Theoretical analysis and control results for the Fitzhugh-Nagumo equation

Theoretical analysis and control results for the FitzHugh-Nagumo equation Paulo M. D. Magalh˜aes∗ , Adilson J. V. Brand˜ao† , Enrique Fern´andez-Cara‡and Marko Antonio Rojas-Medar§ Abstract In this paper we are concerned with some theoretical questions for the FitzHugh-Nagumo equation. First, we present a simple proof of the existence and uniqueness of strong solution. We also consider an optimal control problem for this system. We prove the existence of optimal state-control pairs and, as an application of the Dubovitski-Milyoutin formalism, we deduce the corresponding optimality system. We also connect the optimal control problem with a controllability question and we construct a sequence of controls that produce solutions that converge strongly to a desired global state. Finally, we present some open questions related to the control of this equation. 1 Introduction and main results Let Ω ⊂IRNbe a bounded open set with smooth boundary ∂Ω (N= 1,2 or 3) and let T > 0 be a finite number. We will set Q= Ω×(0, T) and Σ = ∂Ω×(0, T) and we will denote by |·|(resp. (·,·)) the usual norm (resp. scalar product) in L2(Ω). In the sequel, Cdenotes a generic positive constant. Let ψ1,ψ2and ψ3be three given functions in L∞(Q). We will consider the FitzHugh-Nagumo equation        ut−∆u+v+F0(x, t;u) = g, vt−σu +γv = 0, u(x, t)|Σ= 0, u(x, 0) = u0(x), v(x, 0) = 0, (1) ∗DEMAT/ICEB/UFOP, Ouro Preto-MG, Brazil. Partially supported by Instituto do Milenio-IM-AGIMB E-mail: [email protected] †DEMAT/ICEB/UFOP, Ouro Preto-MG, Brazil. Partially supported by Instituto do Milenio-IM-AGIMB E-mail: [email protected] ‡Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, SPAIN. Partially supported by grant BFM2003–06446 of the D.G.E.S. (Spain) and 2005/50705-4 of the FAPESP- Brazil. E-mail: [email protected] §IMECC-UNICAMP, CP 6065, 13083-970, Campinas-SP, Brazil. Partially supported by Grant 301354/03-0, CNPq-Brazil and BFM2003–06446 of the D.G.E.S. (Spain). E-mail: [email protected] 1 where g∈L2(Q), σ > 0 and γ≥0 are constants, u0∈L2(Ω) (at least) and F0(x, t;u) is given by F0(x, t, u) = (u+ψ1(x, t))(u+ψ2(x, t))(u+ψ3(x, t)). In this system, gis the control, which is constraint to belong to a nonempty closed convex set Gad ⊂L2(Q) and uand vare the state variables. The FitzHugh-Nagumo system is a simplified version of the Hodgkin-Huxley model, which seems to reproduce most of its qualitative features. The variable u is the electrical potential across the axonal membrane; vis a recovery variable, associated to the permeability of the membrane to the principal ionic components of the transmembrane current; gis the medicine actuator (the control variable), see [11, 10] for more details. Taking into account the role that can be played by actuators in this context (by inhibiting in the case of calmant medicines and by exciting in the case of anti-depressive products), it is natural to consider control questions for this model. An equivalent formulation to (1) is easily obtained by solving the second equation, which gives v(x, t) = σZt 0 e−γ(t−s)u(x, s)ds. (2) We obtain:        ut−∆u+σZt 0 e−γ(t−s)u(s)ds +F0(x, t;u) = g, u(x, t)|Σ= 0, u(x, 0) = u0(x). (3) In the sequel, unless otherwise specified, we will always prefer this shorter formulation of the problem. Accordingly, we will work with couples (u, g) which a posteriori give the secondary variable vthrough (2). This paper deals with several questions concerning systems (1) and (3). First, we will deal with existence, uniqueness and regularity results. In this context, we will provide a simple proof of a known result; a previous proof was given in [12]. The result is the following: Theorem 1 Assume that one has g∈L2(Q)and u0∈H1 0(Ω). Then (1) possesses exactly one solution (u, v), with u∈L2(0, T;H2(Ω)) ∩C0([0, T ]; H1 0(Ω)), ut∈L2(Q),(4) v∈C0([0, T]; H2(Ω)), vt∈L2(0, T;H2(Ω)).