85 (t, 1H, J = 7.6 Hz, H13), 8.05 (d, 1H, J = 8.3 Hz, H15), 8.31 (t, 2H, J = 8.6 Hz, H10 and H12), 8.62-8.64 (m, 1H, H8), 8.71-8.73 and 8.77 (m, 1H and t, 2H, J = 8.1 Hz; H1, H4 and H5), 8.83 (d, 1H, J = 9.4 Hz, H9), 9.66 (s, 1H, H16); 13 C NMR (CDCl3)  123.3 (CH), 123.8 (C), 123.8 (CH), 123.9 (CH), 126.0 (CH, C14), 126.2 (C), 126.3 (CH, C9), 126.8 (CH), 127.0 (C), 127.1 (CH), 127.3 (C), 127.7 (CH), 127.8 (CH), 128.7 (CH, C15), 128.9 Please do not adjust margins Please do not adjust margins (CH), 129.0 (CH), 129.1 (CH), 129.2 (C), 129.4 (C), 130.3 (C), 130.7 (CH), 131.3 (C), 137.0 (CH, C16), 148.0 (C), 149.5 (C); HRMS (MALDI, DCTB matrix), m/z: 330.127 (2 ppm) found (calcd for C25H16N, [M+H] + , requires 330.1277). Crystal data for 2. C25H15N, M = 329.38, monoclinic, P 21/c, a = 12.054(2), b = 10.6413(10), c = 13.5706(19) Å, β = 114.096(9) °, V = 1589.1(4) Å 3 , Z = 4, d = 1.377 g cm -3 , μ = 0.080 mm -1 . A final refinement on F 2 with 3521 unique intensities and 235 parameters converged at ωR(F 2 ) = 0.2049 (R(F) = 0.0563) for 2798 observed reflections with I > 2σ(I). CCDC 2085922. 15-Phenylphenanthro[9,10-b]acridine (4a). It was prepared according to the general procedure by starting from phenyl 2-(2-triphenylenyl)aminophenyl ketone (3a; 0.44 g). Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt 80:20; Rf = 0.40) gave 4a in 57% yield (0.23 g) as an orange solid: mp 254 °C; IR (ATR): 702, 718, 757, 782, 873, 923, 1075, 1262, 1424, 1491, 1509, 1534, 1552, 1626, 2186, 3064 cm - 1 ; 1 H NMR (CDCl3)  7.44-7.63 (m, 5H), 7.65-7.74 (m, 5H), 7.82 (t, J = 8.5 Hz, 2H), 8.23 (d, 1H, J = 6.3 Hz), 8.34 (d, 1H, J = 7.2 Hz), 8.48 (br s, 2H), 8.87 (br s, 1H), 8.91 (s, 1H), 9.50 (s, 1H); 13 C NMR (CDCl3)  121.2 (CH), 122.9 (CH), 123.6 (CH), 123.7 (CH), 124.0 (CH), 124.3 (C), 124.8 (CH), 125.3 (C), 125.7 (CH), 127.1 (CH), 127.8 (CH), 128.1 (CH), 128.3 (CH), 128.8 (C), 128.8 (2CH), 128.8 (CH), 128.9 (CH), 129.5 (CH), 129.8 (C), 129.9 (C), 130.4 (C), 130.5 (CH), 130.8 (C), 130.9 (2CH), 132.9 (C), 136.0 (C), 147.0 (C), 147.7 (C), 149.6 (C); HRMS (MALDI, DCTB matrix), m/z: 406.159 (0 ppm) found (calcd for C31H20N, [M+H] + , requires 406.1590). 13-Chloro-15-phenylphenanthro[9,10-b]acridine (4b). It was prepared according to the general procedure by starting from 5-chloro-2-(2-triphenylenyl)aminophenyl phenyl ketone (3b; 0.48 g). Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt 80:20; Rf = 0.525) gave 4b in 95% yield (0.42 g) as a yellow solid: mp > 260 °C; IR (ATR): 717, 754, 787, 821, 869, 878, 949, 970, 1029, 1067, 1135, 1164, 1256, 1307, 1404, 1421, 1441, 1487, 1507, 1599, 1952, 3064 cm -1 ; 1 H NMR (CDCl3)  7.47 (ddd, 1H, J = 8.2, 7.6 and 0.95 Hz), 7.53-7.57 (m, 3H), 7.61-7.66 (m, 3H), 7.70-7.74 (m, 4H), 8.16 (d, 1H, J = 7.0 Hz, H1), 8.23 (d, 1H, J = 9.1 Hz, H11), 8.40-8.44 (m, 2H, H4 and H5), 8.76 (dd, 1H, J = 6.0 and 3.6 Hz, H8), 8.82 (s, 1H, H16), 9.38 (s, 1H, H9); 13 C NMR (CDCl3)  121.0 (CH), 122.6 (CH), 123.6 (CH), 123.7 (CH), 124.0 (CH), 124.3 (C), 129.2 (CH), 124.8 (CH), 125.2 (CH), 125.3 (C), 127.8 (CH), 128.1 (CH), 128.5 (CH), 129.0 (2CH), 129.1 (CH), 129.3 (C), 129.5 (C), 129.6 (C), 130.4 (C), 130.8 (C), 130.8 (2CH), 130.9 (CH), 131.6 (C), 131.8 (CH), 133.2 (C), 135.3 (C), 146.5 (C), 147.1 (C), 147.3 (C); HRMS (MALDI, DCTB matrix), m/z: 440.120 (0 ppm) found (calcd for C31H19

