Baparou Danhane

Jérôme Lohéac

Ensemble controllability of parabolic type equations

Keywords: Ensemble control, parabolic equations, average control 2020 MSC: 93B05, 93B35, 93C05, 93C20, 93C25, 35K90

In this paper, we extend the concept of ensemble controllability introduced in 2009 by Jr-Shin Li and Navin Khaneja to a class of linear partial dierential equation. More precisely, we consider some abstract parabolic equation, where the system depends on some unknown parameter which is assumed to belong to a compact interval. We investigate the possibility of approximatively reaching (in L 2 -norm) any target state from any initial state, with an open loop control. Here the initial and target states might depend on the unknown parameter, but the control is assumed to be parameter independent.

Introduction

The modeling of physical systems often involves in their dynamics some parameters that are inherent to them. These parameters are generally estimated but not known with precision. For instance, the dynamic of a car depends on its mass, the evolution of the temperature depends on the diusion coecient of the material. . . In the context of classical control theory, it is customary to assume that these parameters are known and xed. Thus, if a system is to be sent from its initial state to a prescribed nal state, the constructed control implicitly depends on the xed parameters. As a consequence, when these parameters are modied, the built control no longer solves the control problems.

Hence, each time a system parameter is modied, a new control has to be Email addresses: bdanhane@laas.fr (Baparou Danhane), jerome.loheac@univ-lorraine.fr (Jérôme Lohéac) Preprint submitted to System & Control Letters November 16, 2023 designed. It therefore seems reasonable to ask whether it is possible to build controls that are independent of the parameters of the system. This is the aim of Ensemble controllability, and this notion is also named (approximate) simultaneous controllability.

The notion of ensemble controllability was rst introduced in [START_REF] Li | Ensemble control of bloch equations[END_REF], where the authors studied the ensemble controllability of Bloch equations with a dispersion in their natural frequencies. A much more elaborate study of the topic discussed in the above reference, using functional analysis approach, was undertaken in [START_REF] Beauchard | Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations[END_REF].

For linear control systems many results exist. From the generic concept of ensemble controllability, many other notions emerged such as Uniform ensemble controllability and L q -ensemble controllability. One can cite for instance [START_REF] Dirr | Uniform and L q -ensemble reachability of parameter-dependent linear systems[END_REF][START_REF] Li | Ensemble control of time-invariant linear systems with linear parameter variation[END_REF][START_REF] Schönlein | Controllability of ensembles of linear dynamical systems[END_REF] where the authors gave necessary and sucient conditions for L q -ensemble controllability and uniform ensemble controllability for linear control systems. In [START_REF] Chen | Controllability issues of linear ensemble systems over multidimensional parameterization spaces[END_REF][START_REF] Danhane | Conditions for uniform ensemble output controllability, and obstruction to uniform ensemble controllability[END_REF], the authors gave some obstructions to L q and uniform ensemble controllability, and in [START_REF] Schönlein | Feedback equivalence and uniform ensemble reachability[END_REF], the author explore how the application of feedback methods enlarges the class of parameter-dependent linear systems that are uniformly ensemble reachable. In [START_REF] Lazar | Control of parameter dependent systems[END_REF][START_REF] Schönlein | Computation of open-loop inputs for uniformly ensemble controllable systems[END_REF], the authors gave some numerical tools to compute optimal controls. We also quote [START_REF] Agrachev | Ensemble controllability by Lie algebraic methods[END_REF],

where the authors investigate ensemble controllability for non-linear systems using the geometrical point of view.

Dealing with feedback stabilization of parameter dependent partial differential equation, we mention the recent papers [START_REF] Guth | Ensemble feedback stabilization of linear systems[END_REF][START_REF] Alleaume | Ensembles of hyperbolic PDEs: Stabilization by backstepping[END_REF]. In [START_REF] Guth | Ensemble feedback stabilization of linear systems[END_REF], the authors investigate the ensemble feedback stabilization of linear systems using a Riccati based feedback mechanism. It should be mentioned that in their paper, the set of parametrization is a nite subset of the real line. More importantly, the feedback control proposed, itself, depends on the parameter but the gain does not. The fact that the gain does not depend on the parameter makes the analysis more delicate. The question of feedback stabilization of parameter dependent system with feedback controls that do not depend on the parameter is still, at our knowledge, an open question, even for linear systems. In [START_REF] Alleaume | Ensembles of hyperbolic PDEs: Stabilization by backstepping[END_REF], the authors use the backstepping method to stabilize an ensemble of hyperbolic partial dierential equations.

To our knowledge, the concept of ensemble controllability has not yet been investigated in the case of partial dierential equations, especially when the set of parametrization is a continuum. It is worth noting, however, that in [START_REF] Guth | Ensemble feedback stabilization of linear systems[END_REF], as an application of their results, the authors considered a Galerkin discretization of a parameterized one dimensional heat equation with inter-nal controls, where the uncertain parameter is the reaction coecient and belongs to a nite subset of the real line. They have shown, under some conditions, that any nite truncation of the discretized system solution is ensemble controllable.

As mentioned above, the ensemble controllability of partial dierential equations when the set of parametrization is a continuum is open, and in this paper, we investigate the L 2 -ensemble controllability of linear parabolic type equations with internal controls.

The paper is organized as follows. In Section 2, we state the controllability problem and give the main result of the paper. This section is followed by Section 3 where we apply our main result to the heat and 2D Grushin equations, and give some relations between ensemble controllability and average controllability. The proof of the main result is given in Section 4.

This section is followed by Section 5 where we made a digression on nite dimensional control systems. Numerical examples are given in Section 6.

Concluding remarks and open problems are provided in Section 7.

The following notations are used.

We set R + the nonnegative real line, N * = N \ {0}, and for κ ∈ N, N κ = {0, 1, . . . , κ}.

Let H be a Hilbert space. The notation •, • H denotes the scalar product of elements of H.

For two Banach spaces X and Y , the space X n 2 n 1

, with n 1 , n 2 ∈ N * , stands for the set of matrices with n 2 rows and n 1 columns with entries in X . The notation • X is used for the norm on X . The space L(X , Y ) stands for the Banach space of linear and continuous map from X into Y , and we set L(X ) = L(X , X ). We identify

L(R n 1 , R n 2 ) to R n 2 n 1
. For G ∈ L(X , Y ), Ran(G) and Ker(G) respectively denote the range and the null space of G, rk(G) is the dimension of Ran(G), and G is the adjoint of G. For X ⊂ X , X is the closure of X in X . For an operator A, D(A) denotes its domain. The matrix A ∈ R n n is said Hurwitz if all its eigenvalues have a negative real part.

For a family of vectors (e k ) k∈N , the notation Span K {e k , k ∈ N} stands for the set of linear combinations of these vectors over the eld K. For a Borel measure space M = (M, T , ν), the space L 2 (M , X ) which is the space of ν-measurable

functions f satisfying M f (σ) 2 X dν(σ) < ∞ is simply denoted by L 2 (M, X ). Moreover, if X = R k for some k ∈ N * then L 2 (M, R k ) is identied to L 2 (M, R) k ,
and denoted by L 2 (M) k . The space L 2 loc (R + , X ) stands for the one of measurable functions from R + to X that are square integrable on any bounded interval of R + . We similarly dene the sets of continuous function, C(M, X ) and C(M) k . The notation supp(f ) stands for the support of the function f .

Problem statement and main result

Let (Θ, B, µ) be a Borel measure space where Θ is a compact subset of R, and let X, U and Y be three Hilbert spaces. For every θ ∈ Θ, we consider the Cauchy problem

ẏ(t, θ) = α(θ)Ay(t, θ) + B(θ)u(t) (t > 0), (2.1)
y(0, θ) = y 0 (θ).

In the above equation and what follows, ẏ stands for the derivative of y with respect to the time variable.

We make the following assumptions on the above system.

(hp 1 ) (A, D(A)) generates an analytic exponentially stable semigroup (T t ) t 0 on X;

(hp 2 ) For µ-almost every θ ∈ Θ, B(θ) ∈ L(U, X) and Θ B(θ) 2 L(U,X) dµ(θ) < ∞;
(hp 3 ) There exist L ∈ L(U, Y ) and C ∈ L(X, Y ) and β ∈ L 2 (Θ) such that for every ϕ ∈ L 2 (Θ, X)

LB (θ)ϕ(θ) = β(θ)Cϕ(θ) (θ ∈ Θ,
T t z ∈ D(A ) (z ∈ X, t > 0).
Remark 2.2. The Assumption (hp 5 ) implies that

Span R θ ∈ Θ → β(θ) α(θ) k , k ∈ N = L 2 (Θ). ( 2
Θ = [1, 2], α(θ) = θ and β(θ) = 1, (2.
2) is automatically fullled thanks to the Weierstrass approximation theorem, see for instance [START_REF] Borwein | The full Müntz theorem in C[0, 1] and L 1 [0, 1[END_REF][START_REF] Erdélyi | The full Müntz theorem in L p [0, 1] for 0 < p < ∞[END_REF].

The rst remark to be made is that since (A, D(A)) generate T t which is analytic and exponentially stable, it follows from (hp 5 ) that for every θ ∈ Θ, (α(θ)A, D(A)) also generate T α(θ)t which is analytic and exponentially stable.

Therefore, from [19, Proposition 4.2.5] and (hp 2 ), we deduce that for every u ∈ L 2 loc (R + , U ), every y 0 ∈ L 2 (Θ, X), and almost every θ ∈ Θ, (2.1) admits a unique solution given by

y(t, θ) = T α(θ)t y 0 (θ) + (Φ t u)(θ) (t > 0), (2.3)
where

(Φ t u)(θ) = t 0 T α(θ)(t-τ ) B(θ)u(τ ) dτ. (2.4)
Dening y : (t, θ) ∈ R + × Θ → y(t, θ) as a solution of (2.1), we can prove the following lemma.