(5) In the sequel, H1,2(Q) stands for the Hilbert space H1,2(Q) = {w∈L2(0, T;H2(Ω)) : w= 0 on Σ, wt∈L2(Q)}. 2 Our second goal in this paper is to study an optimal control problem for (3). We will mainly deal with the cost functional J(u, g) = 1 2ZZQ |u−ud|2dx dt +a 2ZZQ |g|2dx dt, (6) where a > 0. In particular, we will deduce the optimality system for (3), (6) following the Dubovistky-Milyutin formalism (see [8]). Definition 1 Let Qbe the set Q={(u, g)∈H1,2(Q)×L2(Q) : (3) is satisfied },(7) Then the admissibility set for (3),(6) is Uad ={(u, g) : (u, g)∈ Q, g ∈ Gad }.(8) It will be said that (ˆu, ˆg)is a global optimal state-control if (ˆu, ˆg)∈ Uad and J(ˆu, ˆg)≤ J (u, g)∀(u, g)∈ Uad. It will be said that (ˆu, ˆg)is a local optimal state-control if (ˆu, ˆg)∈ Uad and there exists ε > 0such that, whenever (u, g)∈ Uad and ku−ˆukH1,2(Q)+kg−ˆgkL2(Q)≤ ε, one has J(ˆu, ˆg)≤ J (u, g). Let us state our second main result: Theorem 2 Assume that u0∈H1 0(Ω) and Gad ⊂L2(Q)is a nonempty closed convex set. Then there exists at least one global optimal state-control (ˆu, ˆg). Furthermore, if (ˆu, ˆg)is a local optimal state-control of (3),(6) and we assume that Gad has nonempty interior and J0(ˆu, ˆg)does not vanish, there exists ˆp∈ H1,2(Q)such that the triplet (ˆu, ˆp, ˆg)satisfies (3) with greplaced by ˆg, the linear backwards system        −ˆpt−∆ˆp+σZT t e−γ(s−t)ˆp(s)ds +DuF0(x, t; ˆu) ˆp= ˆu−ud, ˆp(x, t)|Σ= 0, ˆp(x, T ) = 0 (9) and the additional inequalities ZZ Q (ˆp+aˆg)(g−ˆg)dx dt ≥0∀g∈ Gad,ˆg∈ Gad.(10) Roughly speaking, in order to apply the Dubovistky-Milyutin formalism, we first reformulate the control problem in the form ½Minimize J(u, g) subject to (u, g)∈ Q, g ∈ Gad,(11) 3 where we recall that the set Gad is a nonempty closed convex subset of L2(Q) (the control constraint set) and Qis given by an equality constraint: Q={(u, g)∈H1,2(Q)×L2(Q) : M(u, g) = 0} for a suitable operator M. Assume that (ˆu, ˆg) is a local minimizer of (11). Then we associate to (ˆu, ˆg) the cone K0of decreasing directions of J, the cone K1of feasible directions of Gad and the tangent subspace K2to the constraint set Q. We have the following (geometrical) necessary condition of optimality: K0∩K1∩K2=φ. Accordingly, there exist continuous linear functionals f0,f1and f2, not simultaneously zero, such that fi∈K∗ ifor i= 1,2,3 and f0+f1+f2= 0 (this is the Euler-Lagrange equation for the previous extremal problem). From this equation we obtain the optimality system (3) (with greplaced by ˆg), (9), (10). A large family of control problems involving partial differential equations can be solved by this method. In particular, several interesting generalizations and modified versions of (3), (6) can be considered: other non-quadratic functionals, control problems with constraints on the state, multi-objective control problems, etc. Remark 1 When int Gad is the empty set, to determine the cone K1is more complicated. In this case, we can argue as in [7] to obtain a similar result. For simplicity, we will not give the details. Remark 2 When J0(ˆu, ˆg) = (0,0), it is natural to look for second-order optimality conditions. This