THE MAXIMUM PRINCIPLE FOR LINEAR INFINITE DIMENSIONAL CONTROL SYSTEMS WITH STATE CONSTRAINTS

. We prove a version of Pontryagin’s maximum principle for linear inﬁnite dimen- sional control systems (including point target conditions and state constraints). This result covers some examples for which no nonlinear theory is available at present.


Introduction.
Let E be a Banach space, A the infinitesimal generator of a strongly continuous semigroup S(t), t ≥ 0. In recent years, optimal control problems for semilinear control systems y (t) = Ay(t) + f (t, y(t), u(t)) have been the object of a number of papers; these include the treatment of control constraints u(t) ∈ U , state constraints y(t) ∈ M and target conditions y(t) ∈ Y at the end of the trajectory. See [22], [23] for results, bibliography, and information on other ways to approach these problems that do not use semigroup theory. A common feature of all theories is that point target conditions Y = {ȳ} are difficult to treat except in exceptional cases.
We present in this paper a linear-convex theory (for the system y (t) = Ay(t) + Bu(t) with A the infinitesimal generator of a compact analytic semigroup) that can handle some interesting problems with state constraints for abstract parabolic systems with point targets; nonlinear perturbations of these problems are not amenable at present to nonlinear theories (see Remark 10.2 for an explanation of this last point). The theory is expounded in detail for time optimal problems, but it can handle as well other cost functionals (see Section 9). Applications to various distributed parameter systems are presented in Sections 6 and 8.
2. Preliminaries and existence results. The system is y (t) = Ay(t) + Bu(t) , y(0) = ζ , (2.1) with A the infinitesimal generator of a compact, analytic semigroup S(·) in a Banach space E and B ∈ (F, E) = {space of all linear bounded operators from F into E}.
In order to deal with parabolic equations in the most interesting cases -L 1 spaces and spaces of continuous functions -we use the theory of Phillips or -adjoints. Recall that if S(·) is an arbitrary strongly continuous semigroup in a Banach space E its adjoint semigroup S(·) * may not be strongly continuous in E if E is not reflexive. This motivates [26] the introduction of the Phillips adjoint semigroup S (t), the restriction of S(t) * to the subspace E ⊆ E * whose elements make S(t) * y continuous in t ≥ 0. The space E coincides with the closure of D(A * ) in E * and A , the infinitesimal generator of S (·) is the restriction of A * to E with domain D(A ) = {y * ∈ D(A * ); A * y ∈ E } (see [26,Chapter XIV] or [18,Chapter 2] for complete details). The space E is determining in the sense that the norm y = sup z∈E | z, y | is equivalent to the original norm of E; in particular z, y = 0 for all z ∈ E implies y = 0 ( · , · is the duality pairing of E and E * ). When S(·) is continuous in the norm of (E, E) for t > 0 S(t) * is continuous in the norm of (E * , E * ) for t > 0. Moreover Application of the -adjoint theory to S (t) produces the space E = (E ) , the semigroup S (t) = (S ) (t) and its infinitesimal generator A = (A ) .
and this property implies Application of (2.2) to S (t) and A produces For existence reasons we assume F = X * (2. 6) with X a separable Banach space. B is a linear bounded operator from X * into E and (a) B * : E → X, (b) E is -reflexive and E, E are separable. The state and control constraint for (2.1) are respectively. Existence results below are based on Theorem 2.1. L 1 (0, T ; X) * = L ∞ w (0, T ; X * ). The space L ∞ w (0, T ; X * ) consists of all X * -valued functions u(·) such that u(·), x is measurable and a.e. bounded for all x ∈ X ("a. e." depending on x). If X is separable, elements of L ∞ w (0, T ; X * ) have measurable norm (although they may not be strongly measurable) and the norm of L ∞ w (0, T ; X * ) is the essential supremum norm. The duality pairing of L 1 (0, T ; E) and L ∞ w (0, T ; X * ) is u(·), f (·) = u(t), f (t) dt (the proof of Theorem 2.1 follows from the Dunford-Pettis theorem and is essentially contained in [11]; see [21] for other references). If X * itself is separable, then L ∞ w (0, T ; X * ) = L ∞ (0, T ; X * ) . (2.8) Given an arbitrary Banach space, we denote below by C(0, T ; U ) the space of all continuous E-valued functions f (·) defined in 0 ≤ t ≤ T equipped with the supremum norm and by Σ(0, T ; E * ) the space of all E * -valued countably additive measures µ defined in the field Φ generated by all subintervals [a, b], (a, b], [a, b), (a, b). The space is equipped with the total variation norm µ = sup Σ µ(e j ) , supremum taken over all finite collections of pairwise disjoint elements of Φ. With this norm, Σ(0, T ; E * ) becomes a Banach space and The duality pairing is given by For a proof and references see [21, §4]. The integral is defined as (a particular case of) the Bartle integral [3]. For general information on vector valued measures and integrals, see [10].