Lemma 2.3. Under Assumptions (hp 1 ), (hp 2 ) and (hp 5 ), for every T > 0,

the map Φ T dened by (2.4) belongs to L(L 2 (R + , U ), L 2 (Θ, X)), and for every y 0 ∈ L 2 (Θ, X), the solution y dened by (2.3) belongs to C(R + , L 2 (Θ, X)).

Proof. Using the fact that T t is exponentially stable, it follows from [14, Theorem 4.3] that there exist two constants M 1 and δ > 0 such that T t L(X) M e -δt for every t 0. Since α is a positive function of θ, it therefore holds that for every t 0 and every θ ∈ Θ, T α(θ)t L(X)

M e -α(θ)δt . Setting α = min θ∈Θ α(θ), we have

T α(θ)t L(X) M e -αδt (t 0, θ ∈ Θ).
(2.5) Using (2.5), we have for every τ ∈ [0, t]

T α(θ)(t-τ ) B(θ)u(τ ) X M e -αδ(t-τ ) B(θ) L(U,X) u(τ ) U .
From the above inequality, we obtain by integrating with respect to τ on the time interval (0, t) and using Cauchy-Schwarz inequality

t 0 T α(θ)(t-τ ) B(θ)u(τ ) X d(τ ) M √ 2δα B(θ) L(U,X) u L 2 ((0,t),U ) .
From this observation, it follows that the map τ → T α(θ)(t-τ ) B(θ)u(τ ) belongs to L 1 ([0, t], X) and by Bochner integrability, we have for every t 0 and every θ ∈ Θ,

t 0 T α(θ)(t-τ ) B(θ)u(τ ) d(τ ) X M √ 2δα B(θ) L(U,X) u L 2 ((0,t),U ) .
The preceding inequality leads to

Θ t 0 T α(θ)(t-τ ) B(θ)u(τ ) d(τ ) 2 X dµ(θ) M 2 γ 2δα u 2 L 2 ((0,t),U ) . (2.6)
where γ = Θ B(θ) 2 L(U,X) dµ(θ) < ∞, thanks to Assumption (hp 2 ). For every t 0, the linearity of Φ t being obvious, we deduce from (2.6) that Φ t ∈ L(L 2 (R + , U ), L 2 (Θ, X)). Moreover, using estimations (2.5) and (2.6), it follows that for every t 0,

y(t) L 2 (Θ,X) M e -δαt y 0 L 2 (Θ,X) + M γ 2δα u L 2 ((0,t),U ) .
Therefore, for any u ∈ L 2 loc (R + , U ), y ∈ C(R + , L 2 (Θ, X)).

The Lemma 2.3 therefore lends legitimacy to the following denition.

Denition 2.4. The system (2.1) is said L 2 (Θ, X)-ensemble controllable in time T > 0 if for every y 0 , y 1 ∈ L 2 (Θ, X) and every ε > 0, there exists a parameter independent open loop control u ∈ L 2 ([0, T ], U ) such that the solution of (2.1) with y 0 as initial state datum and u as input satises

y(T ) -y 1 L 2 (Θ,X)
ε.

(2.7)

In this denition, functions of L 2 ([0, T ], U ) are seen as functions of L 2 (R + , U ) supported on [0, T ].
The main result of this paper is the following. Theorem 2.5. Under the Assumptions (hp 1 ) to (hp 6 ), the system (2.1) is L 2 (Θ, X)-ensemble controllable in any time T > 0.

Let us make the following remarks.

Remark 2.6. Observe that if the system (2.1) is L 2 (Θ, X)-ensemble controllable in time T > 0, then for every λ ∈ R, the system

ż(t, θ) = (α(θ)A -λI)z(t, θ) + B(θ)v(t) (2.8) is also L 2 (Θ, X)-ensemble controllable in time T > 0. Indeed, for every θ ∈ Θ, (α(θ)A, D(A)) generates the semigroup T α(θ)t , then (α(θ)A -λI, D(A)) generates the semigroup e -λt T α(θ)t . Therefore, for every u ∈ L 2 ([0, T ], U ), the solution of (2.8) with initial condition z 0 (θ) is given by z(t, θ) = e -λt (T α(θ)t z 0 (θ) + (Φ t u λ )(θ)) = e -λt y(T )
where u λ (t) = e λt u(t), and where y solves (2.1) with control u λ and initial condition z 0 (θ).

If (2.1) is L 2 (Θ, X)-ensemble controllable in time T > 0, then for every ε > 0 there exists u λ ∈ L 2 ([0, T ], U ) such that, y(T ) -e λT z 1 L 2 (Θ,X) e λT ε, that is to say that z(T ) -z 1 L 2 (Θ,X) ε, i.e., (2.8) is L 2 (Θ, X)-ensemble controllable.
Remark 2.7. As it will be seen in the proof of Theorem 2.5 in Section 4, this theorem is still valid for unbounded operators B(θ) ∈ L(U, X -1 ) provided that we replace Assumption (hp 1 ) by the fact that for µ-almost every θ ∈ Θ, B(θ) is admissible for T t and that the admissibility constant, which depends on θ, belongs to L 2 (Θ). These hypotheses ensure that Lemma 2.3 still holds with an unbounded control operator. In this case, we should have in Assumption (hp 3 ) C ∈ L(X κ 1 , Y ) where X κ 1 is the dual space of X -1 with respect to the pivot space X (see [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]), and C should be required to be admissible for the semigroup T t . It goes without saying that in the proof, one should adjust what should be adjusted. However, we do not have example with unbounded control operator such that Assumption (hp 4 ) is satised. Hence, we do not detail the case of unbounded control operators.

Applications

Application to the heat equation

Let Ω be a smooth domain of R d (d 1), X = U = L 2 (Ω) and consider the system

ẏ(t, x, θ) = α(θ)∆y(t, x, θ) + β(θ)χ ω(θ) (x)u(t, x), (t, x) ∈ R + × Ω, y(t, x, θ) = 0, (t, x) ∈ R + × ∂Ω, y(0, x, θ) = y 0 (x, θ), x ∈ Ω, (3.1)
where θ ∈ Θ, with Θ a compact interval of R. In this system, the control is acting in a domain ω(θ) ⊂ Ω that might depend on the parameter θ. We also assume that there exists a nonempty open subset ω 0 such that ω 0 ⊂ θ∈Θ ω(θ).

System (3.1) corresponds to (2.1) with A = ∆, D(A) = H 2 (Ω) × H 1 0 (Ω) and B(θ) = β(θ)χ ω(θ) (x).
Obviously, in this context, Assumption (hp 1 ) is satised, and Assump-

tion (hp 2 ) is satised, if β ∈ L 2 (Θ). Also, A = A and for every θ ∈ Θ, B(θ) = B(θ). Choosing Y = L 2 (ω 0 ), L = C = χ ω 0 , Assumptions (hp 3
) and (hp 4 ) are sat- ised. Since the heat equation is approximatively controllable in any time T > 0 from any non-empty open subset of Ω, it follows by duality that (A , C) is also approximately observable in any time, that is to say that Assumption (hp 5 ) is also satised. As a consequence of Theorem 2.5, we have the following result.

Proposition 3.1. If α and β satisfy Assumption (hp 6 ), and if there exists

a non-empty open subset ω 0 such that ω 0 ⊂ ω(θ) for every θ ∈ Θ, then the system (3.1) is L 2 (Θ, L 2 (Ω))-ensemble controllable.

Application to the Grushin equation

Let γ > 0 and Ω = (-1, 1) × (0, 1), we consider the 2D Grushin equation

ẏ(t, x, v, θ) = α(θ) ∂ 2 x y(t, x, v, θ) + |x| 2γ ∂ 2 v y(t, x, v, θ) +β(θ)χ ω(θ) (x, v)u(t, x, v), (t, x, v) ∈ R + × Ω, y(t, x, v, θ) = 0, (t, x, v) ∈ R + × ∂Ω, y(0, x, v, θ) = y 0 (x, v, θ), (x, v) ∈ Ω, (3.2)
where, as in Section 3.1, θ ∈ Θ, with Θ a compact interval of R. In this system, the control is acting in a domain ω(θ) ⊂ Ω that might depend on the parameter θ. We also assume that there exists a nonempty open subset ω 0 such that ω 0 ⊂ θ∈Θ ω(θ).

We set X = U = L 2 (Ω). For every θ ∈ Θ, we set B(θ [START_REF] Heler | Spectral theory and its applications[END_REF]Section 4.3], or [21, Section 2.1]) of the operator A, is extended to a selfadjoint operator with dense domain D(A) in L 2 (Ω), and we refer to [START_REF] Morancey | Approximate controllability for a 2d Grushin equation with potential having an internal singularity[END_REF] for a precise description of D(A). Furthermore, this operator (still denoted by A) is the generator of an analytic semi-group on L 2 (Ω), see [21, Section 2.1].

) = β(θ)χ ω(θ) (x, v). If β ∈ L 2 (Θ), then Assumption (hp 2 ) is satised. For every f ∈ C ∞ 0 (Ω), we dene A by Af = ∂ 2 x f + |x| 2γ ∂ 2 v f . Using Friedrichs extension (see
In addition, using Poincaré inequality, we get that the semi-group generated by A is exponentially stable. That is to say that Assumption (hp 1 ) is fullled.

As for Section 3.1, Assumption (hp 3 ) is satised with Y = L 2 (ω 0 ), L = C = χ ω 0 , and Assumption (hp 4 ) is also trivially satised. According to [START_REF] Morancey | Approximate controllability for a 2d Grushin equation with potential having an internal singularity[END_REF], the pair (A * , C) is approximatively observable, i.e., Assumption (hp 5 ) is satised. We also refer to [START_REF] Laurent | Tunneling estimates and approximate controllability for hypoelliptic equations[END_REF] for survey results on approximate controllability for hypoelliptic equations, and let us quote that depending on ω 0 and γ, the pair (A, C * ) might not be null controllable, see [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Koenig | Non-null-controllability of the Grushin operator in 2d, C. R[END_REF][START_REF] Dardé | Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with nonrectangular control sets[END_REF]. In conclusion, Theorem 2.5 leads to the following result. Proposition 3.2. If α and β satisfy Assumption (hp 6 ), and if there exists a non-empty open subset ω 0 such that ω 0 ⊂ ω(θ) for every θ ∈ Θ, then the system (3.2) is L 2 (Θ, L 2 (Ω))-ensemble controllable, for every γ > 0.