can be made following the results in [1] for this and many other problems. An analysis of this situation will be given in a next paper. Our third goal in this paper is to analyze the behavior of the solutions to problems of the kind (3), (6) as a→0+. It is well known that this is a way to pass from the optimal control to a controllability approach. More precisely, if Gad =L2(Ω), it is expected that the solutions (ˆu, ˆg) of (3), (6) satisfy ˆu→ud as a→0+. A result of this kind in established in our next theorem. In order to give the statement, we have to introduce a new function: H0(x, t;s) =    F0(x, t;s)−F0(x, t; 0) sif s6= 0, DuF0(x, t; 0) otherwise. Then we have: 4 Theorem 3 Assume that u0= 0,Gad =L2(Q)and ud∈Lr(Q), where r≥4. For each n= 1,2, . . ., let (un, pn, gn)be a solution of the coupled problem                          un t−∆un+σZT t e−γ(s−t)un(s)ds +F0(x, t;un)pn=gn, −pn t−∆pn+σZT t e−γ(s−t)pn(s)ds+H0(x, t;un)pn=|un−ud|r−2(un−ud), un(x, t)|Σ=pn(x, t)|Σ= 0, un(x, 0) = 0, pn(x, T) = 0, pn+1 ngn= 0. (12) Then un→udstrongly in Lr(Q)as n→ ∞. In this way, for any target ud∈Lr(Q) we can construct a sequence of (possibly unbounded) controls gnand associated states unthat converge to ud. The proof of this theorem will be given below. It relies on some estimates for unin Lr(Q) and pnin L2(Q). Remark 3 This result is inspired by the ideas of J.-L. Lions in the context of the approximate controllability of linear parabolic equations; see [13, 9]. Remark 4 The equation satisfied by pnin (12) is not exactly the same satisfied by ˆpin (9). First, we have a different right hand side. This is motivated by the search of a good estimate for un. Indeed, it will be seen in Section 4 that the term |un−ud|r−2(un−ud) with r≥4 is needed to bound unin Lr(Q) and then H0(·;un) in L2(Q). The second difference is that the coefficient of pnin (12) is H0(x, t;un) and not DuF0(x, t;un). This is also needed to estimate un. This paper is organized as follows. Sections 2 to 4 are respectively devoted to the proofs of theorems 1, 2 and 3. Then, in Section 5 we present several additional remarks and open questions. Among other things, we will address some controllability questions. It will be seen there that, unfortunately, very few is known on the subject. 2 Existence, uniqueness and regularity results Assume that g∈L2(Q) and u0∈H1 0(Ω) in (3). Notice that (3) can be written in the form    ut−∆u+G(u) + F(u) = g, u(x, t)|Σ= 0, u(x, 0) = u0(x), (13) where we have set G(u)(x, t)≡σZt 0 e−γ(t−s)u(x, s)ds (14) 5 and F(u)(x, t)≡F0(x, t;u(x, t)).(15) We will first prove that (13) possesses at least one solution u∈H1,2(Q) with the help of the Leray-Schauder’s principle. Thus, let us consider for each λ∈[0,1] the auxiliary problem    ut−∆u=λ(g−G(u)−F(u)), u(x, t)|Σ= 0, u(x, 0) = u0(x). (16) Also, let us introduce the mapping Λ : L6(Q)×[0,1] 7→ L6(Q), with u= Λ(w, λ) if and only if uis the unique solution to    ut−∆u=λ(g−G(w)−F(w)), u(x, t)|Σ= 0, u(x, 0) = u0(x). (17) We will prove the following: Lemma 1 The mapping Λ : L6(Q)×[0,1] 7→ L6(Q)is well-defined, continuous and compact. Lemma 2 All functions usuch that Λ(u, λ) = ufor some λare uniformly bounded in L6(Q). In view of the Leray-Schauder’s principle, this suffices to affirm that (3) possesses at least one solution. Proof of lemma 1: First, notice that for any w∈L6(Q) we have F(w)∈ L2(Q) and G(w)∈L∞(0, T;L6(Ω)). Furthermore, the mappings w7→ F(w) and w7→ G(w) are continuous. Consequently, it is obvious that (w, λ)7→ Λ(w, λ) is well-defined and