NULL CONTROLLABILITY FOR DISTRIBUTED SYSTEMS WITH TIME-VARYING CONSTRAINT AND APPLICATIONS TO PARABOLIC-LIKE EQUATIONS

. We consider the null controllability problem fo linear systems of the form y (cid:48) ( t ) = Ay ( t ) + Bu ( t ) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapola- tion space containing Y . The control u is constrained to lie in a time-varying bounded subset Γ( t ) ⊂ U . From a general existence result based on a selec- tion theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set Γ( t ) contains the origin in its in-terior at each t > 0, the local constrained property turns out to be equivalent to a weighted dual observability inequality of L 1 type with respect to the time variable. We treat also the problem of determining a steering control for gen- eral constraint sets Γ( t ) in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that Γ( t ) is a closed ball centered at the origin and its radius is time-varying.


(Communicated by Michael Malisoff)
Abstract. We consider the null controllability problem fo linear systems of the form y (t) = Ay(t) + Bu(t) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y . The control u is constrained to lie in a time-varying bounded subset Γ(t) ⊂ U . From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set Γ(t) contains the origin in its interior at each t > 0, the local constrained property turns out to be equivalent to a weighted dual observability inequality of L 1 type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets Γ(t) in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that Γ(t) is a closed ball centered at the origin and its radius is time-varying.
1.1. Problem formulation and reference to the literature. In this paper, we treat the problem of null controllability for linear infinite dimensional distributed control systems such as y (t) = Ay(t) + Bu(t), y(0) = y 0 .
We consider the situation where, at each instant t > 0, the control u is constrained to take on values in a preassigned subset Γ(t) of the control space U and the following issues will be studied: (a) Does there exist a constrained control u steering the system to the origin at some time T > 0 ? (b) Assuming that such a control exists, can we characterize it by specific properties ? (c) Otherwise, can we characterize an appropriate control which steers the system (1) as close as possible to 0 at time T ? Such problems are naturally motivated by real world applications involving actuators with amplitude and rate limitations and challenging related issues have not been investigated clearly so far, even for the case where the time-varying we mention the practical case where the controls are supposed to be positive ( [33]).
• Under the assumption that Γ(t) contains 0 in its interior, we establish in the time-varying constraint context the fact that the constrained controllability can be reduced to some kind of weighed observability relative to the uncontrolled adjoint system. • Exploiting the properties dealing with the constrained null controllability, we present some methods characterizing the appropriate control solution of one of the problems (a), (b), and (c) stated above. Note that the study of the existence of such a control has independent interest. By using nonsmooth convex analysis tools, we shall see that for general time-varying constraint sets Γ(t), such a control satisfies a maximum principle property. We consider also the associated time optimal control question in a general setting. Furthermore, when Γ(t) contains 0 in its interior, the maximum principle property turns out to be a general time-varying bang-bang property. On the other hand, the applications treated in the present paper are drastically different from the ones given in [6] as follows.
• While the applications in [6] concern hyperbolic-like equations such as wave and beams equations, here we focus our study to the heat equation with distributed and boundary control. • In our general approach, we recover and extend similar results which are mainly established in the case where Γ(t) coincides with a closed ball centered at the origin and having (eventually) time-varying radius. The plan of the paper is as follows. In the remaining part of this section, we introduce our general abstract framework. The notion of admissible unbounded control operator B is given. The corresponding notion of admissible observation operator which is needed for its adjoint B * is deduced by duality. We recall also the main results on unconstrained null controllability given in observability inequality form and we precise the various notions relative to the constrained null controllability. Moreover, we present a variational characterization of any control steering the system from a given initial state to the origin. In section 2 we treat Problem (a) and we extend the existing results to general time-varying constraint sets Γ(t) with unbounded admissible control operator. The problems (b) and (c) are addressed in the third section. The fourth section deals with the application of the results obtained in the preceding section to specific partial differential equations (PDEs) including heat equation with distributed and boundary control. In the last section, some further comments and concluding remarks are provided.
1.2. Basic concepts and auxiliary results. Let Y , U be two Hilbert spaces denoting the state space and the control space respectively. Also, let A be the infinitesimal generator of a linear C 0 -semigroup on Y denoted by (e tA ) t≥0 . Finally, let B denote the linear control operator which may be unbounded in the sense that it is bounded from U into a space larger than the state space Y as follows. We define the space Y 1 to be D(A) with the norm . 1 given by for some β ∈ ρ(A), the resolvent set of A, and the space Y −1 to be the completion of Y with respect to the norm It is easy to verify that the norm . 1 is equivalent to the graph norm on D(A).