  ; 1 H NMR (CDCl3)  7.55 (td, 1H, J = 7.6 and 1.0 Hz), 7.64-7.76 (m, 4H), 7.89 (td, 1H, J = 7.7 and 1.2 Hz), 7.95 (d, 1H, J = 8.2 Hz), 8.18 (d, 1H, J = 9.2 Hz), 8.34 (d, 1H, J = 8.6 Hz), 8.55 (d, 1H, J = 8.7 Hz), 8.57-8.60 (m, 1H), 8.68 (d, 1H, J = 8.4 Hz), 8.72 (d, 1H, J = 9.2 Hz), 8.73-8.76 (m, 1H); 13 C NMR (CDCl3)  121.4 (C), 123.3 (CH), 123.4 (CH), 123.9 (CH), 125.0 (C), 125.1 (CH), 125.7 (CH), 125.7 (C), 126.6 (CH), 126.9 (CH), 127.3 (CH), 127.6 (CH), 127.9 (CH), 128.7 (CH), 128.7 (C), 128.9 (C), 129.3 (CH), 129.6 (C), 130.0 (C), 130.7 (CH), 131.1 (C), 131.5 (CH), 141.2 (C), 148.1 (C), 149.9 (C); HRMS (MALDI, DCTB matrix), m/z: 364.089 (1 ppm) found (calcd for C25H15N 35 Cl, [M+H] + , requires 364.0888).

  (m, 2H), 7.27-7.37 (m, 5H), 7.54 (ddd, 1H, J = 8.2, 7.5 and 1.2 Hz), 7.59-7.64 (m, 3H), 7.98 (dd, 1H, J = 7.8 and 1.6 Hz), 8.24 (d, 1H, J = 8.2 Hz), 8.26 (d, 1H, J = 2.3 Hz, H1'), 8.51-8.53 (m, 2H), 8.61-8.63 (m, 2H), 10.33 (s, 1H, CHO); 13 C NMR (CDCl3)  116.4 (CH), 122.8 (CH), 123.1 (CH), 123.3 (2CH), 123.4 (CH), 123.45 (CH), 123.45 (CH), 123.5 (CH), 125.0 (CH), 125.4 (C), 125.9 (CH), 126.8 (CH), 127.2 (CH), 127.5 (CH), 127.6 (CH), 129.3 (CH), 129.3 (C), 129.3 (C), 129.6 (CH), 129.7 (C), 129.9 (2CH), 130.3 (C), 131.3 (C), 132.1 (C), 135.7 (CH), 148.0 (C), 148.9 (C), 150.3 (C, C1), 190.8 (CH, CHO); HRMS (MALDI, DCTB matrix), m/z: 423.167 (12 ppm) found (calcd for C31H21NO, M +• , requires 423.1618).

  C NMR (CDCl3)  117.1 (2CH), 121.8 (2CH), 122.1 (C), 123.7 (CH), 123.7 (CH), 123.8 (CH), 123.8 (CH), 125.2 (CH), 126.7 (CH), 127.6 (2CH), 127.8 (CH), 127.9 (CH), 128.4 (CH), 128.4 (CH), 128.4 (CH), 128.9 (C), 129.1 (C), 130.4 (C), 130.5 (C), 130.8 (C), 132.4 (C), 133.5 (2CH), 137.9 (C), 143.5 (C), 178.4 (C, C=O); HRMS (ASAP), m/z: 422.1538 (0 ppm) found (calcd for C31H20NO, [M+H] + , requires 422.1539).