continuous from L6(Q)×[0,1] into L6(Q). The compactness of Λ is a consequence of parabolic regularity. Indeed, if (w, λ)∈L6(Q)×[0,1] and u= Λ(w, λ), we have u∈L2(0, T;H2(Ω)) ∩C0([0, T ]; H1 0(Ω)), ut∈L2(Q), i.e. u∈H1,2(Q) (we are using here that u0∈H1 0(Ω)). Moreover, the estimates we will prove in lemma 2 show that, whenever (w, λ) belongs to a bounded set of L6(Q)×[0,1], the associated ubelongs to a bounded set in H1,2(Q). Since this space is compactly embedded in L6(Q) for N= 1,2 or 3, we deduce that Λ : L6(Q)×[0,1] 7→ L6(Q) is compact. Proof of lemma 2: Let us assume that λ∈[0,1], u∈L6(Q) and Λ(u, λ) = u, i.e. usolves (16). We will prove that, for some constant C > 0 independent of λand u, one has kukL6(Q)≤C. (18) 6 In fact, we will directly prove much more: that uis uniformly bounded in H1,2(Q). Let us rewrite (16) in the form        ut−∆u+λv +λF(u) = λg, vt+γv −σu = 0, u(x, t)|Σ= 0, u(x, 0) = u0(x), v(x, 0) = 0. (19) Then, by multiplying by u(resp. λ σv) the first equation (resp. the second equation), integrating in Ω and adding the resulting identities, we get 1 2 d dt|u|2+λ 2σ d dt|v|2+|∇u|2+λγ σ|v|2+λ(F(u), u) = λ(g, u) (20) in (0, T). Notice that, for any ε > 0, there exists Cεsuch that (F(u), u)≥(1 −ε)kuk4 L4−Cε.(21) Indeed, we have for instance ¯¯¯¯ZΩ ψju3dx¯¯¯¯ ≤CkψjkL∞kuk3 L4≤ε 8kuk4 L4+Cε for any j= 1,2,3. In view of (20) and (21), we have: 1 2 d dt|u|2+λ 2σ d dt|v|2+|∇u|2+λγ σ|v|2+λ(1 −ε)kuk4 L4≤1 2|∇u|2+λCε. Now, from Gronwall’s Lemma, the following is obtained: (kukL∞(0,T ;L2(Ω)) +kukL2(0,T ;H1 0(Ω)) ≤C, λkvkL∞(0,T ;H1 0(Ω)) +λkukL4(Q)≤C. (22) Let us now multiply by utthe first equation in (19) and let us integrate in Ω. We get: 1 2|ut|2+1 2 d dt|∇u|2+λ(v, ut) + λ(F(u), ut) = λ(g, ut).(23) Now, we have (F(u), ut)≥1 4 d dtkuk4 L4−ε|ut|2−Cε(1 + |∇u|2)−Cεkuk4 L4(24) since, for instance, ¯¯¯¯ZΩ ψju2utdx¯¯¯¯ ≤CkψjkL∞kuk2 L4|ut| ≤ ε 8|ut|4+Cεkuk4 L4 7 for any j= 1,2,3. On the other hand, |(v, ut)| ≤ ε|ut|2+Cε|v|2≤ε|ut|2+Cε.(25) From (23)–(25), we obtain the inequality (1 −2ε)|ut|2+1 2 d dt|∇u|2+λ 4 d dtkuk4 L4≤Cε(1 + |∇u|2) + λCεkuk4 L4 and, from Gronwall’s Lemma, we find: kutkL2(Q)+kukL∞(0,T ;H1 0(Ω)) +λkukL∞(0,T ;L4(Ω)) ≤C. (26) Notice that we have used here again the fact that u0∈H1 0(Ω). Finally, let us multiply by −∆uthe first equation in (19) and let us integrate in Ω. This time, we obtain the following identity: 1 2 d dt|∇u|2+|∆u|2+λ(v, −∆u) + λ(F(u),−∆u) = λ(g, −∆u).(27) It is not difficult to check that (F(u),−∆u)≥3 4|∇(u2)|2−ε|∆u|2−Cε(1 + |u|2+kuk4 L4).(28) Consequently, 1 2 d dt|∇u|2+|∆u|2+3λ 4 d dt|∇(u2)|2≤ε|∆u|2+λCε and using again Gronwall’s Lemma we find that kukL2(0,T ;H2(Ω)) +kukL∞(0,T ;H1 0(Ω)) ≤C. (29) This ends the proof of the Lemma. Let us now see that the solution we have found is unique. Thus, let u1and u2be two solutions (in H1,2(Q)) of (3) and let us set u=u1−u2. Let us also introduce v=v1−v2=σZt 0 e−γ(t−s)(u1(s)−u2(s)) ds. Then the following holds:        ut−∆u+v+F(u1)−F(u2) = 0, vt+γv −σu = 0, u(x, t)|Σ= 0, u(x, 0) = 0, v(x, 0) = 0. Consequently, by multiplying the first and second equations respectively by u and 1 σvand integrating in Ω, we get: 1 2 d dt|u|2+1 2σ d dt|v|2+|∇u|2+γ σ|v|2+ (F(u1)−F(u2), u) = 0.