Using the graph norm topology in D(A * ), we can as well define the analogue spaces (Y ) 1 and (Y ) −1 . By duality, we can check that ( [32], Remark 2.1) Exploiting these facts, we note that the duality product ., . Y ,Y can be expressed by the duality product ., . (Y )1,Y−1 as follows for some β ∈ ρ(A), where β stands for the conjugate of the complex number β. It is clear that both duality pairings coincide when y ∈ Y 1 and y ∈ (Y ) 1 so that they both will be denoted by ., . throughout this paper. We consider the abstract control system (1) where u ∈ L p loc (0, ∞; U ), 1 < p < ∞, y 0 ∈ Y and B is bounded from U to D(A * ) . Let y(.; y 0 , u) denote the solution of (1). Considering formally the variation of constant formula we introduce for any T > 0 the operator L T defined by It is clear that L T ∈ L(L p (0, T ; U ), D(A * ) ). Throughout this paper, we suppose that the control operator B is p-admissible in the sense that it satisfies for some (and hence any) T > 0 and for any u ∈ L p (0, T ; U ) one of the following equivalent admissibility conditions for some positive constant C T . It is well known that (6) or (7) ensures the existence of a unique solution for (1) in the class y ∈ C(0, T ; Y ) ( [48]). Moreover, we get from [48] and [49] the fact that whenever (6) holds for any u ∈ L 1 (0, T ; U ), then the control operator B is necessarily bounded from U to Y . On the other hand, identifying U with its dual, we have B * ∈ L(D(A * ), U ). Let 1 < q < ∞ be the conjugate exponent of p satisfying Then, considering B * as observation operator, we introduce the dual q-admissibility condition ( [49]) NULL CONTROLLABILITY WITH TIME-VARYING CONSTRAINT 3279 for some positive constant C T . Note that the output function is well defined for ϕ 0 ∈ D(A * ). The admissibility condition (9) implies that for any T > 0, the expression in (10), considered as a function of the variable ϕ 0 , can be extended to a linear continuous operator from Y to L q (0, T ; U ). It is well known that the p-admissibility of B as control operator is equivalent to the dual q-admissibility of B * as observation operator ( [49]). Moreover, it is easy to see that the adjoint operator L * T : D(A * ) → L q (0; T ; U ) is given by Then, (11) means that L * T admits a continuous extension, still denoted by L * T , from Y to L q (0, T ; U ). In particular, given a prescribed time T > 0, we shall be concerned with the final state and the following space of reachable states from y 0 in time T Then the notion of unconstrained null controllability is defined as follows.
Definition 1.1. The system (1) (or the pair (A, B)) is p-null controllable in time T if given any initial state y 0 , the set of reachable states R(T, y 0 , p) contains 0.
Combining the admissibility of the operators B and B * as control and observation operators respectively, we state the following variational formulation of the fact that the control u ∈ L p loc (0, ∞; U ) steers the system to the origin. See, for instance, [6]. The origin is reached from y 0 in time T by a control u ∈ L p (0, T ; U ) if and only if, The following remark precises the fact that the notion of controllability can be related to an observability inequality. We shall obtain a similar property in the context of constrained control. Remark 1. It is well known that the p-null controllability is equivalent to the following final state L q -observability inequality. See ( [10], Theorem 3.25 (d) , p. 71) and ( [11], Theorem 2.3) for some positive constant c q .
Let us precise the framework of constrained controllability related to the family of constraint subsets Γ(t) introduced above. We suppose that this family satisfies the assumption (H Γ ) specified by • Γ(t) is closed convex for each t > 0, • there exists a closed bounded subset Γ ∞ in U such that Γ(t) ⊂ Γ ∞ for all t > 0.

LARBI BERRAHMOUNE
Let Γ p denote the set of admissible control functions u in L p loc (0, ∞; U ) such that u(t) ∈ Γ(t) almost everywhere in (0, ∞). Also, Γ p (T ) will denote the restriction of Γ p to the interval [0, T ]. Any control function in Γ p or Γ p (T ) will be referred to as (Γ(.), p)-admissible control. We introduce the following set of reachable states with (Γ(.), p)-admissible controls Then we define the constrained null controllability notions as follows. The following result gives basic properties fulfilled by the set of admissible controls Γ p (T ) and the corresponding set of reachable states R Γ (T, y 0 , p). These properties will be used throughout this paper. Note that the case p = ∞ can be seen as a degenerate one for which the conclusions of the lemma are not verified. Proof. Clearly (H Γ ) implies that Γ p (T ) is convex and bounded. Also, Γ p (T ) is strongly closed. Indeed, Let {f n } be sequence in Γ p (T ) such that f n → f strongly in L p (0, T ; U ). It is well known that, for some subsequence still denoted by {f n }, we have f n (t) → f (t) almost everywhere in (0, T ). Since f n (t) ∈ Γ(t) almost everywhere in (0, T ) and Γ(t) is closed, it follows that f n (t) ∈ Γ(t) almost everywhere in (0, T ) so that f ∈ Γ p (T ). Moreover, as a convex subset, Γ p (T ) is weakly closed. Then the weak compactness property for Γ p (T ) is a ready consequence of the facts that 1 < p < ∞ and L p (0, T ; U ) is a reflexive Banach space. On the other hand, the p-admissibility condition of the control operator B formulated by (7) implies that L T given by (5) is linear continuous from L p (0, T ; U ) to Y . Thus, L T is also linear continuous with respect to the corresponding weak topologies, which yields the weak compactness of L T ( Γ p (T )) and its translate R Γ (T, y 0 , p).