Application to average controllability

In this paragraph, we link the concept of ensemble controllability to the one of Average controllability. As far as we know, the concept of average controllability has been introduced in [START_REF] Zuazua | Averaged control[END_REF] with the aim of controlling in average the dynamics of systems that depend on parameters with parameter independent open loop controls. Many results on partial dierential equations have been obtained and can be found for instance in [START_REF] Lü | Averaged controllability for random evolution Partial Dierential Equations[END_REF][START_REF] Avetisyan | Averaged controllability of Euler-Bernoulli beams with random material characteristics: the Green's function approach[END_REF][START_REF] Bárcena-Petisco | Averaged dynamics and control for heat equations with random diusion[END_REF][START_REF] Coulson | On average controllability of random heat equations with arbitrarily distributed diusivity[END_REF][START_REF] Marín | Robust averaged control of vibrations for the Bernoulli-Euler beam equation[END_REF]. Dealing with the numerical aspect of averaged controllability, we refer to [START_REF] Lazar | Control of parameter dependent systems[END_REF][START_REF] Abdelli | Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation[END_REF].

In this paper, the following denition borrowed from [START_REF] Zuazua | Averaged control[END_REF] will be considered. Denition 3.3. Let M = (Θ, T , µ) be a probability measure space. The system (2.1) is Approximately controllable in average in time T > 0 if for every y 0 , y 1 ∈ L 2 (Θ, X) and every ε > 0, there exists a parameter independent open loop control u ∈ L 2 ([0, T ], U ) such that the solution of (2.1) with y 0 as initial state datum and u as input satises

Θ y(T, θ) -y 1 (θ) dµ(θ) X ε. (3.3)
The next result follows directly from Theorem 2.5.

Corollary 3.4. Under the Assumptions (hp 1 ) to (hp 6 ), the system (2.1) is approximately controllable in average in any time T > 0.

It is well-known that average controllability does not imply ensemble controllability. Indeed, consider, for instance, the system (2.1) with

M = ([0, 1], T , µ), α(θ) = 1, A = 1, B(θ) = θ.
In this example, we take µ to be the Lebesgue measure and X = U = R. It is easily seen through [26, Theorem 1] that this system is approximatively controllable in average in any time, but, according to [3, Proposition 7], it cannot be L 2 (Θ)-ensemble controllable in any time T > 0.

Proof. We aim to show that for every y 0 , y 1 ∈ L 2 (Θ, X) and every ε > 0, there exists a parameter independent open loop control u ∈ L 2 ([0, T ], U ) such that (3.3) is satised.

Let y 0 , y 1 ∈ L 2 (Θ, X) and take any ε > 0.

According to Theorem 2.5, the system is L 2 (Θ, X)-ensemble controllable in any time T > 0. Hence, writing down Denition 2.4, there exists an open

loop control u ∈ L 2 ([0, T ], U ) such that y(T ) -y 1 L 2 (Θ,X) ε. (3.4)
Using Cauchy-Schwarz inequality (recall that µ(Θ) = 1), we get

Θ y(T, θ) -y 1 (θ) dµ(θ) X Θ y(T, θ) -y 1 (θ) X dµ(θ), Θ y(T, θ) -y 1 (θ) 2 X dµ(θ) 1 2 y(T ) -y 1 L 2 (Θ,X)
ε.

As a consequence of Corollary 3.4, the heat equation considered in (3.1) is approximately controllable in average in any time T > 0. This result can be found in [27, Section 3], see also [START_REF] Bárcena-Petisco | Averaged dynamics and control for heat equations with random diusion[END_REF]. In fact, it has been proved that the system (3.1) is null controllable in average in any time T > 0.

An other consequence is that the Grushin equation considered in (3.2) is also approximately controllable in average in any time T > 0. As far as we know, the averaged controllability for the Grushin equation has never been considered.

Proof of Theorem 2.5

In this section, we provide a proof to our main result.

By Denition 2.4, system (2.1) is L 2 (Θ, X)-ensemble controllable in a time T > 0 if and only if Ran(Φ T ) = L 2 (Θ, X), (4.1)
relation which is equivalent to

Ker(Φ T ) = {0}. (4.2)
We now compute the adjoint of Φ T . Since Φ T is a linear and bounded operator from L 2 (R + , U ) into L 2 (Θ, X), Φ T is a linear and bounded operator from L 2 (Θ, X) into L 2 (R + , U ). Therefore, for every ϕ 1 ∈ L 2 (Θ, X) and every u ∈ L 2 (R + , U ), we have, by denition

Φ T ϕ 1 , u L 2 (R + ,U ) = Θ ϕ 1 (θ), (Φ T u)(θ) X dµ(θ) = Θ ϕ 1 (θ), T 0 T α(θ)(T -τ ) B(θ)u(τ ) dτ X dµ(θ) = T 0 Θ B(θ) T α(θ)(T -τ ) ϕ 1 (θ) dµ(θ), u(τ ) U dτ.
It therefore follows that for every ϕ 1 ∈ L 2 (Θ, X),

Φ T ϕ 1 (t) =    Θ B(θ) T α(θ)(T -t) ϕ 1 (θ) dµ(θ), if t ∈ (0, T ), 0, if t T. (4.3)
Due to (4.2), we now aim to show the following unique continuation property,

Θ B(θ) T α(θ)(T -t) ϕ 1 (θ) dµ(θ) = 0, ∀t ∈ [0, T ] ⇒ ϕ 1 = 0. (UCP) Let ϕ 1 ∈ L 2 (Θ, X) such that for every t ∈ [0, T ], Θ B(θ) T α(θ)(T -t) ϕ 1 (θ) dµ(θ) = 0. (4.4) Setting ϕ(t, θ) = T α(θ)(T -t) ϕ 1 (θ), ϕ solves φ(t, θ) = -α(θ)A ϕ(t, θ) (t ∈ (0, T )), ϕ(T, θ) = ϕ 1 (θ). (4.5) Setting again φ(t, θ) = ϕ(T -t, θ), it follows that φ solves φ(t, θ) = α(θ)A φ(t, θ) (t ∈ (0, T )), φ(0, θ) = ϕ 1 (θ). (4.6) Equation (4.6) shows that φ(t, θ) = T α(θ)t ϕ 1 (θ). Since T t is an analytic semigroup, (UCP) is equivalent to Θ B(θ) φ(t, θ) dµ(θ) = 0, ∀t 0 ⇒ ϕ 1 = 0, (UCP')
where φ solves (4.6). Using (hp 1 ) and applying the operator L dened in (hp 3 ) to the above equation, we get Θ LB(θ) φ(t, θ) dµ(θ) = 0 (t 0).

Assumption (hp 3 ) ensures that L(B ϕ)(θ) = β(θ)Cϕ(θ) for every ϕ ∈ L 2 (Θ, X). Hence, it follows that C Θ β(θ) φ(t, θ) dµ(θ) = 0 (t 0), (4.7)
that is to say,

Θ β(θ) φ(t, θ) dµ(θ) ∈ Ker(C) (t 0).
Therefore, using (hp 1 ), we have, according to Remark 2.1,

Θ β(θ) φ(t, θ) dµ(θ) ∈ Ker(C) ∩ D(A ) (t > 0).
and Assumption (hp 4 ) leads to

C Θ β(θ)A φ(t, θ) dµ(θ) = 0 (t > 0). (4.8)
But φ is solution of (4.6), hence, we have

A φ(t, θ) = 1 α(θ) φ(t, θ) (t > 0),
this combined with (4.8) gives us

C Θ β(θ) α(θ) φ(t, θ) dµ(θ) = 0 (t > 0).
For every t > 0, let us now integrate the above equation on [t, +∞), to obtain

C Θ β(θ) α(θ) +∞ t φ(τ, θ) dτ dµ(θ) = 0 (t > 0). (4.9)
Using the fact that for every θ ∈ Θ, T α(θ)t is exponentially stable, we deduce that +∞ t φ(τ, θ) dτ = -φ(t, θ) (t > 0, θ ∈ Θ).

Equation (4.9) therefore reads

Θ β(θ) α(θ) C φ(t, θ) dµ(θ) = 0 (t > 0). (4.10)
Applying the above process k times, we end up with

Θ β(θ) α(θ) k C φ(t, θ) dµ(θ) = 0 (t > 0, k ∈ N).
Using Assumption (hp 6 ) and Remark 2.2, we deduce that for µ-almost every θ C φ(t, θ) = 0, ∀t > 0.

Finally, we use Assumption (hp 5 ) to deduce that for µ-almost every θ, ϕ 1 (θ) = 0.

This proves (UCP), and hence (2.1) is L 2 (Θ, X)-ensemble controllable in time T .

Digression on ordinary dierential equations

In this section we consider the case where X = R n , U = R m and Y = R q , with n, m and q three integers. In this framework (2.1) is just an ordinary dierential equation and the Assumptions (hp 1 ) to (hp 6 ) read:

(as 1 ) The matrix A belonging to R n n is Hurwitz;

(as 2 ) The control operator B belongs to L 2 (Θ) n m ;

(as 3 ) There exist L ∈ R q m , C ∈ R q n and β ∈ L 2 (Θ) such that for every ϕ ∈ L 2 (Θ) n , LB (θ)ϕ(θ) = β(θ)Cϕ(θ);

(as 4 ) Ker(C) is stable by A ; (as 5 ) (A , C) is observable;

(as 6 ) The function α ∈ C(Θ) is positive, injective and non-vanishing, and β ∈ L 2 (Θ) is such that β(θ) = 0 for µ-almost every θ ∈ Θ.