  General information and crystallographic details; Experimental procedures and analyses of intermediate compounds; NMR spectra of the new compounds as well as numbering used in the experimental section; Highest occupied Hückel molecular orbitals (HOMO); Qualitative evaluation of two polycycles as matrices for MALDI-MS. CCDC 2085922, 2085923. For ESI and crystallographic data in CIF, see DOI: 10.1039/x0xx00000x Please do not adjust margins

	a. Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) -UMR 6226,
	F-35000 Rennes, France. E-mail: florence.mongin@univ-rennes1.fr
	b. Laboratoire de Synthèse Organique Appliquée, Faculté des Sciences Exactes et
	Appliquées, Université d'Oran 1 Ahmed Ben Bella, BP 1524 El M'Naouer, 31000
	Oran, Algeria. E-mail: bentabedg@gmail.com
	c. Univ Rennes, CNRS, ScanMat -UMS 2001, Centre Régional de Mesures Physiques
	de l'Ouest -CRMPO, F-35000, Rennes, France. E-mail: nicolas.leyondre@univ-
	rennes1.fr
	† Electronic Supplementary Information (ESI) available:

Table 1

 1 Absorption and emission properties of phenanthroacridines 2

, 4, 7, 8 and acridones 11, 13 in

  toluene at 25°C.

	Compound	abs (nm)	max (M -1 cm -1 )	em (nm)	F
	2	392	9480	447	0.06
	4a	414	14300	465, 495	0.30
	4b	419	8700	469, 499	0.30
	4c	412	12000	456, 485, 520	0.14
	7	406	11500	467	0.02
	8	399	11000	463	0.05
	11	436	4760	445, 473	0.08
	13	391	8940	398, 420	0.03

  Me), 7.56 (t, 1H, J = 7.6 Hz), 7.63-7.68 (m, 4H), 7.79 (t, 1H, J = 7.5 Hz), 8.27 (d, 1H, J = 8.7 Hz), 8.30 (d, 1H, J = 8.8 Hz), 8.52-8.53 (m, 2H), 8.77 (d, 1H, J = 7.5 Hz), 8.84-8.86 (m, 1H), 9.44 (s, 2H); 1 H NMR (CDCl3) at higher concentration (about 2 mg in 0.5 mL)  3.21 (s, 3H, Me), 7.49 (t, 1H, J = 7.6 Hz), 7.58-7.63 (m,

				NMR (CDCl3) at a lower concentration (about 1 mg in 0.5 mL) 
				3.35 (s, 3H, 4H), 7.74 (t, 1H, J = 7.5 Hz), 8.20 (t, 2H, J = 8.7 Hz), 8.43 (d, 2H, J
				= 6.8 Hz), 8.63 (d, 1H, J = 6.8 Hz), 8.76 (d, 1H, J = 4.9 Hz), 9.25 (s,
				1H), 9.31 (s, 1H); 13 C NMR (CDCl3)  14.0 (CH3), 118.6 (CH), 123.5
				(CH), 123.7 (CH), 123.8 (CH), 124.5 (C), 124.6 (CH), 124.8 (CH),
				125.3 (CH), 125.5 (C), 127.8 (CH), 128.0 (CH), 128.2 (CH), 128.2
				(C), 128.7 (CH), 129.8 (C), 130.0 (C), 130.2 (CH), 130.4 (C), 130.6
				(C), 132.4 (C), 3C not seen, 3CH not seen; HRMS (MALDI, DCTB
				matrix), m/z: 344.144 (2 ppm) found (calcd for C26H18N, [M+H] + ,
				requires 344.1434).
					16-Chlorophenanthro[9,10-a]acridine (7). It was prepared
				by	adapting	a	literature	procedure. 37	To	2-(2-
				triphenylenyl)aminobenzoic acid (6; 0.36 g, 1.0 mmol) was
				added a large excess of POCl3 (4.0 mL). The solution was stirred
				at reflux under argon for 72 h and then concentrated to dryness.
				The residue was dissolved in CH2Cl2 and the solution slowly
				poured onto ice-cooled 15% aqueous NH4OH solution. After
				decantation, the organic phase was dried over MgSO4, filtered,
				and concentrated under reduced pressure. Purification by
				column chromatography on silica gel (eluent: petroleum ether-
				AcOEt 80:20; Rf = 0.475) gave 7 in 84% yield (0.31 g) as a yellow
				solid: IR (ATR): 719, 741, 750, 819, 839, 909, 955, 1016, 1197,
				1246, 1260, 1344, 1383, 1415, 1462, 1494, 1533, 1597, 1721,
				2917 cm -1
		35 ClN, [M+H] + ,
	requires 440.1200).
	15-Methylphenanthro[9,10-b]acridine	(4c).	It	was
	prepared according to the general procedure by starting from
	2-(2-triphenylenyl)aminoacetophenone	(3c;	0.36	g).
	Purification by recrystallization from AcOEt gave 4c in 90% yield
	(0.31 g) as an orange solid: mp > 260 °C; IR (ATR): 717, 751, 857,
	947, 1052, 1261, 1424, 1509, 1551, 1664, 2166, 2921 cm -1 ; 1 H