2.1. Preliminaries, measurable selection theorem. We recall some basic ideas on calculus of measurable maps in the context defined by the time-varying constraint subsets {Γ(t)} t>0 . To this end, we shall treat these subsets as a set-valued map denoted by Γ(.) : (0, ∞) U . If such a map has closed images, then it is said to be measurable if the inverse of each open set in U is a measurable set in (0, ∞). Then we introduce for each t > 0 the support function of Γ(t) defined by S Γ (t, .) : U → R which for any u ∈ U is given by The resulting function S Γ (., .) : (0, ∞)×U → R possesses the following properties.
Then the function S Γ is a Carathéodory map on (0, ∞) × U in the sense that for each t > 0, the function S Γ (t, .) : u → S Γ (t, u) is continuous on U and for each u ∈ U , the function S Γ (., u) : t → S Γ (t, u) is measurable on (0, ∞). Moreover, for any measurable function w : (0, ∞) → U , the function defined by t → S Γ (t, w(t)) is measurable on (0, ∞).
is also measurable on (0, ∞). This completes the proof of the lemma.
Taking into account the admissibility condition (9), the function is well defined in L 1 loc (0, ∞) for any ϕ 0 ∈ Y . We shall use a general version of the selection theorem in the infinite dimensional context. Note that the selection theorem stated in [12] deals with finite dimensional spaces and it has been exploited for constrained controllabilty purposes in [35]. Many versions of the selection theorem are disseminated in various references. Below we shall present the one which will be convenient in the context of unbounded admissible control operator. Lemma 2.2. Suppose that Γ(.) satisfies the assumption (H Γ ). Then, for any ϕ 0 ∈ Y , there exists a measurable function v such that v ∈ Γ T (p) and Proof. Let us consider the functional g : (0, ∞) × U → R given by For each u ∈ U , the real valued function t → g(t, u) is measurable. Furthermore, by the admissibility condition (9) we have after extension to Y , B * e (T −t)A * ϕ 0 ∈ U for almost all t > 0 so that the function u → g(t, u) is continuous on U . Hence, the function g is a Carathéodory map. On the other hand, the function h introduced in (17) is a measurable function and, at each instant t > 0, Γ(t) is closed, bounded and convex so that it is also weakly compact in U . Moreover since, for almost all 0 < t < T , the linear functional is continuous on U with respect to the weak topology, it follows that for some u t ∈ Γ(t). Thus, h(t) ∈ g(t, Γ(t)) for almost all t ≥ 0. By using Filippov's theorem in ( [4], p. 316), it follows that there exists a measurable selection v(t) ∈ Γ(t) such that This yields (18). Obviously, v ∈ Γ T (p) and this completes the proof of the lemma.

2.2.
Null controllability under general time-varying constraint sets. The measurable selection theorem formulated in Lemma 2.2 enables us to establish the following (Γ(.), p)-null controllability result under general varying constraint sets Proof. From Lemma 1.5 it follows that R Γ (y 0 , T, p) is a closed bounded convex subset of Y . On the other hand, the system (1) is (Γ(.), p)-null controllable at y 0 in time T if and only if 0 ∈ R Γ (y 0 , T, p). Then we use a corollary of Hahn-Banach separation theorem stated in terms of inequality between appropriate support functionals. Let S R : Y → R and S 0 : Y → R denote the support functionals associated to R Γ (T, y 0 , p) and {0} respectively. Reducing the problem to the inclusion {0} ⊂ R Γ (y 0 , T, p) and using ( [9], Proposition 2.42, p. 43), we obtain that the sys- This inequality means that It follows that the proof of the theorem is complete provided that By using the admissibility conditions (7), (8) and (9), the equality (25) amounts to In order to establish (26), let us consider the functional F T : L p (0, T ; U ) → R given by The admissibility condition (9) implies that F T is linear continuous on L p (0, T ; for some u 0 ∈ Γ T (p). Furthermore, clearly we have On the other hand, Lemma 2.2 yields for some v ∈ Γ T (p) Thus (26) is verified and this completes the proof of the theorem.
Remark 2. (i) Given any subset Λ ⊂ Y containing the origin in its interior, it is easy to see by a positive homogeneity argument that the system (1) Let us anticipate by considering the particular case where Λ coincides with the closed unit ball in Y defined by It follows that the system (1) This characterization will play a crucial role for various questions related to the steering control.
(ii) By virtue of (5) and (11), the characterization property in Theorem 2.3 can be expressed by Corollary 1. Suppose that the assumptions of Theorem 2.3 hold. Then the system Proof. From Theorem 2.3, the system (1) is locally (Γ(.), p)-null controllable in time T > 0 if and only if for some r > 0 and any initial state satisfying y 0 Y ≤ r, we have Remark 3. From the proof, it turns out that the constant c in (32) can have the following interpretation. It can be seen as the radius r of a closed ball in Y centered at the origin and contained in the set of initial states which can be steered to the origin in time T by (Γ(.), p)-admissible controls.
Let us consider a useful situation where global (Γ(.), p)-null controllability holds whenever the system is locally (Γ(.), p)-null controllable. Recall that the semigroup (e tA ) t≥0 is said to be stable if for any z ∈ Y , e tA z Y → 0 as t → +∞.