In this case, the proof of Theorem 2.5 can be given by classical arguments.

Since this classical arguments reveal that certain assumptions are not necessary for controllability we give the main lines. Indeed, thank to (UCP), let

ϕ 1 ∈ L 2 (Θ) n such that Θ B(θ) e α(θ)(T -t)A ϕ 1 (θ) dµ(θ) = 0 (t ∈ [0, T ]).
(5.1)

Using Assumption (as 3 ), we have C Θ β(θ)e α(θ)(T -t)A ϕ 1 (θ) dµ(θ) = 0 (t ∈ [0, T ]).

(5.2)

Computing the derivatives of the above equation with respect to t, and evaluating the obtained equation at time t = T , we deduce

CA k Θ β(θ)α(θ) k ϕ 1 (θ) dµ(θ) = 0 (k ∈ N).
(5.3)

We now observe that Assumptions (as 4 ) and (as 5 ) imply Ker(C) = {0}. Indeed, Assumption (as 5 ) (i.e., the pair (A , C) is observable) is equivalent to the pair (A, C ) is controllable. Therefore, the space N = {x ∈ R n | ∀k ∈ N, (A ) k x ∈ Ker(C)} is reduced to the null subspace of R n . But Assumption (as 4 ) (i.e., Ker(C) is stable by A ) ensures that N = Ker(C), and hence, Ker(C) = {0}. Consequently, (5.3) implies

A k Θ β(θ)α(θ) k ϕ 1 (θ) dµ(θ) = 0 (k ∈ N).
(5.4)

Thanks to Assumption (as 1 ), A is invertible, hence, Θ β(θ)α(θ) k ϕ 1 (θ) dµ(θ) = 0 (k ∈ N).

(5.5)

Assumption (as 6 ) together with Remark 2.2 leads to

Span{θ ∈ Θ → β(θ)α(θ) k , k ∈ N} = L 2 (Θ),
and it follows that ϕ 1 = 0 µ-almost everywhere on Θ. This ends the proof of Theorem 2.5 in the case X = R n , U = R m and Y = R q .

We now give some comments on these lines.

Remark 5.1. We have seen in the above lines that Assumption (as 1 ) is not necessary in the case of ordinary dierential equations. In fact, we only need

A to be invertible. One can see from the proof we provided in Section 4 that the exponential stability character of the semigroup was used to pass from (4.9) to (4.10). Therefore, this constraint was imposed by our proof.

Theorem 2.5 might hold without the exponential stability character of the operator (A, D(A)). However, this has resisted to several attempts of proof.

Remark 5.2. Observe that since Ker(C) = {0}, it follows that q n. Moreover, from this condition, it follows from (as 1 ) that for µ-almost every θ ∈ Θ, rk(B(θ)) = n. In particular, we have m n. Note that unlike the state controllability of linear time invariant systems where it is natural to consider m n, it makes fully sense here to consider systems with m n since the state space is an innite dimensional space, here L 2 (Θ) n .

Remark 5.3. Assumption (as 3 ) can be useful in practice when the control operator does not look nice. For instance, if one wants to check the L 2 (Θ) 2ensemble controllability of system (2.1) with

Θ = [0, 1], α(θ) = θ 2 , A = 0 -1 1 0 , B(θ) = 0 2 √ θ + 1 1 1 + θ √ θ + 1 0 1 + θ 3 θ 2 , (5.6)
this assumption tells us that it is not necessary to use ve sources (inputs or controls) in system (5.6) since only two sources are sucient. Indeed, taking in Assumption (as 3 )

L = 1 0 0 0 0 1 0 0 , C = 0 1 2 0 and β(θ) = √ θ + 1,
one can infer from [START_REF] Schönlein | Controllability of ensembles of linear dynamical systems[END_REF]Theorem 5] (or [4, Theorem 1]) that system (5.6) is L 2 (Θ) 2 -ensemble controllable. One should rst note that the multiplication operator

A β : L 2 (Θ) → L 2 (Θ) dened for every f ∈ L 2 (Θ) by (A β f )(θ) = β(θ)f (θ) is an isomorphism of L 2 (Θ), when β ∈ L ∞ (Θ) sat-
ises Assumption (as 6 ). Using therefore the fact that the system (2.1) with

Θ = [0, 1], α(θ) = θ 2 , A = 0 -1 1 0 , B(θ) = C = 0 2 1 0 , is C(Θ) 2 -
ensemble controllable, here comes [5, Theorem 5], we deduce from the density of continuous functions in L 2 (Θ) that the system given by (5.6) is L 2 (Θ) 2 -ensemble controllable.

Numerical aspects

The aim of this paragraph is to illustrate numerically the L 2 (Θ, X)ensemble controllability of system (3.1) in one dimensional case. More precisely, we take Θ a compact interval of R * + , µ the Lebesgue measure,

Ω = (0, π), ω(θ) = ω = [ω, ω] ⊂ (0, π) independent of θ, X = L 2 (0, π), U = L 2 (ω)
, and consider the system ẏ(t, x, θ) = α(θ)∂ x y(t, x, θ) + β(θ)χ ω (x)u(t, x), (t, x) ∈ R + × (0, π), y(t, x, θ) = 0, (t, x) ∈ R + × {0, π}, y(0, x, θ) = y 0 (x, θ),

x ∈ (0, π), (6.1) where α and β satisfy Assumption (hp 6 ). We recall that there exist an increasing sequence

(λ k ) k 1 ∈ (R + ) N * such that λ k → ∞ when k → ∞ and a Hilbert basis (φ k ) k 1 of L 2 (0, π) such that (φ k ) k 1 ⊂ H 1 0 (0, π) and -∂ 2 x φ k = λ k φ k .
In the present case, we have

λ k = k 2 and φ k (x) = 2 π sin(kx) (k ∈ N * , x ∈ (0, π)).

System discretization

Given any y 0 ∈ L 2 (Θ, X) and every u ∈ L 2 ([0, T ], L 2 (0, π)), we write the solution y(t, θ) = ∞ k=1 y k (t, θ)φ k where y k (t, θ) = y(t, θ), φ k L 2 (0,π) solves, for every k 1,

ẏk (t, θ) = -α(θ)λ k y k (t, θ) + β(θ)u k (t) (t > 0), y k (0, θ) = y 0 k (θ). (6.2)
where we have set u k (t) = χ ω u(t), φ k L 2 (0,π) and y 0 k (θ) = y 0 (θ), φ k L 2 (0,π) . We now use nite element method by subdividing the interval ω in M+2 points (x k ) k∈N M+1 where x k = ω + kh with h = (ω -ω)/(M + 1), to approximate the control u by

u M (t, x) = M =1 v (t)ψ (x), (6.3) where ψ (x) =      x-x -1 x -x -1 , if x ∈ [x -1 , x ], x +1 -x x +1 -x , if x ∈ [x , x +1 ], 0 otherwise ( ∈ N * M ), Since by construction u M is supported in ω, we have, by setting u M k (t) = χ ω u M (t), φ k L 2 (0,π) , u M k (t) = χ ω u M (t), φ k L 2 (0,π) = M =1 Γ k, v (t), with Γ k, = φ k , ψ L 2 (0,π) . A direct computation leads to Γ k, = -1 hk 2 (φ k (x -1 ) -2φ k (x ) + φ k (x +1 )) (k ∈ N * N , ∈ N * M ).
Setting Y(t, θ) = (y 1 (t, θ), . . . , y N (t, θ)) , U(t) = (v 1 (t), . . . , v M (t)) , we deduce from (6.2) that

Ẏ(t, θ) = α(θ)AY(t, θ) + β(θ)BU(t), (6.4)
where

A =    -λ 1 . . . -λ N    ∈ R N N and B =    Γ 1,1 . . . Γ 1,M . . . . . . Γ N,1 . . . Γ N,M    ∈ R N M . (6.5)
From the above, each solution y(t, θ) = ∞ k=1 y k (t, θ)φ k can be found as a limit of y N (t, θ) = N k=1 y k (t, θ)φ k when N tends to innity, where (y 1 (t, θ), . . . , y N (t, θ))

= Y(t, θ) solves (6.4) with Y(0, θ) = (y 0 1 (θ), . . . , y 0 N (t, θ)) . Let us observe that the system (6.4) is L 2 (Θ) N -ensemble controllable.

More precisely, we have the following proposition. Proposition 6.1. Assume that (hp 6 ) holds. For every N ∈ N * and every M 3N -1, the system (6.4) is L 2 (Θ) N -ensemble controllable, where A and B are given by (6.5).

Proof. Let us show that Assumptions (as 1 ) to (as 5 ) are satised. We clearly have that Assumptions (as 1 ) and (as 2 ) are satised. Assumption (as 3 ) is also satised with C = B * and L = I M .

We then aim to prove that for M large enough, we have rk(B) = N. This will ensure Assumptions (as 4 ) and (as 5 ), and the proof will be concluded using Theorem 2.5 (see also Section 5).

Let us assume by contradiction that rk(B) < N, then there exists α 1 , . . . , α N ∈ R, which are not all trivial, such that

0 = N k=1 α k φ k , ψ L 2 (0,π) ( ∈ N * M ). (6.6)
Let us then dene f (x) = N k=1 α k φ k , there exists a polynomial P of degree at most N -

1 such that f (x) = sin(x)P (cos(x)) for every x ∈ [0, π]. More precisely, P = N k=1 α k U k-1
, where U k is the second kind Tchebychev polynomial of degree k. This ensures that f admits at most N -1 zeros on (0, π). Consequently, there exist an interval I ⊂ ω of length greater than (ω -ω)/N where f is of constant sign. But, for M 3N -1 (i.e., 3 ω-ω M+1 ω-ω N ), there exists ∈ N * M such that x -1 , x , x +1 ∈ I, that is to say that supp ψ ⊂ I. This leads to a contradiction with (6.6), since ψ and f have a constant sign on supp ψ , and do not vanish in the interior of supp ψ .