Given ζ ∈ E and u(·) ∈ L ∞ w (0, T ; X * ) the solution of (2.1) is The admissible control space C ad (0,t; U ) is the subset of L ∞ w (0, T ; X * ) whose elements satisfy the control constraint (2.7) a.e. The reachable set R(t, ζ; U ) consists of the instantaneous values y(t, ζ; u) of trajectories for all u(·) ∈ C ad (0, T ; U ). We denote by R(t, ζ; U ) ⊆ C(0,t; E)×E the set of all elements (y(·, ζ, u), y(t, ζ, u)) ∈ In what follows K is a linear operator with dense domain D(K) and range in E. We assume that K has a bounded everywhere defined inverse and that it commutes with S(t), that is, is continuous in the norm of (E, E) in t > 0. Since K −1 is bounded and everywhere defined K * (K −1 ) * = I; on the other hand, (K −1 ) * K * ⊆ I, so that, although K * may not be densely defined, (K −1 ) * = (K * ) −1 . Taking adjoints we obtain The space E * (K) consists of the completion of E * with respect to the norm (K −1 ) * z ; obviously we have the embedding E * → E * (K) and, almost by definition, (K −1 ) * : E * (K) → E * is an isometric isomorphism Lemma 2.3. S(t) * can be extended to E * (K) and S(t) * E * (K) ⊆ E for t > 0.
Proof. The extension is We need the backwards adjoint equation in 0 ≤ t ≤t with a forcing term µ in Σ(0,t; E * ) and z ∈ E * (K). By definition, the solution of (2.11) is The first term has been given sense in Lemma 2.3. The integral is understood as follows: for each s, z i (s) is the unique element of E * such that Moreover, For the proof (of a more general result) see [22,Theorem 4.3]. We note that the "final condition" z(t) = z in (2.11) should not be taken seriously, even as a left-sided weak limit; if µ has mass att, the limit will not equal z.
Proof. That z(·) is E-weakly left continuous was proved in Lemma 2.4. If we prove that z(s) takes values in E almost everywhere, then since E is separable, z(·) is almost separably valued, thus strongly measurable ( [25]). Since the measure µ is countably additive in Φ, it cannot have more than a countable number of atoms s j (points where µ(s j ) = 0). Assuming s is not an atom, write We have (2.14) In view of the uniform continuity of S(t) for t > 0, the max on the right side tends to zero as h → 0, so that S(·) * z i,ε (s) is continuous at t = 0, proving that z i,ε (s) ∈ E . On the other hand, estimating in a similar fashion, Since z i,ε (s) ∈ E and E is closed, z i (s) ∈ E and the proof is finished.
Lemma 2.6. Let g(·) be a E -weakly measurable (E ) * -valued essentially bounded function and let µ ∈ Σ(0, T ; E * ). Then where in the last equality we have used the fact that S(t − s) * µ(dt) takes values in E except perhaps in a countable set (thus almost everywhere). We take now limits under the integrals as h → 0. On the right we use the dominated convergence theorem. On the left we use the fact that h(t) = S (t − s) * g(s)ds is continuous in 0 ≤ t ≤t, hence its range is compact, thus S(h)h(t) → h(t) uniformly in 0 ≤ t ≤t.