  successively added activated Cu[START_REF] Vogel | Vogel's textbook of practical organic chemistry[END_REF] (0.20 equiv, 13 mg, 0.20 mmol) and K2CO3 (2 equiv, 0.29 g, 2.0 mmol). The mixture was degassed and refluxed under argon for 24 h. During this time, activated Cu 51 (4 x 13 mg, 4 x 0.20 mmol) was added after 2, 4, 6 and 8 h of heating. After cooling to room temperature, the mixture was concentrated. Purification by column chromatography on silica gel (eluent: petroleum ether-AcOEt 90:10; Rf = 0.21) gave 12 in 57% yield (0.24 g) as a yellow solid: mp 126-132 °C; IR (ATR): 696, 721, 752, 821, 906, 1051, 1097, 1155, 1191, 1241, 1265, 1305, 1348, 1390, 1436, 1453, 1476, 1490, 1592, 1613, 1685, 1947, 2251, 2748, 2849, 3061 cm -1 ; 1 H NMR (CDCl3)  7.09 (t, 1H, J = 7.3 Hz), 7.15-7.17

	(CH), 123.4 (CH), 123.6 (CH), 123.7 (CH), 123.8 (CH), 123.9 (CH),
	124.0 (CH), 124.0 (CH), 124.1 (CH), 124.8 (C), 125.4 (CH), 127.0
	(CH), 127.4 (CH), 127.4 (CH), 127.8 (CH), 127.9 (CH), 128.0 (CH),
	128.0 (CH), 128.5 (CH), 128.5 (CH), 128.6 (CH), 128.7 (CH), 128.8
	(C), 129.1 (C), 129.2 (C), 129.3 (C), 130.0 (C), 130.4 (C), 130.6 (C),
	130.9 (C), 131.3 (C), 132.7 (C), 133.9 (CH), 134.6 (C), 138.0 (C),
	142.1 (C), 144.1 (C), 178.8 (C, C=O); HRMS (ASAP), m/z:
	572.2012 (0 ppm) found (calcd for C43H26NO, [M+H] + , requires
	572.2009).
	N-Phenyl-N-(2-triphenylenyl)-2-aminobenzaldehyde (12).
	It was prepared under an argon atmosphere by adapting a
	procedure	reported	previously. 30	To	2-(2-
	triphenylenyl)aminobenzaldehyde (1; 0.35 g, 1.0 mmol) and
	iodobenzene (0.24 g, 0.13 mL, 1.2 mmol) in degassed Bu2O (1
	mL) were

Scheme 3 Synthesis and ORTEP diagram (30% probability) of phenanthro[9,10-a]acridine (2).

Scheme 8 Amplitudes of the highest occupied molecular orbital (HOMO) coefficients and charges (in brackets) obtained by using the HuLiS calculator27 to rationalize the regioselective formation of the acridones 11 and 13.

Acknowledgements

We thank the Université de Rennes 1 and the Centre National de la Recherche Scientifique (F. M.). We acknowledge the Fonds Européen de Développement Régional (FEDER; D8 VENTURE Bruker AXS diffractometer). We thank Joshua M. Sims for helpful discussions, H. T. Huynh for providing paracetamol, and F. Lassagne for his help all along the study.

Notes and references

Conflicts of interest

There are no conflicts to declare.