Minimal L 2 -norm control

Given T > 0, y 0 , y 1 ∈ L 2 (Θ, L 2 (0, π)) and ε > 0, we aim to minimize the following problem

min 1 2 T 0 ω |u(t, x)| 2 dx dt s.t. θ π 0 |y(T, θ, x) -y 1 (θ, x)| 2 dx dθ ε, (6.7)
where y is the solution of (6.1) with control u and initial condition y 0 . Using Fenchel-Rockafellar duality, see e.g. [START_REF] Borwein | Techniques of variational analysis[END_REF][START_REF] Ekeland | Convex analysis and variational problems[END_REF], we obtain that the minimizer of the above problem is given by

u(t, x) = Θ β(θ)ϕ(t, θ, x) dθ (t ∈ (0, T ), x ∈ ω),
where ϕ solves the adjoint problem

-φ(t, θ, x) = α(θ)∂ 2 x ϕ(t, θ, x) (t ∈ (0, T ), x ∈ (0, π), θ ∈ Θ), ϕ(T, θ, x) = ϕ 1 (θ, x) (x ∈ (0, π), θ ∈ Θ). with ϕ 1 ∈ L 2 (Θ, L 2 (0, π)) the minimizer of J(ϕ 1 ) = 1 2 T 0 ω Θ β(θ)ϕ(t, θ, x) dθ 2 dx dt - Θ π 0 y 1 (θ, x)ϕ 1 (θ, x) dx dθ + Θ π 0 y 0 (θ, x)ϕ(T, θ, x) dx dθ + ε Θ π 0 |ϕ 1 (θ, x)| 2 dx dθ.
Considering the discretization proposed in Section 6.1, the minimization problem (6.7) becomes

min 1 2 T 0 v(t) Mv(t) dt s.t. θ |Y(T, θ) -Y 1 (θ)| 2 dθ ε, (6.8)
where Y is solution of (6.4), with initial condition Y 0 (θ), and where for ı ∈ {0, 1}, Y ı (θ) = y ı (θ), φ 1 L 2 (0,π) , . . . , y ı (θ), φ N L 2 (0,π) , and where M ∈ R M M is given by M k, = ψ k , ψ L 2 (0,π) . Note that one can also choose M = hI M .

The minimizer is obtained through the adjoint system, i.e.,

v(t) = M -1 B Θ β(θ)p(t, θ) dθ (t ∈ (0, T )), with p solution of -ṗ(t, θ) = α(θ)A p(t, θ) (t ∈ (0, T ), θ ∈ Θ), p(T, θ) = p 1 (θ) (θ ∈ Θ), with p 1 ∈ L 2 (Θ) N the minimizer of J(p 1 ) = 1 2 T 0 B Θ β(θ)p(t, θ) dθ M -1 B Θ β(θ)p(t, θ) dθ dt - Θ Y 1 (θ), p 1 (θ) dθ + Θ Y 0 (θ), p(T, θ) dθ + ε Θ |p 1 (θ)| 2 dθ.
Remark 6.2. It is expected (but not proved yet) that ϕ 1,N , dened by ϕ 1,N (θ, x) = N k=1 p 1 (θ) sin(kx), goes to ϕ 1 as N and M goes to ∞.

Numerical examples

To numerically approximate a control, we nally consider a discretization of Θ and numerically compute the integral over Θ, similarly, we use a numerical scheme to approximate the solutions of (6.4) and of the adjoint system.

In practice, we have used the mid point rule on a uniform discretization of Θ to compute the integrals over Θ, and the Crank-Nicolson method, with a uniform discretization of [0, T ] to compute the solutions of the direct and adjoint systems.

For the numerical simulation below, we have considered Θ = [1, 2], α(θ) = θ, β(θ) = 1 and ω = (0.5, 0.8). We have also considered N = 21, M = 200, and the discretization step is 5.10 -3 for Θ and 5.10 -4 for the time interval [0, T ]. The nal time is set to T = 1, and we have set ε = 10 -1 . We will consider two dierent initial and target states:

(i) y 0 (θ, x) = 2θx(π -x)(x -θ) and y 1 (θ, x) = 0; (ii) y 0 (θ, x) = θ if x < θ, -θ if x > θ
and y 1 (θ, x) = (2 -θ) sin(3x). On Figures 1 and5, we display the obtained controls. On Figures 2 and6, we show the L 2 -norms of y(t) -y 1 with respect to time, at the nal time T , they are equal to ε (up to a numerical tolerance which as been set to 10 -4 ), and for the two examples, the functions (θ, x) → y(T, θ, x) are plotted on Figures 3 and7. Finally, we have shown on Figures 4 and9 the time evolution of the N Fourier coecients used for the simulations.

Concluding remarks

In this paper, we have studied the L 2 (Θ, X)-ensemble controllability of some class of parabolic equations. But there still exist many practical questions to be addressed. Out of the one raised in Remark 5.1, we list below some of them.

• In Assumption (hp 3 ) we have assumed a kind of decoupling condition on the control operator, mainly the separation between the space variable and the parameter one. What if this splitting is not possible? For instance, in (3.1), one might think of the control operator of the form • Can we consider α depending on the space variable in (2.1)? Similarly, can we consider an operator A that depend on θ? For instance the ensemble controllability of (3. 1) is open if we replace θ∆ by div (M (θ)∇), with M (θ) a positive matrix (which is not of the form α(θ)I).

B(θ, x) = χ ω (x)ρ(θ, x) instead of B(θ, x) = χ ω (x)ρ(θ).
• The question of C(Θ, X)-ensemble controllability, that is uniform ensemble controllability of system (2.1) is open. 8: Error to the target, i.e., plot of (θ, x) → y(T, θ, x) -y 1 (θ, x). See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii).

ensemble controllability is also impossible. However, this is not clear in higher space dimension. In fact, we cannot end up with a contradiction with the Hautus test, as it is done in [START_REF] Dirr | Uniform and L q -ensemble reachability of parameter-dependent linear systems[END_REF].

• Dealing with the numerical approach: what are the required relations between N, M and the discretization of [0, T ] and Θ, to ensure that the discrete control u M and the discrete adjoint ϕ 1,N goes to u and ϕ 1 as N → ∞?

In addition, we chose here to approach y with a nite combination of eigenfunctions of the Dirichlet-Laplace operator. This was to ensure the ensemble controllability of the discretized system (see Proposition 6.1).

Even if other discretization are possible, it is not clear that the discretized system will be ensemble controllable. For instance, if we use classical nite dierences, we will necessarily have M < N when ω is strictly included in Ω, and hence rk(B) < N. Thus, the controllability of the discretized system is no more a direct application of Theorem 2.5.

Figures 1 to 4 (

 4 Figures 1 to 4 (respectively Figures 5 to 9) tackle the case where the initial and target state are given by (i) (respectively (ii)).

Figure 1 :

 1 Figure 1: Control (L 2 -norm: 4.230773 × 10 2). See Section 6.3 for the parameters used for this simulation, initial and target states are given by (i).

  ) L 2 (Θ,L 2 (0,π))

Figure 2 :

 2 Figure 2: Time evolution of the L 2 -norm of the solution. See Section 6.3 for the parameters used for this simulation, initial and target states are given by (i).

Figure 3 :

 3 Figure 3: Parameter dependent state at final time. See Section 6.3 for the parameters used for this simulation, initial and target states are given by (i).

Figure 4 :

 4 Figure 4: Time evolution of the 21 Fourier coefficients used in the simulation. See Section 6.3 the parameters used for this simulation, initial and target states are given by (i).

Figure 5 :-y 1 L 2

 512 Figure 5: Control (L 2 -norm: 1.945471 × 10 3 ). See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii).

Figure 6 :

 6 Figure 6: Time evolution of the L 2 -norm of the solution. See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii).

Figure 7 :

 7 Figure 7: Parameter dependent state at final time. See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii). See also Figure 8 for a visualization of the distance to the target state.

Figure

  Figure8: Error to the target, i.e., plot of (θ, x) → y(T, θ, x) -y 1 (θ, x). See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii).

Figure 9 :

 9 Figure 9: Time evolution of the 21 Fourier coefficients used in the simulation. See Section 6.3 for the parameters used for this simulation, initial and target states are given by (ii).

•

  Last but not least, in this paper we have considered the problem of internal controllability. The case of boundary controllability is open.

					Y1(t)								Y2(t)								Y3(t)
	-10 -5 0 5								0 -10 -5 5 10								-5 0 5
	-20 -15								-25 -20 -15								-15 -10
		0	0.2	0.4	t	0.6	0.8	1		0	0.2	0.4	t	0.6	0.8	1		0	0.2	0.4	t	0.6	0.8	1
					Y4(t)								Y5(t)								Y6(t)
	0 -5 -3 -2 1 -1 -4 -6	0	0.2	0.4	t	0.6	0.8	1	1 0 -1 -2 -3	0	0.2	0.4	t	0.6	0.8	1	4 3 2 1 0 -1 -2 -3 -4	0	0.2	0.4	t	0.6	0.8	1
					Y7(t)								Y8(t)								Y9(t)
	2 3 4								1.5 2 2.5 3								1 1.5
	0 -2 -1 1								-1 -0.5 0 0.5 1								-0.5 0 0.5
		0	0.2	0.4	t	0.6	0.8	1		0	0.2	0.4	t	0.6	0.8	1		0	0.2	0.4	t	0.6	0.8	1
					Y10(t)								Y11(t)								Y12(t)
	0.5 1 1.5								0 0.5 1								-0.4 -0.2 0 0.2 0.4
	0 -0.5	0	0.2	0.4	t	0.6	0.8	1	-0.5 -1	0	0.2	0.4	t	0.6	0.8	1	-0.6 -0.8 -1 -1.2	0	0.2	0.4	t	0.6	0.8	1
					Y13(t)								Y14(t)								Y15(t)
	0 -0.6 0.2 0.4 -0.2 -0.4 -0.8	0	0.2	0.4	t	0.6	0.8	1	0 0.2 -0.2 -0.4 -0.6 -0.8	0	0.2	0.4	t	0.6	0.8	1	0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7	0	0.2	0.4	t	0.6	0.8	1
					Y16(t)								Y17(t)								Y18(t)
	0.3								0.25
	0.2								0.2								0.3
	0.1 0 -0.2 -0.1 -0.3 -0.4	0	0.2	0.4	t	0.6	0.8	1	0.1 0.15 0.05 0 -0.05 -0.1	0	0.2	0.4	t	0.6	0.8	1	0.2 0.1 0 -0.1	0	0.2	0.4	t	0.6	0.8	1
					Y19(t)								Y20(t)								Y21(t)
	0.4								0.25								0.1
	0.3								0.2								0.08
	0 -0.1 0.1 0.2	0	0.2	0.4	t	0.6	0.8	1	0.15 0.1 0.05 0 -0.05	0	0.2	0.4	t	0.6	0.8	1	0.06 0.04 0.02 0 -0.02 -0.04	0	0.2	0.4	t	0.6	0.8	1
			In fact, even if, as stated in Remark 2.7, the results are still valid with Color used for the dient values of θ
			unbounded control operators, the verication of Assumption (hp 4 ) is 1 1.2 1.4 1.6 1.8 2
			not easy, and we do not have examples of unbounded control operators θ
			where Assumption (hp 4 ) is fullled.
			For a one dimensional heat equation as considered in (3.1) but this
			time with boundary controls, we can prove using [3] that the C(Θ, X)-
			ensemble controllability cannot hold, and we believe that the L 2 (Θ, X)-