Proof. If the system (1) is locally (Γ(.), p)-null controllable in some time T > 0, then for some positive constant δ > 0, the system is (Γ(.), p)-null controllable at y 0 in time T whenever y 0 < δ. Then given an arbitrary initial state y 0 ∈ Y , let T 1 = T 1 (y 0 ) > 0 be such that e T1A y 0 < δ and consider the steering control v such that y(T ; e T1A y 0 , v) = 0. Let us introduce the control given by Note that the assumption 0 ∈ Γ(t) for all t > 0 and the fact that the familly (Γ(t)) t>0 is increasing imply that this control satisfies u ∈ Γ T1+T (p). Moreover, we have This completes the proof of the theorem.

2.3.
Time optimal control under time-varying constraints. Given an initial state y 0 , the time optimal control problem that we shall consider amounts to the existence of a final minimal time T * such that there exists a (Γ(.), p)-admissible steering control u satisfying y(T * ; y 0 , u) = 0 and for any T < T * , 0 / ∈ R Γ (T, y 0 , p), where R Γ (T, y 0 , p) is the set of reachable states defined by (15). Below we give an existence result for the time optimal control under general time-varying constraints.
Proof. Assume that (34) holds for some T f > 0. Let us introduce the set of instants given by Clearly T is a nonempty set and it has 0 as lower bound so that its infimum T * := inf T satisfies T * ≥ 0. We shall see later on that actually we have T * > 0. Let (T n ) n≥1 a decreasing sequence in T such that T n → T * as n → ∞. For a given ϕ 0 ∈ Φ 1 , we deduce from the proof of Lemma 2.2 that for any n ≥ 1 there exists v n ∈ Γ Tn (p) such that we get for all n ≥ 1 Then we consider the decomposition and we denote by v * n the restriction of v n to the interval (0, T * ). Since v * n ∈ Γ T * (p) for any n ≥ 1, it follows by using Lemma 1.5 that for some subsequence still On the other hand, we have The q-admissibility of the observation operator B * implies that as n → ∞, Combined with the extended q-admissibility condition (9) this equality gives Since the sequence {v * n } n is bounded and Moreover, and an elementary change of variable gives Hence for any n ≥ 1 we obtain for some positive constant C. By construction and following the assumption (H Γ ), there exists clearly some positive constant C ∞ such that for any Consequently, taking into account (38) and (39) we deduce that as n → ∞ so that by using (36) we get Furthermore, the inequality T * > 0 holds since otherwise we get the excluded trivial case e T * A y 0 = 0 resulting from ϕ 0 , e T * A y 0 ≥ 0 for all ϕ 0 ∈ Φ 1 .
The converse part of the theorem is obvious. This completes the proof of the theorem.
2.4. The case 0 ∈ int(Γ(t)). Let us examine more closely the special case which arises under the strengthened assumption 0 ∈ int(Γ(t)) in the prototypical situation where at each instant t > 0, Γ(t) is a closed ball centered at the origin. To this end, we introduce and we define for a given function m ∈ L ∞ + (0, ∞) the time-varying constraint control set denoted by Γ m(t) and given by Then clearly Γ m(.) satisfies (H Γ ) and, for each t > 0, the analogue of the corresponding support function in (17), denoted by S Γ m(t) , is Hence the corresponding functional J Γ m(.) is given by Hence, the results concerned with Γ m(.) -null controllability can be stated as follows.
Let us introduce for any r > 0 the closed ball In the case where at each instant t > 0, Γ(t) contains Γ r for some r > 0, then the assumption (H Γ ) implies that there exists R > 0 such that so that at any t > 0 for all u ∈ U . Then we get easily from Corollary 2 that for (Γ (.) , p)-null controllability question, the structure of the sets {Γ(t)} t>0 will not matter. Hence, our time-varying (Γ(.), p)-null controllability problem can be reduced to a canonical one in which the constraint set coincides with the closed unit ball Γ 1 . Moreover, it will be interesting to reformulate the results above in this situation. Indeed, in this case, the support function of the canonical constraint set Γ 1 has the more familiar expression S Γ1 (u) = u U (49) so that the corresponding functional J Γ1 is given by Hence, the results concerned with (Γ 1 , p)-null controllability can be stated as follows.

Null controllability from measurable set in time.
Here we shall consider an interesting situation where, beside the time-varying control constraint sets Γ(t), the constrained control is activated only over a measurable subset E ⊂ (0, T ) with measure µ(E) > 0. Hence we are led to consider the system where χ E denotes the characteristic function of E. Then it follows that the constrained null controllability for (53) can be reduced to the one for (1) under appropriate time-varying control constraint sets denoted by Γ E (t). Indeed, it is easy to see that if we set then the resulting constrained null controllability of the system (53) amounts to the (Γ E (.), p)-null controllability of the system (1). We can proceed as in subsections 2.1 and 2.2, by introducing for each t > 0 the corresponding support function of Γ E (t) given by S Γ E (t, .) : U → R and Then the corresponding functional J Γ E (.) (.; T, y 0 ) : Y → R is clearly defined by for all ϕ 0 ∈ Y .
Remark 6. We can as well treat the case where the control from a measurable set in time E is contained in closed balls with time-varying radius so that at each instant t ∈ E, we have u(t) ∈ Γ m(t) with Γ m(t) defined by (41) and (42). Hence, we are led to introduce for m ∈ L ∞ + (0, ∞) the associated function m E defined by Then the resulting constrained null controllability of the system (53) amounts to the (Γ m E (.) , p)-null controllability of the system (1). In particular, the system (1) is locally (Γ m E (.) , p)-null controllable in time T > 0 if and only if for some for all ϕ 0 ∈ Y .