(30) 8 We have (F(u1)−F(u2), u) = ZΩ£(u1+ψ1)(u1+ψ2)(u1+ψ3)−(u2+ψ1)(u2+ψ2)(u2+ψ3)¤u dx =I0+ 3 X j=1 Ij+X 1≤j<k≤3 Ij,k, where we have used the notation I0=ZΩ¡(u1)3−(u2)3¢(u1−u2)dx, Ij=ZΩ ψj(u1+u2)|u1−u2|2dx for 1 ≤j≤3 and Ij,k =ZΩ ψjψk|u1−u2|2dx for 1 ≤j < k ≤3. Since I0≥0, we find that 1 2 d dt|u|2+1 2σ d dt|v|2+|∇u|2+γ σ|v|2 ≤CZΩ (1 + |u1|+|u2|)|u|2dx ≤ kβ(t)kL∞|u|2, where the function βbelongs to L2(0, T;L∞(Ω)). Since u(x, 0) ≡0 and v(x, 0) ≡ 0, we deduce that uvanishes identically, whence u1=u2. Hence, (3) possesses exactly one solution in H1,2(Q). Remark 5 Instead of (1), we could have started from the more general system        ut−∆u+v+F0(x, t;u) = g, vt−σu +γv = ˜g, u(x, t)|Σ= 0, u(x, 0) = u0(x), v(x, 0) = v0(x), (31) where ˜g∈L1(0, T;L2(Ω)), v0∈L2(Ω). Then, the problem is reduced again to a system of the form (3), with greplaced by g=g−v0(x)e−γt −Zt 0 e−γ(t−s)˜g(s)ds. Indeed, the solution of (31) is given by the couple (u, v), where uis the solution of (3) with this right hand side and v=v0(x)e−γt +Zt 0 e−γ(t−s)˜g(s)ds +σZt 0 e−γ(t−s)u(s)ds. 9 and we have ZZ Q |u−ud|rdx dt +nZZ Q |p|2dx dt ≤C, (56) where the constant Cis independent of λand n. From (56), arguing as in the proof of Lemma 2, we deduce that uis in fact bounded in L6(Q) by a constant that can depend on n. Obviously, we also obtain from (56) that the norm of pin L2(Q) is uniformly bounded. Then, arguing as in the proof of Lemma 3, the same is found for p. This ends the proof. Let us now finish the proof of Theorem 3. For each n, let (un, pn, gn) be a solution of (12). Then, the identity (55) and the estimate (56) hold for (un, pn):        ZZ Q |un−ud|rdx dt +nZZ Q |pn|2dx dt =−ZZ Q |un−ud|r−2(un−ud)uddx dt −ZZ Q F(0) pndx dt (57) and ZZ Q |un−ud|rdx dt +nZZ Q |pn|2dx dt ≤C. (58) Accordingly, unis uniformly bounded in Lr(Q) and pn→0 strongly in L2(Q). Let us look at the equation satisfied by pnin Q: −pn t−∆pn+σZT t e−γ(s−t)pn(s)ds +H(un)pn=|un−ud|r−2(un−ud). In the left hand side, the first three terms converge to zero in the distribution sense. This is also the case for the fourth one, since H(un) is uniformly bounded in L2(Q) (it is just at this point where we use that r≥4). Consequently, the right hand side also converges to zero. Since it is bounded in Lr0(Q), it converges weakly to zero in this space (r0is the conjugate exponent of r). But this implies that unconverges strongly to udin Lr(Q). Indeed, from (57), the weak convergence of |un−ud|r−2(un−ud) and the fact that ud∈Lr(Q), we see that ZZ Q |un−ud|rdx dt +nZZ Q |pn|2dx dt →0. This ends the proof. 5 Some final remarks and open problems This Section is devoted to discuss some additional facts concerning the control of (3). Some of them lead to open problems that, in our opinion, are of considerable interest. 16 5.1 Other optimal control problems There are many other optimal control problems that can be considered for systems of the kind (3). Let us mention one of them. Thus, consider the new cost functional L, where L(u, g) = 1 2ZΩ |u(x, T)−u1(x)|2dx +a 2ZZ Q |g|2dx dt (59) and u1∈L2(Ω) is a given function. The following result holds: Theorem 4 Assume that u0∈H1 0(Ω) and Gad ⊂L2(Q)is a nonempty closed convex set. Then there exists at least one global optimal state-control (ˆu, ˆg) of (3),(59). Furthermore, if (ˆu, ˆg)is a local optimal state-control, Gad has nonempty interior and L0(ˆu, ˆg)does not vanish, there exists ˆp∈H1,2(Q)such that the triplet (ˆu, ˆp, ˆg)satisfies (3) with greplaced by ˆg, the linear backwards system        −ˆpt−∆ˆp+σZT t e−γ(s−t)ˆp(s)ds +DuF0(x, t; ˆu) ˆp= 0, ˆp(x, t)|Σ= 0, ˆp(x, T ) = ˆu(x, T)−u1(x) (60) and the additional inequalities ZZ Q (ˆp+aˆg)(g−ˆg)dx dt ≥0∀g∈ Gad,ˆg∈ Gad.