Acknowledgements

The authors thank Martin Lazar for helpful discussions and comments on this topic. The authors are supported by the French ANR project TRECOS, ANR20CE400009. The second author is also supported by the ANR project NICETWEET, ANR20CE480009. This work has been initiated while the rst author was member of Université de Lorraine, CNRS, Inria, IECL, Nancy, and he thanks IECL for hosting and support.


Erfan Dashtimoghadam

Mitchell Maw

Andrew N Keith

Foad Vashahi

Yulia D Gordievskaya

Elena Yu Kramarenko

Egor A Bersenev

Evgeniia A Nikitina

Dimitri A Ivanov

Yuan Tian

Andrey V Dobrynin

Mohammad Vatankhah-Varnosfaderani

Sergei S Sheiko

Super-soft, firm, and strong elastomers toward replication of tissue viscoelastic response

Polymeric networks are commonly used for various biomedical applications, from reconstructive surgery to wearable electronics. Some materials may be soft, firm, strong, or damping however, implementing all four properties into a single material to replicate the mechanical properties of tissue has been inaccessible. Herein, we present the A-g-B brush-like graft copolymer platform as a framework for fabrication of materials with independently tunable softness and firmness, capable of reaching a strength of ~10MPa on par with stress-supporting tissues such as blood vessel, muscle, and skin. These properties are maintained by architectural control, therefore diverse mechanical phenotypes are attainable for a variety of different chemistries. Utilizing this attribute, we demonstrate the capability of the A-g-B platform to enhance specific characteristics such as tackiness, damping, and moldability.

New concepts

Soft tissues are protected from accidental trauma by two intrinsic defense mechanisms: strain-adaptive stiffening and shock absorbance. Although many synthetic materials can replicate tissue softness, matching tissue strength and viscoelastic response remains challenging. We report the A-g-B brush-like graft copolymer platform for the design of thermoplastic elastomers as a framework for full replication of tissue softness, firmness, strength, and energy dissipation. While the bottlebrush B-block facilitates material softness, microphase separation of randomly grafted Ablocks yields a reversible physical network that concurrently enhances mechanical resilience and damping. Unlike the conventional "one molecule -one strand" approach to the network construction, one A-g-B molecule spans multiple meshes, which reinforces the integrity of the stress-supporting scaffold. Furthermore, the design-by-architecture approach empowers architectural programming of mechanical properties at a given chemical composition by adjusting dimensions of the A and B blocks. Reciprocally, the platform allows tuning of the A-g-B chemistry at a given architecture for a desired mechanical profile to satisfy application specific needs, such as moldability, tackiness, and controlled swellability. The synergistic combination of the architectural and chemical control enables precise and predictable property regulation of elastomeric materials for a broad range of practical applications including but not limited to biomedical devices, pressure-sensitive adhesives, and additive manufacturing.

Introduction

Combining softness, firmness, strength, and damping into a neat material is incompatible with current polymer network designs, yet such a combination is commonplace in biological tissues. [START_REF] Griffith | [END_REF][2][3][4][5][6][7][8][9][10][11] Initially very soft tissues (Young's modulus: ~ 10 2 -10 5 Pa) stiffen rapidly with deformation, empowering up to a 1000-fold modulus increase (aka firmness) and the ability to withstand >10 MPa stress-at-break for protection of delicate organs against accidental rupture. Additional protection is provided a relatively high damping factor ( ) ranging from 0.1 of skin to 0.7 of brain tissue, which allows for absorbing shocks and vibrations in a broad frequency range. This combination of distinct mechanical traits is enabled by a hierarchical scaffold of collagen fibers that endorse a cascade of molecular deformation mechanisms activated at different stress levels.

12 Implementation of such self-organization principles in synthetic materials is challenging. Despite recent advances in replicating tissue's softness and strain-stiffening, [13][14][15][16][17][18][19] integrating softness with tissue-comparable strength remains elusive. Successful efforts to increase the strength and toughness of elastomers through double networks typically lead to a The mesh interconnectivity is highlighted by a bold backbone of an A-g-B macromolecule. Interplay of multiple architectural parameters, such as length and grafting density of side chains in the B block as well as A block dimensions, permits unparalleled control of equilibrium and viscoelastic mechanical properties. Chemistries chosen for application specific functions, such as adhesion, moldability, or water uptake, can be implemented within specific mechanical phenotypes. (b) Stress-elongation curves of selected covalent bottlebrush elastomer ( kPa and ) and a thermoplastic ABA brush copolymer ( = 13.3 kPa and ) samples. Although, the ABA system demonstrate very high firmness thay may reach 0.9 on par with skin (dashed line), both systems exhibit low strength ( ). 15 (c) Self-assembled networks of A-g-B brush-like graft copolymers display a unique combination of softness, firmness, and strength (ESI, Table S1). A sample of PBA-ran-PMMA-g-(PDMS/PS) graft copolymer exhibits strength ~8 MPa on par with aorta (dashed line), surpassing many other strong tissues.

Please do not adjust margins Please do not adjust margins considerable modulus increase ( > 10 6 Pa). [20][21][22] On the other hand, advancements in the design of super-soft elastic materials suffer from consequentially low strength. 23,24 Herein, we present the bottlebrush graft copolymer (A-g-B) platform, which delivers robust physical networks where a single molecule connects multiple meshes (as designated by the bold backbone) empowering a unique combination of softness, firmness, and strength (Fig. 1a). Furthermore, this platform accommodates broad chemical diversity of the A and B blocks to satisfy needs of specific applications such as tissue-mimetic elastomers for biomedical devices, soft robotics, wearable electronics, pressure-sensitive adhesives, and additive manufacturing.

Incorporation of bottlebrush macromolecules into elastomers has led to a breakthrough in mechanical property control of polymer networks. 25,26 Due to the architectural disentanglement of brush-like strands, it became possible to prepare super-soft elastomers with a modulus down to 100 Pa (Fig. 1b). 27 However, brush networks with covalent crosslinks show relatively low firmness, , defined by the strainstiffening parameter  a ratio of the mean square end-to-end distance, , of undeformed strands to their contour length, . 14,28,29 Firmness was subsequently increased through the design of ABA block copolymer networks (middle panel, Fig. 1a), where the bottlebrush backbones get additionally extended by microphase segregation of A and B blocks yet remain stretchable due to the "hidden length" of coiled A-blocks inside the network nodes. 13,30 These thermoplastic elastomers with bottlebrush strands demonstrated unprecedented firmness up to on par with biological tissues, while maintaining the tissue-like softness ( ) and extensibility ( ) (Fig. 1b). However, the ABA systems possess a relatively low strength of 0.5 MPa, which is ~10 times weaker than that of stress-supporting tissues such as skin and blood vessels (ESI, Table S1). 13,15 The low strength is ascribed to the "one strand -one molecule" construction, where the deformationcaused withdrawal of an A block from a network node leads to its coiling and strand removal from the load-bearing scaffold. 13,30 To address this issue, we introduce A-g-B bottlebrush graft copolymers ("-g-" denotes long A-blocks randomly grafted to a bottlebrush backbone of block B), where one brush molecule may span multiple network cells to enhance network resilience (Fig. 1c). When an A-block is dislodged from an A-domain during deformation, the corresponding strand remains strained, which concurrently maintains the load-bearing scaffold, improves tension distribution, and allows for re-association of the loose A-block with neighboring A-domains. Even though the A-g-B networks are less perfectly organized than the ABA networks, they demonstrate ~10-100 strength enhancement compared to ABA systems, attaining 8 MPa greater than human aorta (Fig. 1c, ESI, Table S1). 31 Furthermore, self-assembled Ag-B networks can be reversibly disassembled either by heating above the order-disorder temperature or dissolution in a good solvent, which facilitates materials processability.