Lemma 2.7. The operator [21,Theorem 9.5]. We look at the time optimal problem with target condition Recall that, since X is separable L 1 (0, T ; X) is separable as well, thus the L 1 (0, T ; X)weak topology of L ∞ w (0, T ; X * ) is given by a metric; we don't need generalized sequences.
We add to (a) and (b) the assumption (c): The admissible control space Hypothesis (c) is satisfied, for instance if U is the unit ball of X * ; in this case C ad (0, T ; U ) is the unit ball of L ∞ w (0, T ; X * ), which is L 1 (0, T ; X)-weakly compact by Alaoglu's Theorem. It is easy to see that Hypothesis (c) implies automatically that U is convex and closed which, in turn, implies convexity of R(t, ζ; U ). Theorem 2.8. Assume that the state constraint set M and the target set Y are closed. Then, if there exists a control driving ζ to Y for some t > 0 and satisfying the state constraints, the time optimal problem has a solution.
The proof is standard. The hypotheses imply that a minimizing sequence {u n (·)}, u n (·) ∈ C ad (0, t n ; U ) exists, where t n ≥t = optimal time. Extend u n (·) (with the same name) to [0, T ] (T = t 1 ) setting u n (t) = u = fixed element of U in t ≥ t n . Selecting if necessary a subsequence and using Lemma 2.7 we may assume that u n (·) is L 1 (0, T ; X)-weakly convergent in L ∞ (0, T ; X * ) toū(·) ∈ C ad (0, T ; U ) and that y(t, ζ; u) is uniformly convergent to y(t, ζ; u), so that the target condition and the state constraint are satisfied in the limit. Using essentially the same argument we prove.
3. The maximum principle for the time optimal problem. The operator K below satisfies the assumptions in last section. The domain D(K) is equipped with the norm y D(K) = Ky , which makes it a Banach space. The dual D(K) * can be identified with the space E * (K), the duality pairing given by In fact, (3.1) defines a bounded linear functional in D(K) for any z ∈ E * (K).
We see easily that the identification of D(K) * is an algebraic and metric isomorphism.
Given a closed convex set Y in a Banach space E and an element y ∈ Y the tangent cone T Y (y) to Y at y is The results in this section are on the set target condition (in particular, on the point target condition y(t, ζ, u) =ȳ ∈ D(K)). Assumptions (a), (b) and (c) in last section are in force.
Theorem 3.1. Letū(·) be a time optimal control with optimal timet. Assume that U, M and Y are convex and closed with Int(U ) = φ in X * and Int(M ) = φ in E, and that ) (the latter normal cone defined according to the duality (3.1)) and such that a.e. in 0 ≤ s ≤t,z(·) the solution of (2.11) for µ, z.
In Theorem 3.1, "Y is closed" means "Y is closed in E". The proof follows from a chain of auxiliary results.
Assume that the ∆ n contain a common ball for n large enough. Then every weakly convergent subsequence of {z n } has a nonzero limit.
Lemma 3.2 is a particular case of Lemma 3.5 in [20]. If the conclusion is not true, passing to a subsequence we may assume that For n large enoughỹ + δx n = y n ∈ ∆ n . On the other hand, (3.6) gives z n ,ỹ = z n , y n − δ z n , x n ≤ ε n − δc/2 < 0 for n large enough, a contradiction.
The proof of Lemma 3.3 follows covering Q with a finite number of balls of arbitrarily small diameter. (3.9) and the result follows from continuity of KS(t) in t > 0.