Remark 7. Beside the time optimal control problem introduced in subsection 2.3, there exists another one in the situation where we try to reach the origin at a fixed time T > 0 while delaying the activation of the constrained control until some instant 0 < τ < T . Hence the system under study would have the form where χ (τ,T ) denotes the characteristic function of the subinterval (τ, T ). The system (61) has the form (53) if we set E = (τ, T ). There is a kind of time optimal control problem whose aim is to delay initiation of active constrained control in (61) as late as possible, such that the corresponding solution reaches the origin by a fixed ending time T . Considering the associated functional given by then by proceeding as in Theorem 2.5, it is easy to establish that this second version of time optimal control problem admits a solution if and only if there exists some initial activating time 0 < τ 0 < T such that Moreover, the optimal time, denoted by τ * , is given by 3. Steering control.

Preliminaries.
Beside the results on constrained null controllability, we shall examine the problem of characterizing the appropriate control steering y 0 to the origin. We note that, assuming that such a control exists, a classical strategy has consisted of reducing the question to the two following problems. The first one is the time optimal control problem already introduced in section 2. The second one is the so-called norm optimal control which consists of determining the steering control which minimizes an appropriate norm. To best of our knowledge, at general abstract level, the solutions obtained for these problems concern the case where the control operator B is bounded. See [13] and [14]. The case where B is unbounded has been treated mainly in the context of boundary control problems for heat equations with L ∞ setting with respect to the time variable. See, for instance, [22], [23] and [34].
Here, we shall not consider directly these issues. Instead, we shall consider the case where B is (eventually) unbounded and, taking into account from the results stated in section 2 the fact that the existence of the steering control depends closely on the initial state y 0 and the related final time T = T (y 0 ), we shall formulate our problem in the following general form: (P) Given a final time T > 0, characterize any control steering the system (1) as close as possible to 0 in time T .
We shall treat this problem by using a variational method based on minimizing an appropriate convex functional. Having in mind the existing literature which makes use of dual convex arguments and introduces backward adjoint system, it turns out that the null control problem is reduced to the minimization of a dual conjugate functional with respect to the final condition of the adjoint state. The form of this functional is closely related to the type of system under study. The functional which has been used in the context of time-invariant constraint control set Γ for systems of hyperbolic type is defined by J H (.; y 0 , T ) : Y → R and where K H : U → R + is convex and lower semicontinuous. It has been established that the steering control can be obtained in constructive way provided that K H is differentiable, locally Lipschitz and satisfies K (U ) ⊂ Γ. This result covers even the case of unconstrained null controllability where Γ = U . See [5] and [6] for details. Inspired by the treatment of the null controllability issue performed in [46] in the unconstrained control context for the heat equation, it seems that, in the context of parabolic-like equations, the most convenient functional valid even in the time-varying constraint sets can be defined by J P (.; y 0 , T ) : where K P : R + → R + is an appropriate convex function. In particular, following the approach initiated in [35] for multivariable systems, a natural choice for a given general time-varying control constraint sets Γ(t) will consist of taking K P as the identity function so that J P (.; y 0 , T ) coincides with the functional J Γ(.) (.; y 0 , T ) already defined in (22). Note that an immediate difficulty arises because of the lack of differentiability of J Γ(.) (.; y 0 , T ) resulting from the sup-operation present in the support function S Γ (., .). Here, using nonsmooth analysis tools, we shall extend the results in [6] and [35] to the case of time-varying constraints Γ(t) by minimizing J Γ(.) (.; y 0 , T ) on the closed unit ball Φ 1 ⊂ Y introduced in (29). Moreover, we shall consider carefully the case of the constraints Γ m(.) defined by (41) and (42) in the context of parabolic-like equations. Roughly speaking, by such a context, we mean that the following assumptions hold: (H P 1 ) (Regularizing property) For each ϕ 0 ∈ Y , we have e tA * ϕ 0 ∈ D(A * ) for all t > 0 and the function t → e tA * ϕ 0 is continuous from (0, ∞) to D(A * ).
Note that these assumptions are readily verified by diffusion processes characterized by analyticity properties of the associated semigroups. This fact will be made more pecise in Section 4. Furthermore, our problem (P) is contained in the optimal target control class. In the context of the heat equation, various results enable us to relate this problem to both optimal time control and optimal norm control problems. See, for instance, [41], [42] and [46].