Results and discussion

Synthesis A-g-B brush-like graft copolymers are defined by a set of six architectural parameters [ ], where is the degree of polymerization (DP) of the side-chains in the brush B block, is the DP of backbone spacer between side chains, and are respectively the DP and volume fraction of linear A-blocks, is the DP of a brush backbone, and corresponds to the backbone DP between A-blocks equivalent to the DP of network strand (Fig. 2). For convenience of discussion, we also introduce parameter , which approximately corresponds to an average number of A-blocks per bottlebrush macromolecule. To study the effect of mesh interconnectivity on elastomer strength, we synthesized a series of A-g-B graft copolymers (similar to "Janus graft block copolymers") 32 with different using a combination of reversible addition-fragmentation chain transfer (RAFT) copolymerization of polydimethylsiloxane (PDMS) and Brterminated poly(ethylene glycol) (PEG) macromonomers with consecutive atom transfer radical polymerization (ATRP) of poly(methyl methacrylate) (PMMA) A-block grafted from the terminal bromine, yielding poly[MMA-g-(PDMS/PMMA)] bottlebrush graft copolymers with (Fig 2, ESI, Table S2, Please do not adjust margins Please do not adjust margins Sections S1.2-S1.4). [33][34][35][36][37][38][START_REF]World Intellectual Property Organization[END_REF] To vary side chain grafting density ( , PDMS macromonomers were co-polymerized with nbutyl acrylate (BA) as backbone spacers between PDMS side chains yielding poly[nBA-ran-MMA-g-(PDMS/PMMA)] graft copolymers with (Fig. 2a). For convenience, bottlebrush (densely grafted) samples will be referred to as PMMA-g-PDMS ( =1), while the loosely grafted structures are named as PMMA-g-PDMS ( =4,8). All synthesized samples are summarized in the (ESI, Table S2), while selected series of A-g-B networks with representative properties and structural characteristics are in Table 1.

To demonstrate the universality of the platform and its modular nature in addressing specific applications, we prepared A-g-B graft copolymers with different chemical compositions of A-blocks and B-side chains. For example, PMMA A-blocks were replaced with polystyrene (PS) (ESI, Sections S1.7-S1.8, Scheme S3) to enable elastomer fluidity at ~100℃ for molding and 3D printing. By substituting PDMS side chains in the B-block with higher glass transition polyisobutylene (PIB) (ESI, Section S1.6, Scheme S2), we augmented viscoelastic dissipation at room temperature and conventional strain rates, thereby enhancing adhesion and vibration damping. For scalability, the controlled radical polymerization (CRP) methods were replaced with free radical polymerization (FRP) to synthesize PS-g-PDMS ( =8) and PSg-PIB ( =8) (Fig. 2b).

Structure

Different techniques were employed for characterization of both molecular structure and morphology of self-assembled Ag-B networks. The copolymer composition [ ] was monitored by [START_REF] Griffith | [END_REF] H-NMR spectroscopy while samples with identical compositions but different were verified by the number average molecular weight from gel-permeation chromatography (ESI, Fig. S5-S12). Atomic force microscopy (AFM) was used for molecular imaging of A-g-B macromolecules to confirm their dimensions and microphase separation (ESI, Section S2.1). Langmuir-Blodgett monolayers demonstrate densely packed worm-like PMMA-g-PDMS ( =1) macromolecules (Fig. 3a, ESI, Fig. S22), where the Fig. 3. Structural characterization of A-g-B brush copolymers by atomic force microscopy (AFM) and ultra-small angle X-ray scattering (SAXS). (a) AFM micrograph of a Langmuir-Blodgett monolayer of PMMA-g-PDMS ( =1) with shows densely packed worm-like macromolecules separated by a distance , where is a contour length of PDMS side chain in the brush block. The cross-sectional profile was measured perpendicular to the molecular orientation (dashed line). (b) AFM micrograph of a sparse monolayer exhibits star-like aggregates due to association of A-blocks. (c) Dimensions of A-g-B network morphology:

-interbrush distance, -A-domain diameter, -interdomain distance, -interfacial area per brush strand at the domain surface, -bottlebrush packing area in the bulk. (d) SAXS curves of PMMA-g-PDMS ( =1) samples with identical [ ] yet different numbers of A-blocks per bottlebrush ( ) as indicated. The variation does not cause any significant effects on the network morphology. The deviation of the green curve is due to , indicating that some molecules may have only one A block (loose ends) that lead to a smaller domain size ( ) and correspondingly smaller distance between the domains ( ) (Table 1). (e) SAXS curves of PS-g-PDMS ( =8) samples with identical [ ] yet different and corresponding values. For this sample series, the bottlebrush peak is broader and shifts towards higher values because of possible interpenetration of side chains and backbone folding inside the bottlebrush envelope promoted by the decrease in grafting density. 40 The increase of at a given results in the corresponding increase in the aggregation number, , and decrease of the interfacial area per brush strand, (Table 1). Please do not adjust margins intermolecular distance of 8.3 ± 1.0 nm is consistent with twofold the side chain contour length, , using and as a projection length of the PDMS repeat unit. The dense monolayer arrangement hinders the microphase separation of the A-blocks, yet starlike aggregates of multiple bottlebrushes are evident in loosely packed films (Fig. 3b).

More insight into bulk morphology of A-g-B networks was obtained by small-angle X-ray scattering (SAXS) (ESI, Section S2.2). First, A-g-B samples of near identical [ , ] yet different , i.e. different numbers of A-blocks per backbone ( ), are shown to produce nearly identical SAXS curves (Fig. 3d), which supports the hypothesis that long A-g-B macromolecules with form topologically similar networks, differing only in mesh interconnectivity. Second, we show that the network morphology depends on the A-g-B architecture. Specifically, we studied (i) variation at a given (Fig. 3e), (ii) variation at a given (ESI, Fig. S20), and (iii) variation of grafting density of B side chains (Table 1, ESI, Fig. S21). SAXS curves elicit three characteristic network dimensions: the interbrush distance ( ), A-domain diameter ( ), and the interdomain distance ( ) (Fig. 3c,d). For densely grafted bottlebrush blocks ( ), corresponds to the brush diameter and the Kuhn length of the bottlebrush backbone (

). 29 From the domain diameter, we obtain two structural parameters: (i) aggregation number and (ii) interfacial area per brush strand , where is mass density of A polymer, -domain volume, and -molecular mass of A-block (Table 1). Unlike the previously studied ABA linear-brush-linear copolymers, 13,15,40,41 each A block in an A-g-B macromolecule anchors two bottlebrush strands to a network node (Fig. 3c), which is accounted by the factor of 2 in the denominator of the equation. As a result, the footprint of bottlebrush strands at the domain surface, , is considerably smaller than the bottlebrush packing area in the bulk as . This suggests strong extension of bottlebrush backbone at the interface, causing additional enhancement of the strainstiffening response of A-g-B elastomers as discussed below. Please do not adjust margins Please do not adjust margins Systematic studies of backbone dilution on mechanical properties of brush networks by increasing , , and have been established previously, 13,14 therefore, our main focus here is twofold: (i) strengthening the network through enhancement of mesh interconnectivity by increasing the number of A-blocks per brush macromolecules ( ) and (ii) architecturally tuning the viscoelastic response. We conducted uniaxial tensile tests (ESI, Section S2.3, Fig. S24) of A-g-B brush networks with varying from ~200 to 2000 and ranging from ~1.4 to 13 accordingly, while keeping the other architectural parameters ( , , , , ) constant. All samples demonstrate similar softness, 50 kPa, and firmness, 0.46-0.54 yet the increases 15-fold from 0.04 MPa to 0.9 MPa (Fig. 4a). This trend is corroborated by coarse-grained molecular dynamics simulations due to a direct increase in energy cost for A-block withdrawal in samples with a higher -parameter (ESI, Fig. S32-34). Although the achieved strengthening is substantial, the absolute values of are relatively low. Additional strengthening was facilitated by concurrently decreasing grafting density of the side chains, , and increasing volume fraction of A blocks, (Fig. 4c). The decrease of grafting density reinforces the network by increasing volume fraction of the backbones. However, decreasing alone leads to firmness decline as observed in covalent brush networks. 14,29 To reverse the trend, A-g-B's with greater volume fraction of A-blocks, , were synthesized. These elastomers witness a dual effect on network structure: (i) higher aggregation number of network nodes ( ), hence higher crosslink functionality and (ii) stronger strand extension, hence higher firmness ( ) evidenced by the decreasing strand footprint ( ) (Table 1). The series of PS-g-PDMS ( =8) brush-like copolymers with ranging from 0.03 to 0.24 demonstrated steady increase of stress-at-break up to 6 MPa (Fig. 4c) which is comparable and even exceeds the strength of many soft tissues such as aorta, blood vessel, skeletal muscle, and even some cartilage tissues (ESI, Table S1).

Mechanical

Concurrent with the elastic response measured a low strain rate of (Fig. 4a,b), which corresponds to the rubber-elastic plateau for the PS-g-PDMS samples, the A-g-B architecture allows tuning the viscoelastic response. The frequency dependence of the storage modulus and damping factor was measured for samples varying (i) at a given (Fig. 4c), (ii) at a given , and (iii) grafting density, (ESI, Fig. S25-26). The lowering of the crosslink density at a given results in two effects (Fig. 4c): (i) decrease of the storage modulus ( ), which is consistent with the stresselongation curves in Fig. 4b, and (ii) lower frequency shift of the elastic plateau onset, which is evidenced by the corresponding increase of the damping (tan ). Other structural levers, such as grafting density, may be utilized as well to regulate viscoelastic behavior (ESI, Fig. S25). However, nearly full replication of tissues viscoelastic response is achieved by leveraging both architecture and chemistry of A-g-B networks. While a PS-g-PDMS ( =8) sample shows near identical Fig. 4. (a) Strength of PMMA-g-PDMS ( =1) elastomers systematically increases with the number of A blocks per A-g-B macromolecules, z, at a constant DP between A blocks of (Table 1). (b) Stress-elongation curves of PS-g-PDMS ( =8) samples with different (and correspondingly ) values as indicated (Table 1). The decrease of at a given results in progressively increasing strength. Please do not adjust margins Please do not adjust margins damping response to ligament, replacing PDMS with a higher glass transition PIB in PS-g-PIB ( =8) shifts the frequency spectrum demonstrating damping similar to porcine brain (Fig. 4d).