Since dist(M K (t n , Y ), R K (t n , ζ; U )) > 0, M K (t n , Y ) ε and R K (t n , ζ; U ) will be disjoint for ε > 0 sufficiently small, where, for any set A, A ε = {y; dist(y, A) < ε}. If A is convex A ε is convex with interior points, thus we can apply the separation theorem and construct (µ n , w n ) ∈ Σ(0, t n ; E * ) × E * , (µ, w n ) = 1 such that for every y(·) ∈ M(t n ), y ∈ Y and u(·) ∈ C ad,K (0, t n ; U ). We extend µ n to [0,t] setting µ n = 0 there and, using Alaoglu's theorem, select a subsequence such that By Lemma 2.7 the set {y(·, ζ, u); u ∈ C ad (0,t; U )} is compact in C(0,t; E), thus the integral on the right side of (3.13) tends to zero uniformly in u. Sending left the second term on the right of (3.13) we obtain Hence we obtain a contradiction using Lemma 3.2, and we conclude that (µ, w) = 0. Inequality (3.13) implies in particular that In view of the arbitrariness of δ, we obtain in this way that where C does not depend on n = 1, 2, . . . . Since w n → w E-weakly and S(t) * is compact, then S(t) * w n → S(t) * w in the norm of E * [13, Theorem 6, p. 486] and it follows from Fatou's theorem that We take limits in (3.13) as n → ∞. Justification is obvious in all terms except the second on the right side. We can obviously take the limit if u(·) vanishes in t − δ ≤ t ≤t; then we write the integral term in the form (3.16), use the fact that which we use thrice. The first time we take y(t) = y(t, ζ,ū) and y = y(t, ζ, u), cross out S(t − t)ζ in both sides and use Lemma 2.6 in both integrals. We obtain in this way wherez(t) is the solution of (2.11) corresponding to µ and z ∈ E * (K) defined by w = (K −1 ) * z; see (2.10) and the proof of is X * -weakly measurable in X, hence strongly measurable since X is separable. On the other hand,ū(·) is X-weakly measurable, so that B * z (·),ū(·) is a bounded measurable function. Call e the set of all Lebesgue points of b * z (·),ū(·) and of B * z (t).
so that, taking limits, (3.5) results. In the second use of (3.18) we take u =ū on both sides, obtaining for all y(·) ∈ X(t), so that µ ∈ N M(t) (y(·, ζ, u)) as claimed. The third time we take y(t) = y(t, ζ,ū) and obtain for all y ∈ Y . This shows that z ∈ N Y (y(t, ζ, u)) and thus completes the proof of Theorem 3.1.
Following finite dimensional usage, we shall callz(t) the costate corresponding to the optimal controlū(t). When Y is "thin" (say, a point), the assumptions in Theorem 3.1 demand that R(t, ζ; U ) contain a ball in D(K). If we define R ∞ (t) as the subspace of E consisting of all elements We may baptize (3.22) "exact reachability to D(K) in timet". It implies in particular that R ∞ (t) is dense in E.
4. The "full control" system. We give this name to with X = E , so that U ⊆ (E ) * and u(·) ∈ L ∞ w (0, T ; (E ) * ). Hypotheses (a) and (b) (with B = I) in §2 and §3 require E and E separable. If hypothesis (c) holds Theorem 3.1 applies with K = A (we may always assume via a translation that A −1 exists) if U , M are convex with nonempty interior. For the point target condition we only have to check that R(t, ζ; U ) contains a ball in D(A) for anyt > 0, and we can limit ourselves to ζ = 0. This was done in [16 p. 167] for U = unit ball. In the general case, we take an interior point u 0 of U and let u 0 (t) ≡ u 0 , y 0 = y(t, 0, u 0 ); a.e. in 0 ≤ s ≤t,z(·) the solution of (2.11).
We note that there exists a version of this result (Theorem 4.1 in [16]) where S(·) is an arbitrary semigroup. See [16] for the exact statement and proof.