Below, we present some nonsmooth analysis notions which will be used with respect to the spaces Y and U (identified with its dual U ). Let f : X → R ∪ {+∞} be a given extended-valued function, where X is a Banach space. The effective domain of f , denoted domf , is the set The function f is called proper when domf = ∅. Let f be a proper function and let x ∈ domf . An element ζ of X is called a subgradient of f at x (in the sense of convex analysis) if it satisfies the following subgradient inequality : The (eventually empty) set of all subgradients of f at x is denoted by ∂f (x), and referred to as the subdifferential of f at x. Note that if f is convex and x ∈ domf is a point of continuity of f , then ∂f (x) is nonempty and weakly compact ( [9], p. 62). Let P be a convex subset of X. The tangent cone to P at a point x ∈ P , denoted T P (x), consists of all points v ∈ X expressible in the form where (x n ) n is a sequence in P converging to x and (t n ) n is a positive sequence decreasing to 0. The normal cone to P at x ∈ P , denoted N P (x), is the subset of the dual space X defined by It can be shown that the normal cone can be characterized by ( [3], Proposition 4, p. 168) The indicator function of P is the function I P : X → R ∪ {+∞} which has value 0 on P and +∞ elsewhere. Let us note that when P is a convex subset of X, then ∂I P (x) = N P (x); that is, the subdifferential of the indicator function is the normal cone ( [9], p. 61).
The conjugate function f * : If g : X → R ∪ {+∞} is a proper function, its conjugate g * : For any nonempty closed convex subset P of X , we have S * P = I P ; that is, the conjugate function of its support function S P is given by its indicator function ( [9], p. 70). When we take g to be f * , then we obtain the biconjugate of f , namely the function f * * : X → R ∪ {+∞} defined as follows (when f * is proper): Given a proper function f : X → R ∪ {+∞}, it is well known that f is convex and lower semicontinuous, if and only if f * is proper and f = f * * ( [9], p. 69). Moreover, for such a function, we have the subdifferential inversion formula ( [31], p. 35) ζ ∈ ∂f (x) ⇔ x ∈ ∂f * (ζ).
3.2. Steering control in nonsmooth analysis setting. The following lemma gives an identification of the subdifferential of J Γ(.) (.; y 0 , T ).
which is equivalent to In order to precise the subdifferential above, we introduce the functional R : L q (0, T ; U ) → R given by Then by applying the result in Theorem 23 of [31], page 62, it follows that u ∈ ∂R(v) if and only if, or, equivalently, by using (73) v(t) ∈ ∂I Γ(t) (u(t)) = N Γ(t) (u(t)) a.e on (0, T ).
On the other hand, we note that , and by using ( [9], Theorem 4.13, p. 64), we obtain ∂ (R(L * T ϕ 0 )) = L T ∂R (L * T ϕ 0 ) . Hence, ψ 0 − e T A y 0 has the form ψ 0 − e T A y 0 = L T u, for some u ∈ ∂R (L * T ϕ 0 ). Taking into account (80), u satisfies the maximum principle property (75). This completes the proof of the lemma. Remark 8. In principle, the subdifferential of J Γ(.) (.; y 0 , T ) is a multi-valued function. In the constraint case Γ m(.) , the subdifferential of J Γ m(.) (.; y 0 , T ) at any ϕ 0 ∈ Y consists of ψ 0 ∈ Y satisfying (74) with for almost all t ∈ (0, T ). Hence, if then ψ 0 is uniquely determined by (74) and (81) so that the subdifferential of J Γ m(.) (.; y 0 , T ) will correspond to the gradient J Γ m(.) (ϕ 0 ; y 0 , T ). It is easy to check that this fact is true whenever ϕ 0 = 0 and the system (1) is a parabolic-like equation satisfying the assumptions (H P 1 ), (H P 2 ) and (H P 3 ). The following theorem gives a strategy guaranteeing a control which steers the system (1) as close as possible to 0 at time T by minimizing J Γ(.) (.; T, y 0 ) over the closed unit ball Φ 1 in Y .
Proof. We introduce two functionals F : L p (0, T ; U ) → R and G : Y → R given by We now apply Rockafellar's extension duality theory in the sense of Fenchel ([12], chap. 3) and [30]. The functionals F and G are, respectively, proper convex and concave functions. Moreover, for any u ∈ Γ T (p), F is finite at u and G is continuous at L T (u). Hence the problem (84) is "stably set" so that the duality method implies that the problem min To this end, we have to express J Γ(.) (ϕ 0 ; y 0 , T ) over Φ 1 as appropriate combination of the conjugate functions F * and G * . For each ϕ 0 ∈ Φ 1 we have By using (11) and (28) we obtain On the other hand, the conjugate function of G over Φ 1 is given by Indeed, for any It follows that By applying Theorem 3 of [30] and using the facts that F and G coincide with their respective biconjugates so that (88) holds. Furthermore, the "extremality condition" in [30] implies a necessary condition which must be satisfied by all solution pairs ϕ 0 , solving (87) and u solving (86). This condition would read Since F coincides with the indicator function of Γ T (p), the condition (92) means that the normal cone of Γ T (p) at u, denoted by N ( u), contains the function L * T ( ϕ 0 ). This amounts to This is possible only if µ = u(t) achieves the supremum of µ, B * e (T −t)A * ϕ 0 ) for almost all t ∈ [0, T ]. Equivalently, we must have (82). The remaining part of the proof is a ready consequence of Lemma 3.1.
We note by using Theorem 3.2 that the obvious minimizer ϕ 0 = 0 is useless since it leads to a steering control function characterized by the trivial fact However, when the control constraint sets are the closed balls Γ m(t) given by (41) and (42), we can obtain an abstract time-varying bang-bang property verified by any steering control.
so that (81) and Theorem 3.2 imply that any steering control satisfies (95).