The effect of deformation on A-g-B network morphology was studied by in-situ SAXS which allows instantaneous monitoring of changes in the , , and spacings during the mechanical experiment (Fig. 4a). The in-situ uniaxial deformation employed had the strongest impact on the interdomain distance

, which progressively increased along the stretching axis while decreasing in the perpendicular directions. The 1D SAXS curves corresponding to the different deformations are shown in Fig. 5a together with selected 2D patterns given in the insets. Quantitatively the change in the -peak position as a function of deformation is displayed in Fig. 5b, where a significant shift in the -peak occurs with λ such that the regular network of nodules undergoes extension in the stretching direction which is proportional to the deformation applied. Similar to the inter-domain distance, the A-domain exhibits anisotropic variations: increasing along the stretching direction and decreasing in the perpendicular directions (Fig. 5c). The glassy PMMA nodules begin to deviate from solid spheres upon stretching suggesting the onset of A- Please do not adjust margins Please do not adjust margins block withdrawal. 13,15 Stretching also affects the brush diameter corresponding to the -peak position (Fig. 5d). It can be seen that the molecules oriented in the stretching direction (i.e. the ones generating the scattering intensity in the plane perpendicular to the stretching axis) become thinner. This is ascribed to unravelling of the backbone inside bottlebrush envelopes which contributes to strain-stiffening. 15

Chemistry variation and application

In addition to controlling mechanical properties of thermoplastic elastomers, the modular nature of the A-g-B platform allows a broad range of chemical compositions for A and B blocks. The chemical variability accommodates specific functions such as thermal stability, adhesion, and molding (Fig. 1), while their mechanical properties (softness, firmness, and strength) are regulated by network architecture. To that end, we report exemplary A-g-B chemical structures that target specific application needs. (Fig. 6a). First, using PDMS side chains and PMMA A-blocks in the above discussed PMMA-g-PDMS system (Fig. 6b) is beneficial for thermal stability of mechanical properties. The combination of a low glass transition temperature of PDMS ( = -124℃) and highly cohesive PMMA domains maintains nearly constant storage and loss moduli within broad temperature (<100C) and frequency ranges (10 -2 -10

2 Hz), which is valuable for devices subjected to considerable thermal fluctuations. However, other applications, e.g., injection molding and 3D-printing, require fluidity at moderate temperatures. This was achieved by replacing PMMA with less cohesive PS as A-block, resulting in a storage modulus, , decrease above the PS glass transition temperature of 105C and demonstrating an elastomer-to-melt transition at 150C (Fig. 6b). The enhanced fluidity of the PS block enables injection molding and 3D printing of various shapes (Fig. 6c, ESI, Section 2.5) that match the deformation response (modulus, strength, and elongationat-break) of solution cast samples (ESI, Fig. S29).

In other domains of practical applications, pressure sensitive adhesives and vibration damping, materials are expected to demonstrate a particular viscoelastic variance as a function of frequency (within the 10 -2 to 10 2 Hz range) at room temperature. 42 This feature can be addressed by substitution of PDMS with polyisobutylene (PIB) side chains (Fig. 6a), which shifts the material relaxation dynamics toward the Rouse regime at room temperature (Fig. 6d). The combination of lowering the storage modulus ( < 0.1 MPa at 0.01 Hz) and enhanced viscoelasticity (increase of from ~0.1 to ~1 range) facilitates substrate wetting and energy dissipation at a debonding rate of ~ 1 Hz. This results in the increase of the overall work of adhesion ( ) of the material by probe tack testing showing potential as a pressure sensitive adhesive (Fig. 6e, ESI, Section S2.6). The chemistries addressed are merely a subset of synthetic and application pathways capable of implementing A-g-B brush architecture and can be greatly expanded.

Conclusions

The A-g-B brush platform enables immense expansion of the mechanical property scope of thermoplastic elastomers by independently tuning elastic and viscoelastic properties. Specifically, a unique combination of softness, firmness, and strength was attained through coordinated variation of the side chain length, grafting density, volume fraction of A blocks, and interconnectivity network cells. Furthermore, specific properties like molding, damping, and pressure sensitive adhesion were adjusted by varying the chemical composition of the A and B blocks. Specifically, A-g-B materials achieved strength of ~10 MPa, exceeding that of blood vessels, and closely replicated frequency dependence of the damping factor of super-soft brain and super-tough ligament tissues. The thermoplastic nature of A-g-B networks combined with reduced viscosity of brush-like macromolecules enables injection molding and 3D printing of shapes with molecularly tunable tissue-mimetic elastic and dynamic mechanical properties. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins Table 1. X-ray scattering data of brush-like graft copolymers.

(1)

(2) (3) (4) (5) (

, , (1) Number average DP of PS or PMMA side chains as determined by 1 H-NMR. (2) Volume fraction of PS or PMMA, , , .

(3) Number of spacer repeat units between A blocks. (4) Number average DP of brush backbone between PS or PMMA side chains. (5) DP of total brush backbone in the A-g-B macromolecule. (6) Young's modulus determined either as tangent of a stress-strain curve at →1 or from the fitting equation S2 at =1 as . (7) Strain-stiffening parameter obtained by fitting stress-strain curves with equation S1. (8) Aggregation number.

Area per brush strand at the A/B interface calculated from the aggregation number as . (10) Apparent cross section area of bottlebrush cylinder assuming hexagonal packing.

Fig. 1 .

 1 Fig. 1. (a) Evolutionary augmentation of mechanical properties starting with super-softness of covalent bottlebrush elastomers through adding firmness in self-assembled ABA networks to superior strength of molecularly interconnected A-g-B elastomers.The mesh interconnectivity is highlighted by a bold backbone of an A-g-B macromolecule. Interplay of multiple architectural parameters, such as length and grafting density of side chains in the B block as well as A block dimensions, permits unparalleled control of equilibrium and viscoelastic mechanical properties. Chemistries chosen for application specific functions, such as adhesion, moldability, or water uptake, can be implemented within specific mechanical phenotypes. (b) Stress-elongation curves of selected covalent bottlebrush elastomer ( kPa and ) and a thermoplastic ABA brush copolymer ( = 13.3 kPa and ) samples. Although, the ABA system demonstrate very high firmness thay may reach 0.9 on par with skin (dashed line), both systems exhibit low strength ( ).15 (c) Self-assembled networks of A-g-B brush-like graft copolymers display a unique combination of softness, firmness, and strength (ESI, TableS1). A sample of PBA-ran-PMMA-g-(PDMS/PS) graft copolymer exhibits strength ~8 MPa on par with aorta (dashed line), surpassing many other strong tissues.

Fig. 2 .

 2 Fig. 2. Polymerization of A-g-B brush-like graft copolymers with controlled grafting density of the brush (B) block and fraction of linear A blocks distributed as long side chains in the brush block. a) The controlled radical polymerization (CRP) of assorted macromonomers yields poly[nBA-ran-MMA-g-(PDMS/PMMA)] brush copolymers, while b) the free radical polymerization (FRP) produces poly[nBA-ran-MMA-g-(PIB/PS)] brush copolymers with controlled DP of the brush block. See ESI for complete synthetic procedures. The parameter is defined as a ratio of molar ratio of A and B macromonomers, including spacers.

Properties 6 |

 6 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

  Fig. 4. (a)Strength of PMMA-g-PDMS ( =1) elastomers systematically increases with the number of A blocks per A-g-B macromolecules, z, at a constant DP between A blocks of (Table1). (b) Stress-elongation curves of PS-g-PDMS ( =8) samples with different (and correspondingly ) values as indicated (Table1). The decrease of at a given results in progressively increasing strength. (c) Frequency sweeps of the storage modulus ( ) and damping factor ( ) of the PS-g-PDMS samples with different and values from panel b. (d) PS-g-PDMS elastomer ( , , ) replicates the frequency dependence of the paracardium and ligament damping factors. Replacing of PDMS with PIB in the PS-g-PIB elastomer ( , , ) allows closely matching the damping of brain tissue.

Fig. 5 .

 5 Fig. 5. In situ SAXS monitoring A-g-B elastomer deformation of [PMMA-g-PDMS ( =1), , ]. (a) SAXS curves captured during uniaxial extension at different elongation ratios , as indicated. (b) Normalized inter-domain distance as a function of deformation. (c) Normalized domain diameter upon the onset of deformation mechanism. d) Normalized brush diameter as a function of deformation.

Fig. 6 .

 6 Fig. 6. (a) Copolymer structure of distinct A-g-B macromolecules utilized as mesoblocks for enhancing thermostability, moldability at lower temperatures, and adhesive and damping applications. (b) Temperature dependence of storage ( ) and loss ( ) moduli of A-g-B elastomers with chemically different A-blocks (PMMA and PS) yet the same bottlebrush B block with PDMS side chains ( ). Unlike the PS-g-PMMA sample, PS-g-PDMS ( =8) elastomers undergoes melting at ~150℃. (c) Fused filament fabrication 3D printing was performed on a PS-g-PIB ( =8) sample at 150C. (d) Time-dependent and independent A-g-B brush copolymers in the PSA viscoelastic window. (e) Probe tack test of the PS-g-PIB and PS-g-PDMS samples ( ℃, , ) reveals a considerable difference in the work of adhesion of and , respectively.

Acknowledgements

The authors gratefully acknowledge funding from the National Science Foundation (DMR 1921835, 1921858, 1921923, DMR 2049518, and DMR 2004048). Y.D.G. and E.Y.K. acknowledge the Russian Science Foundation (RSCF grant 19-13-00340-П).

Author Contributions

E.D, M.M., and A.N.K. equally contributed to this study, synthesized and characterized A-g-B brush copolymers; F.V., M.M., and E.D. performed fused filament fabrication 3D printing; Y.D.G. performed computer simulations, E.Y.K. conceived the idea of injection molding; E.A.B., E.A.N., and D.A.I. performed SAXS experiments and analysis; Y.T, and A.V.D. provided theoretical analysis of mechanical properties and computer simulations of self-assembled network deformations; M.V.V. and S.S. conceived the idea of the A-g-B platform; M.M. and S.S.S. were primary writers of the manuscript. All authors discussed the results and provided feedback on the manuscript.

Conflicts of interest

None.