When E is reflexive, X = E * . A subclass of the class of reflexive spaces is that of ζ-convex spaces introduced in [7], which are exactly the spaces where the Hilbert transform is a bounded operator from L p (−∞, ∞; E) into itself for 1 < p < ∞ [6]. It was proved in [12] that if E is ζ-convex and A is the infinitesimal generator of an analytic semigroup such that then the solution y(·) of (4.1) with u(·) ∈ L p (0, T ; E), 1 < p < ∞ belongs to D(A) a.e. (and satisfies This can be used as follows: ifū(t) is an arbitrary time optimal control in 0 ≤ t ≤t thenū(·) is as well time optimal in any interval 0 ≤ t ≤t <t (this is the well known optimality principle valid for time invariant equations like (4.1)). We can then take an increasing sequence {t n } ⊆ [0,t) with t n →t and such that y n = y(t n , ζ,ū) ∈ D(A) and apply Theorem 4.1 in each subinterval 0 ≤ t ≤ t n . We obtain a sequence (µ n , z n ) ∈ Σ(0,t; E * ) × E * 1 (A, I), (µ n , z n ) = 0 such that µ ∈ N M(t n ) (y(·, ζ,ū)) and a.e. in 0 ≤ t ≤ t n ,z n (·) the solution of (2.11) with z n , µ n (it does not seem obvious that one can "take limits as n → ∞" in (4.7) in any useful way). Hilbert spaces are ζ-convex, as are L p (Ω) spaces, 1 < p < ∞. The spaces L 1 (Ω) and C(Ω) are not ζ-convex. The Hilbert space particular case of (4.6) was proved much earlier in [9] without Assumption (4.5), and is used in [16] to show a result of the type of (4.7).
5. Nontriviality of the minimum principle and saturation of state constraints. There exists the possibility that the costatez(t) in Theorem 3.1 is trivial in the sense that B * z (t) ≡ 0; this empties the maximum principle. The fact that (µ, z) = 0 does not necessarily prevent this (for instance, take µ = −δ(t − t)z, z ∈ E * , z = 0). We believe that nontriviality of the costate cannot be prevented in general, but it still holds in a number of particular cases. The first result has to do with this, but it also has independent interest. For the proof (of a much more general theorem) see [22, Lemma 6.3]. Lemma 5.1 applied to the maximum principle implies that the measure µ vanishes on the set where the optimal trajectory y(t, ζ; u) does not saturate the state constraint (saturation means y(t, ζ, u) hits the boundary of M ). In particular, µ = 0 if the state constraint is never saturated (or if there is not state constraint; M = E). In this case, we obtain the "ordinary" maximum principle which is (sometimes) guaranteed to be highly nontrivial, as seen below.

is either empty or consists of a sequence {t n }; if {t n } is infinite, its only limit point may bet.
We prove first , thus by analyticity for all t > 0, and w, y = lim t→0+ S(t) * w, y = 0, so that w = 0, which implies z = 0.

Lemma 5.4 [15]. R ∞ (t) is dense in E if and only if
is not dense in E is and only if there exists z ∈ E, z = 0 with z, y = 0 (y ∈ R ∞ (t)), and Proof of Theorem 5.2. We show first that n(t, z) = [0,t]. Note first that ift <t implies S(t −t) * z = 0, thus z = 0 by Lemma 5.3. The rest of Theorem 5.2 follows from the fact that S(t − t) * z is analytic in t <t.
Result like Theorem 5.2 give meaning to the name switching points of an optimal control.ū(·). To fix ideas, assume that F is a Hilbert space and that the control set U is the unit ball of F . Then in the intervals (t n , t n+1 ) between points of the sequence {t n } in Theorem 9.3. The optimal control is then analytic between "switching points" t n .

Moreover, u(·) has at most a countable number of switching points in e(y(·, ζ,ū) (possibly with infinitely many accumulation points) and is (as) smooth (as the norm) in between.
In fact, let [t 0 , t 1 ] be a component interval of e(y(·, ζ,ū)). Divide it into a (at most) countable union of intervals [t n , t n+1 ] with t n < t n+1 and y(t n+1 , ζ, u) ∈ D(A) and apply Theorem 3.1 in each of these subintervals.
In the set target case R(t, ζ; U ) − Y rather than R(t, ζ, U ) contains a ball in D(K) and there is no analogue of Theorem 5.2, except in certain particular cases; one is equation (4.1). In fact, it follows as in Lemma 5.3 that A * S(t − t) * z = 0 in 0 ≤ t ≤t.