(61) Unfortunately, it is not easy to obtain a result similar to Theorem 3 in this context. By analogy with that theorem, it is maybe reasonable to expect that, for some large r, the sequence (un, pn, gn) given by                          un t−∆un+σZT t e−γ(s−t)un(s)ds +F0(x, t;un)pn=gn, −pn t−∆pn+σZT t e−γ(s−t) pn(s)ds+H0(x, t;un)pn= 0, un(x, t)|Σ=pn(x, t)|Σ= 0, un(x, 0) = 0, pn(x, T) = |un(x, T)−u1(x)|r−2(un(x, T)−u1(x)), pn+1 ngn= 0. (62) satisfies un(·, T)→u1in some sense as n→ ∞. But this is unknown. 5.2 Further comments on controllability In general terms, the controllability approach for an evolution partial differential equation or system consists in trying to drive the system from a prescribed initial 17 state at time t= 0 (u0in our case) to a desired final state (or, at least “near” a desired final state) at time t=T. In the interesting case, the control is supported by a set of the form ω×(0, T), where ω⊂Ω is a nonempty (small) open set. At present, controllability problems are relatively well understood for linear and semilinear parabolic equations; see for instance [4, 6, 3]. Unfortunately, this is not the case for the integro-differential system (3), not even for simplified (linearized) similar problems. For instance, consider the linear system        ut−∆u+σZt 0 e−γ(t−s)u(s)ds +α(x, t)u=g1ω, u(x, t)|Σ= 0, u(x, 0) = u0(x), (63) where α∈L∞(Q) and 1ωis the characteristic function of ω. It is said that this system is approximately controllable in L2(Ω) at time T if, for any u1∈L2(Ω) and any ε > 0, there exists g∈L2(ω×(0, T )) such that the corresponding solution satisfies |u(·, T)−u1| ≤ ε. At present, it is unknown whether (63) is approximately controllable. This is the case if αis independent of t; see [2]. Of course, the question is completely open in the case of the nonlinear system (3). References [1] E.P. Avakov, Necessary extremum conditions for smooth abnormal problems with equality and inequality type constraints, Lenin State Pedagogical Institute, Moscow. Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 3–11, 1989. [2] V. Barbu, M. Ianelli, Controllability of the heat equation with memory, Diff. and Integral Eqns., Vol. 13 (10–12), 2000, pp. 1392–1412. [3] A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos, E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41 (2002), no. 3, 798–819. [4] C. Fabre, J.-P. Puel, E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 1, 31–61. [5] T.M. Flett, Differential Analysis. Differentiation, differential equations and differential inequalities, Cambrigde University Press, 1980. [6] A. V. Fursikov, O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series # 34, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, 1996. 18 [7] I. Gayte, F. Guill´en-Gonz´alez, M. A. Rojas-Medar, Dubovitskii-Milyutin Theorem applied to optimal distributed control problems with constraints given by the backward heat equation., to appear. [8] I. V. Girsanov, Lectures on mathematical theory of extremum problem, Lectures Notes in Economics and Mathematical Systems, 67, Springer-Verlag, Berlin 1972. [9] R. Glowinski, J.-L. Lions , Exact and approximate controllability for distributed parameter systems, Acta Numerica 1994, p. 269–378. [10] S. P. Hastings, Some mathematical problems from neurobiology, Amer. Math. Monthly 82 (1975), 881–895. [11] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve J. Physiol. 117 500–44A. [12] D. E. Jackson, Existence and regularity for the FitzHugh-Nagumo equations with inhomogeneous boundary conditions, Nonlinear Anal, T. M. A., Vol. 14, no. 3, pp. 201–216 (1990). [13] J.-L. Lions, Remarks on approximate controllability, J. Analyse Math. 59 (1992), 103–116. 19