(ii) If the system (1) is (Γ m(.) , p)-null controllable at y 0 in time T , then the existence of the minimal time control u * and the associated minimal time 0 < T * ≤ T is guaranteed by Theorem 2.5 so that the corresponding solution, denoted by y * , satisfies y * (T * ; y 0 , u * ) = 0. By contradiction, suppose that for some > 0 and some subset E ⊂ (0, T * ) with positive measure, denoted by |E|, we have Let d 0 = |E| 2 and E = E ∩ (d 0 , T * ). Moreover, taking into account (48) and Corollary 3, it follows that the system (1) is locally (Γ , p)-null controllable in time T over E. Hence, for 0 < t 0 < d 0 small enough, there exists a (Γ , p)-admissible control u ∈ Γ p (T * ) such that the solution of the system can be steered to the origin at time T * so that y(T * ; y(t 0 ) = y 0 − y * (t 0 ), uχ E ) = 0. Moreover, if we set u = u * + uχ E and z = y * + y then we have and z(t 0 ) = y * (t 0 ) + y(t 0 ) = y * (t 0 ) + y 0 − y * (t 0 ) = y 0 .
On the other hand, letting u(t) = u(t+t 0 ) and y(t) = z(t+t 0 ) for 0 < t < T * −t 0 , then since the function m(.) is nonincreasing, it follows that u is (Γ m(.) , p)-admissible and we obtain that and y(T * − t 0 ; y 0 , u) = 0. This contradicts the time minimality of T * with respect to the (Γ m(.) , p)-null controllability for the system (1). This completes the proof of the theorem.

4.
Applications. In this section, we present some applications of the abstract results stated above to systems governed by parabolic partial differential equations (PDEs) by treating the heat equation. Here, we shall treat both distributed and boundary control cases. In what follows, Ω ⊂ R N is an open bounded domain with sufficiently smooth boundary ∂Ω, ω and γ denote non-empty open subsets of Ω and ∂Ω respectively. Moreover, we shall regard L 2 (ω) and L 2 (γ) as subspaces of L 2 (Ω) and L 2 (∂Ω) respectively. We denote by χ ω and χ γ their respective characteristic functions. Along this section, we shall still denote by ∆ the Dirichlet Laplacian operator defined by We shall make use of the well-known identification D((−∆) 1 2 ) ≡ H 1 0 (Ω). See, for instance, [39], p. 93). We shall also denote both the scalar product ., . L 2 (Ω) and the duality product ., .
(103) We shall be concerned with the time-varying saturation constraint specified by the set Γ ω m(t) = u ∈ L 2 (ω) : where m ∈ L ∞ + (0, ∞) has been introduced by (41). Since the control operator B is bounded, it is obviously 2-admissible with respect to the state space L 2 (Ω). Beside (H P 1 ) and (H P 2 ), the assumption (H P 3 ) is a ready consequence of the following observability inequality established in [2] and [28]. For any measurable subset E ⊂ for all ϕ 0 ∈ L 2 (Ω). Moreover, (105) yields for some positive constant c 1 . It follows that the system (102) is locally (Γ 1 , 2)-null controllable in time T over E. We define the analogue of the functional in (22) corresponding to the constraint control sets Γ ω m(.) by J ω m(.) (.; y 0 , T ) : L 2 (Ω) → R and Theorem 3.2 implies that any control u solution to the problem (P) with respect to the state space L 2 (Ω) would satisfy for almost all t ∈ (0, T ), where ϕ satisfies It turns out that, due to the parabolic context, the set of instants 0 < t < T for which e (T −t)∆ ϕ 0 = 0 on ω, is of null measure whenever ϕ 0 = 0. Hence by using Theorem 3.3, we obtain the following facts which are similar to the time-varying bang-bang properties established in [8].