6. Applications to parabolic distributed parameter systems. We consider the uniformly elliptic operator Ay = ΣΣ∂ j (a jk (x)∂ k y) + Σb j (x)∂ j y + c(x)y in a domain Ω ⊆ R m with boundary Γ (a jk = a kj , ΣΣa kj (x)ζ j ζ k ≥ K ζ 2 for x ∈ Ω and ζ ∈ R m , a jk (x) and b j are continuously differentiable inΩ, c continuous inΩ). The domain Ω is bounded and of class C (2) . The formal adjoint is A y = ΣΣ∂ j (a jk (x)∂ k y)−Σ∂ j (b j (x)y)+c(x)y. We denote by A p (β), 1 ≤ p < ∞ the restriction of A in L p (Ω) corresponding to a boundary condition β, either the Dirichlet boundary condition or a variational boundary condition ∂ ν y(x) = γ(x)y(x) (x ∈ Γ) with γ(x) continuously differentiable on Γ and ∂ ν the conormal derivative ΣΣa jk (x)η j (x)∂ k , {η j (x)} the outer normal vector on Γ. The adjoint boundary condition β is β = β for the Dirichlet boundary condition; for a variational boundary condition, We also consider the operator A c (β) determined by A and β in the space C(Ω) of continuous functions inΩ equipped with the supremum norm; for the Dirichlet boundary condition we use C 0 (Ω) ⊆ C(Ω) consisting of all y which are zero at the boundary Γ. Precise characterizations of the domains can be found for instance in [18]. The generation and duality properties of these operators can be summarized as follows. A p (β) generates a compact analytic semigroup S p (t; A, β) in L p (Ω), 1 ≤ p < ∞ and A c (β) generates a compact analytic semigroup S c (t, A; β) in C(Ω) for the Dirichlet boundary condition). If 1 < p < ∞ then L p (Ω) * = L q (Ω) (1/p + 1/q = 1) and We look below at the equations in the space L p (Ω), 1 ≤ p < ∞, and in the space C(Ω) or C 0 (Ω), depending on the boundary condition. Both equations fit into the hypotheses in §3. For 1 < p < ∞ we take E = L p (Ω), X = L q (Ω) with 1/p + 1/q = 1, so that X * = L p (Ω). The control set is U = B(0, 1) = unit ball of L p (Ω).
For p = 1, E = L 1 (Ω), and we take X = C(Ω), so that X * = Σ(Ω). The admissible control space is then the unit ball of L ∞ w (0, T ; Σ(Ω)), so that we rewrite the equation as For (6.5) we take E = C(Ω), X = C(Ω) = L 1 (Ω), so that X * = L ∞ (Ω); the control set is the unit ball of L ∞ (Ω). In all cases we take a closed, convex state constraint set M with nonempty interior. The target condition is The conclusions are based on (4.4). Recalling that the duality set Θ of an element x ∈ X consists of all x * ∈ X * such that ) is a time optimal control such that the final point of the trajectory belongs to D(A p (β)) we havē for a totally arbitrary optimal controlū(·) (no conditions on the target) outside of the set e(y(·, ζ,ū)) where the state constraint is saturated. Example 6.2. The system is (6.4) with p = 1 (or, rather, (6.6)). Given y(·) ∈ C(Ω) the duality set Θ(y) of y is the set of all measures ν ∈ Σ(Ω) supported in the set m(y(·)) = {x ∈Ω; |y(x)| = y C(Ω) } and such that yν is a positive measure and ν Σ(Ω) = y C(Ω) . It follows that ifν(·) is an optimal control and the target belongs to D(A 1 (β)) then there exists then the optimal controlν(t) is supported in the set m(z(t, ·)),z(t, x)ν(t, dx) is a positive measure and ν(t) = 1. The space L 1 (Ω) is not a ζ-space, and we know of no results on arbitrary optimal controls. Example 6.3. The system is (6.5). The duality set of z(·) ∈ L 1 (Ω) consists of all functions y(·) in L ∞ (Ω) with y(x) = z sign z(x) where z(x) = 0 (|y(x)| ≤ z elsewhere). It follows that ifū(t, x) is a time optimal control such that the final point of the trajectory belongs to D(A c (β)) then there exist z ∈ Σ(Ω) 1 (A c (β), I), µ ∈ Σ(0,t; Σ(Ω)), (µ, z) = 0 and such that if z(t, x) is the solution of More information on the measure µ can be obtained using Proof. . Isomorphism (a) is obvious. Then so is (b), since Σ(0,t; Σ(Ω)) and Σ([0,t] ×Ω) are the duals of isometrically isomorphic spaces. The isomorphism is given as follows: if µ ∈ Σ([0,t] ×Ω) we defineμ ∈ Σ(0,t; Σ(Ω)) = C(0,t; C(Ω)) * by Conversely, ifμ ∈ Σ(0,t; Σ(Ω)) we define an element of Σ( Using this identification, we rewrite inequality (3.21) in the form What can be said about the set The motivation for posing this problem is formula (6.14), which determines the control only outside of e. Since not much information on the measure µ is available, nothing obvious can be said of e ∩ supp(µ) (supp (µ) = support of µ), thus the question should be restricted to e\supp(µ).
We contribute a simple result for the case µ = 0, where the objective is to show that the set e ⊆ [0,t] ×Ω has measure zero. The equation is the homogeneous version of (7.2),z

Theorem 7.2. Assume the domain Ω is of class C (∞) , that the operator A(β) is self adjoint (A = A, β = β) with coefficients infinitely differentiable inΩ and that γ(·) is infinitely differentiable on Γ if β is a variational boundary condition. Then
the set e has measure zero.
Proof. . By hypoellipticity of the operator 3) for t <t and (via another replacement oft byt <t) we may assume from the beginning that z ∈ L 2 (Ω) in (7.3). Let {φ n (x)} be the eigenfunctions of A 2 (β) in L 2 (Ω) (corresponding to eigenvalues {−λ n }); they are infinitely differentiable inΩ and φ n C(Ω) = 0(n k ) as n → ∞ (7.4) (see [31]). Write z(t, x) = Σc n (t)φ n (x) (convergence in L 2 (Ω)). We have {c n } the Fourier coefficients of z. In view of (7.4) the series is uniformly convergent inΩ for t <t. Assume now that the set e in (7.2) has positive measure. Then, if χ is its characteristic function, in a set of positive measure, thus, by analiticity, in 0 < t ≤t. However, a Dirichlet series can vanish identically only if all its coefficients are zero [5] so that φ n (x) = 0 for each x ∈ d for all n. If x is a density point of d then all partial derivatives of φ n vanish at x; iterating the argument, all partial derivatives of all orders of φ n vanish in a set of positive measure, in particular, φ n has a zero of infinite order at some point x 0 ∈ Ω. Now, φ n (x) is a solution of the elliptic equation so we can bring to bear results on unique continuation on solutions of elliptic equations having zeros of infinite order [25], [27], [29], [30], [32]. In particular, using [30, Theorem 2.1] we deduce that φ n (x) ≡ 0, which is a contradiction and completes the proof of Theorem 7.2.