We introduce also the Dirichlet map D defined by Then the system (111) can be modeled as an abstract equation similar to (1) and given by ( [39], Proposition 10.7.1, p. 342) Moreover, we recall from elliptic theory in ( [18], Chap. 2) that D is continuous so that the control operator can be defined by B : Hence, this system has the form (1) if we choose U = L 2 (γ) as control space. However, it is well known that there exists u ∈ L 2 (0, T ; L 2 (γ)) for which y(T ; y 0 , u) / ∈ L 2 (Ω) so that the control operator B in (111) is not 2-admissible if we consider L 2 (Ω) as state space. See, for instance, ( [17], p. 202). In order to precise the appropriate state space and the corresponding admissibility condition, we consider two possible situations. The first one is based on the following estimate established in [47]. For some positive constant C > 0, we have for each t > 0 for all u ∈ L 2 (γ) . It follows that y(T ; y 0 , u) ∈ L 2 (Ω) for all u ∈ L p (0, T ; L 2 (γ)) whenever p > 4. Hence the operator defined by (115) is p-admissible with respect to the state space Y = L 2 (Ω) for any p > 4. In other words, we can preserve L 2 (Ω) as state space by smoothing the space of control functions. The second admissibility condition uses the fact that the solution of (111) satisfies y ∈ C(0, T ; H −1 (Ω)) for any u ∈ L 2 (0, T ; L 2 (γ)). See ( [39], p. 343). Hence the operator B defined by (115) is 2-admissible with respect to the state space Y = H −1 (Ω) so that we can use L 2 (0, T ; L 2 (γ)) as control functions space with respect to the larger state space H −1 (Ω). In the sequel, we shall be concerned with this case. In order to precise the corresponding dual 2-admissibility, we note that the operator L T : u ∈ L 2 (0, T ; L 2 (γ)) → y(T ; 0, u) ∈ H −1 (Ω) is linear continuous. Its adjoint L * T : H 1 0 (Ω) → L 2 (0, T ; L 2 (γ)) is defined by = y(T ; 0, u), ϕ 0 . Consider for ϕ 0 ∈ H 1 0 (Ω) the solution of the adjoint equation (101). If we compute the duality product ., . of (101) with y(.; 0, u) and apply the Green's formula, we obtain by integrating by parts where ∂ ν denotes the outward normal derivative to ∂Ω. Hence, L * T has the form As for the constrained null controllability question, let be the time-varying constraint control sets for the system (111). Then beside (H P 1 ) and (H P 2 ), the assumptionn (H P 3 ) is a ready consequence of the following observability inequality established in [2]. For any measurable subset E ⊂ (0, T ) with positive measure, there exists some positive constant c 0 such that the solution of the backward equation (101) satisfies for all ϕ 0 ∈ L 2 (Ω). Note that it follows from Corollary 3 and Corollary 5 that the system (111) is locally (Γ 1 , p)-null controllable over each subset E in time T with respect to the state space L 2 (Ω). In fact, an analogue of this property holds true for the extended state space H −1 (Ω). Proof. Given an arbitrary initial state y 0 ∈ H −1 (Ω), it is well-known that y 0 := e T 2 ∆ y 0 ∈ L 2 (Ω). Let E ⊂ (0, T ) be a subset of positive measure. The system (111) is locally (Γ 1 , p)-null controllable over E ∩ (0, T 2 ) in time T 2 with respect to the space L 2 (Ω) so that for some positive constant δ > 0, the system is (Γ 1 , p)-null controllable over E at y 0 in time T 2 whenever y 0 < δ. Let v be the appropriate steering control such that y( T 2 ; y 0 , v) = 0 and consider the control given by It is easy to check that y(T ; y 0 , u) = 0. On the other hand, taking into account that for some C > 0 ( [26], Theorem 6.13, p. 74) y 0 L 2 (Ωγ ) = e there exists δ > 0 such that y 0 H −1 (Ω) < δ ⇒ y 0 L 2 (Ωγ ) < δ.
This implies that the system (111) is locally (Γ 1 , 2)-null controllable over E with respect to the state space H −1 (Ω) and the proof of the proposition is complete.
We define the analogue of the functional in (22)  dt + e T ∆ ϕ 0 , y 0 .
As in the case of distributed control, the parabolic context implies that the set of time instants 0 < t < T for which ∂ ν e (T −t)∆ ϕ 0 = 0 on γ, is of null measure whenever ϕ 0 = 0. Hence by using Theorem 3.3, we obtain the following similar facts: (i) If the system (111) is not (Γ γ m(.) , 2)-null controllable at y 0 in time T , then the steering control u solution to Problem (P) under the constraints Γ γ m(.) satisfies the following time-varying bang-bang property a.e on (0,T).

5.
Further comments and concluding remarks. This article has provided a general variational approach to constrained null controllability in an abstract setting. Despite the fact that the applications has been focused on heat systems, we conjecture that other potential candidates to these applications are more general parabolic equations and systems, Stokes-like and linearized Boussinesq-like systems, delayed systems. See, for instance, the applications of our method to hyperbolic-like systems under time-invariant constraints developed in [6]. Furthermore, interesting problems could be considered in connection with the results and methods developed in this paper. Among them we mention the following ones.
• An interesting issue would consist on treating the null controllability problem under constraints on the control and the state of the system. We mention particularly other constraint sets of positivity type which are interesting from a practical point of view. In the absence of control saturation, these questions have been treated in [19] and [21] for heat systems. The case of finite dimensional systems is also considered in [20]. • The results given in this paper are theoretical and it would be interesting to address the question of building efficient numerical algorithms to compute the steering control. If we consider the parabolic-like systems modeled by the heat equation, we note that recent developments indicate that the null controllability issue is difficult and still largely open even in the unconstrained control case ( [24], [25]). Recall that such an analysis leads to the necessity of distinguishing two different methods, the continuous and the discrete ones.
In the continuous one, after characterizing the steering control of the system, the emphasis is placed on building efficient numerical methods to approximate it. In the discrete one, one analyzes the null controllability of discrete models obtained after discretizing the state equation by suitable numerical methods and their possible convergence towards the steering control when the meshsize parameters tend to zero. See, for instance, [45]. Moreover, it may happen that one has null controllability in a very general space but not in a classical space of sufficiently smooth functions, then the numerical approximation will necessarily develop singularities ( [7], [15]). This phenomenon, unavoidable even in the finite dimensional systems and whatever be the numerical approximation used, strongly deteriorates the efficiency of the algorithms so that smoothing remedies should be used ( [25], [36]). • The variational characterization of the constrained steering control can be addressed in a semilinear setting. This is still to be performed.