Obviously, there is ample room for improvement in Theorem 7.2. Theorem 2.1 in [30] requires infinitely differentiability from the leading coefficients of the operator but no smoothness from lower order coefficients, thus all that is needed is to justify (7.5). For a unique continuation result that does not require infinitely differentiable leading coefficients, see [25]. If φ n (t) is a sequence biorthogonal to {e −n 2 t } (that is, if φ j (t)e −n 2 t dt = δ jn ) then we obtain a solution of (8.4) in the form

A distributed parameter system with scalar control. The system is
provided that the series is convergent and term-by-term integration can be justified. A biorthogonal sequence satisfying φ n C(0,t) ≤ Cn 3 e πn (8.6) (with C depending on t) was constructed in [17] using previous results in [24] (this sequence is reasonably optimal in that there exists no other biorthogonal sequence with φ n = 0(e an ) for a < π). We may then set (ρ > 0 fixed), the domain D(K) consisting of all y ∈ C 0 [0, π] such that the right side belongs to C 0 [0, π]. There is no simple characterization of the domain in terms of summability conditions on the Fourier coefficients, but K has a bounded inverse given by  (8.9) and it follows that R(t, 0; U ) contains a ball in D(K), thus the same is true of R(t, ζ; U ) for any ζ. We may then apply Theorem 3.1 in the case of point targets y(t, ζ, u) =ȳ ∈ D(K). We do this with control set and state constraint y(t, ·) ∈ M = closed convex set in C[0, π] with nonempty interior; a possible choice is M = unit ball of C[0, π] in which case the state constraint is We deduce existence of a pair (µ, (8.11) z(t, x) the solution of the backwards heat equation with final condition z(t, x) = z. If the constraint is never saturated then Theorem 5.2 shows that σ(t) is nonzero except perhaps in a sequence {t n } whose only point of accumulation, if any, ist. We note that the condition thatȳ ∈ D(K) can be checked without computing sine Fourier coefficients; in fact, it will hold ifȳ(·) can be extended to a functionȳ(x + iξ) (a) 2π-periodic and odd in x, (b) analytic in the strip |ξ| < π, (c) infinitely differentiable in the strip |ξ| ≤ π. (See [17].) Remark 8.1. Theorem 4.4 can be used with other operators K. For instance, we may take (Σa n sin nx) = Σn k a n sin nx and Y a closed ball of centerȳ ∈ D(K) and positive radius in D(K). This situation roughly corresponds to approximation of the target in Sobolev norm of order k.
9. Other cost functionals. We consider the optimal problem for in a fixed or variable time interval 0 ≤ t ≤t with cost functional The problem is a bit of a hybrid: linear system, nonlinear functional. We assume that the Frećhet derivative ∂ y f 0 (y, u) exists in E × U and is continuous in y for u fixed, that f 0 (y, u(·)) is measurable and that ∂ y f 0 (y, u(·)) is strongly measurable (as an E * -valued function) for u(·) ∈ C ad (0,t; U ). Further, for every c > 0 there exists K(c), L(c) such that Let u(·) ∈ C ad (0,t; U ), 0 < s ≤t, v ∈ U , and let u s,v,h (·) be the spike perturbation of u(·). Then where the function S (t − s) * u(τ ) is strongly measurable (see §2). We can then use the theory of Lebesgue points to show.
Convergence is uniform in t ≥ s + ε, and approximations remain bounded, which plays a role in the proof of the result below, which can be found in [19], [20].  y(s, u), v) − f 0 (y(s, u), u(s)) .
Existence results require weak lower semicontinuity of the functional; if {u n (·)} is L 1 (0,t; X)-weakly convergent in L ∞ w (0,t; X * ) toũ(·) then y 0 (t,ũ) ≤ lim inf n→∞ y 0 (t, u n ) (9.4) and the proof of the maximum principle requires convexity: if u, v ∈ U and 0 ≤ α ≤ 1 then The results below are on the target condition y(t, ζ, u) ∈ Y (9.6) Theorem 9.3. Assume that the state constraint set M and the target set Y are closed. Then, if there exists a minimizing sequence {u n (·)} ∈ L ∞ w (0, T ; X * ) driving ζ to Y in time t n satisfying the state constraint and {t n } is bounded the optimal problem has a solution.

Final observations
Remark 10.2. Existing proofs of the maximum principle for nonlinear systems usually rely on Kuhn -Tucker type theorems for infinite dimensional nonlinear programming problems [19], [20], [21]. Applied to a system like (8.1), application of these theorems would require computation of differentials or "directional derivatives" of the solution map u → y(t, ζ, u) in the norm of D(K) rather than in the norm of the original space E. We don't know if these computations are possible; in particular we don't know if any version of the maximum principle is available for nonlinear perturbations of (8.1) (by terms of the form f (y(t, x))).