Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems*

Hanbing LIU Wenqiang LUO Shaohua LI

Abstract In this work， the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation.They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation.Then they show the existence of optimal solution to the optimal control problem， and finally the first order necessary condition is obtained by constructing an appropriate penalty function.

Keywords Fitzhugh-Nagumo equation， Time-periodic， Optimal control

1 Introduction and Main Results

Let Ω ?RNbe a bounded open set with smooth boundary ?Ω(N =1，2 or 3)，and let T ＞0 be a finite number.Set Q = Ω×(0，T) and Σ = ?Ω×(0，T)， and denote by |·| (resp.〈·，·〉)the usual norm (resp.scalar product) in L2(Ω).In the sequel， C denotes a generic positive constant.

Let ψ1，ψ2and ψ3be three given functions in L∞(Q)， ω ?Ω be an open nonempty set.We consider in this work the following controlled time-periodic FitzHugh-Nagumo equation

where g ∈L2(Q)，σ ＞0 and γ ≥0 are constants， and F0(x，t;u) is given by

In the above system， g is the control， and u，v are the state variables.

The FitzHugh-Nagumo model is a simplified version of the Hodgkin-Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron.This model plays an important role in physics， chemistry and mathematical biology.The variable u is the electrical potential across the axonal membrane; v is a recovery variable， associated to the permeability of the membrane to the principal ionic components of the transmembrane current; g is the medicine actuator (the control variable)， see [8， 10] for more details.

Compared with standard semilinear elliptic or parabolic equations， the analysis of the FitzHugh-Nagumo system is more difficult.The analysis of optimal control problems for FitzHugh-Nagumo equations have been already considered in several works.In [5]， the authors have investigated associated problems by the Dubovitskij-Milyutin optimality conditions.In [11]， the time-optimal control problems for a linear version of the FitzHugh-Nagumo equations was studied.The sparse optimal control problems for FitzHugh-Nagumo equations have been investigated in [6–7].

In this work， we shall consider the periodic optimal control problem for the FitzHugh-Nagumo equations.To our best knowledge， the existence of the periodic solution to the FitzHugh-Nagumo equations is not known in the existed literatures.Therefore， we shall firstly apply the Leray-Schauder principle to prove the existence of the periodic solution to the FitzHugh-Nagumo equations.Then， the existence of the optimal solution and the first order necessary condition (maximum principle) will be given.Comparing the optimal control problems considered in the previous mentioned works， the periodic state constraint causes difficulties.We shall construct an appropriate penalty functional to deal with this type of state constraint.For time-periodic optimal control problems for other systems，we cite here(see[3–4，13， 15]).

Now， we present the main results of this work.Concerning the existence and regularity of the periodic solution to (1.1)， we have the the following result.

Theorem 1.1Assume thatg ∈L2(Q).Then(1.1)admits at least one solution(u，v)with

An equivalent formulation to (1.1) is

Our second goal in this work is to study an optimal control problem for (1.2).We will mainly deal with the cost functional

where udis a desired state， and the constant a ＞0.

The second main result in this paper is as follows.

Theorem 1.2There exists at least one global optimal state-control(u，g).Moreover， there existsp ∈H1，2(Q)satisfying

and

About the main results above， we give here several notes.

(i)Theorem 1.1 claims the existence of periodic solution to the FitzHugh-Nagumo equations.Then it is natural to ask whether the solution is unique or not.Such kind of problem for nonlinear parabolic equations has been studied in [1]， wherein the notations of “subsolution”and “supersolution” to (1.1) are introduced.In [1]， using the comparison principle， which is based on the strong maximum principle， the author shows that the periodic solution exists between the subsolution and supersolution (see [1， Theorem 2.1]).Since the nonlinear term in FitzHugh-Nagumo equations is cubic， it is not difficult to see that， for certain specified ψi，i = 1，2，3， (1.1) may possess two subsolutions x1，x2and two supersolutions x1，x2which obey x1＜x1＜x2＜x2.Thus， if we can prove similar results as those in [1] for the FitzHugh-Nagumo equations， then we can show that there may be multiple periodic solutions to (1.1).However， since the FitzHugh-Nagumo equation contains an integral term comparing with the classical parabolic equations， we find that it is not an easy job.Hence， we leave this problem for future research， and deal with the FitzHugh-Nagumo equations by assuming that it may possess multiple periodic solutions.

(ii) In our setting of the control problems， the control is distributedly plugged into the system， and the initial data is not specified but subject to periodic endpoints constraint.In this formulation， the control system might be a kind of multi-response system since the initial data is not specified and the periodic solution to (1.4) might not be unique.Corresponding to the optimal control， there might be multiple state functions satisfying (1.4)， the optimal state function is the one such that the cost functional is minimized.An equivalent setting of the control problem is to treat the initial data as another control variable，which can be realized by impulsive control.Then，the state function is uniquely determined by the two control variables，and the periodic state constraint can be viewed as a mixed control-state constraint.This is somehow the standard formulation for optimal control problems with endpoints state constraint(see[12]).Nevertheless，in our formulation，we can approximate the optimal control problem by an optimization problem， and the optimality condition can be obtained by a very constructive way.It has been used in periodic optimal control problem governed by fluid flows and turns out to be efficient (see [3–4， 13， 15]).

(iii)Now that we obtain the optimality conditions presented in Theorem 1.2，we should try to see that whether we can apply these conditions to numerically approximate the optimal solution.

We can see from Theorem 1.2 that， with the optimality conditions， we can obtain a coupled periodic system with unknowns u and p， which seems not difficult to be solved.However，there are two essential difficulties remain to be overcomed.One is that the system is periodic.Unlike the classical forward-backward evolution systems，solving nonlinear coupled periodic systems is not easy.A monotone sequence method based on comparing principle has been applied in [9]to solve the periodic optimality systems for optimal control of parabolic Volterra-Lotka type equations.For other computing methods of periodic optimal control problems governed by ODEs， we refer to [2， 14] and references therein.Another difficulty is that the optimal control problem is nonlinear， and the optimal solution is not necessarily unique.The convergence of numerical approximation usually requires additional condition， such as the second order sufficient optimality condition for optimal solution(see[6]).These problems will be investigated in future work.

2 Existence and Regularity of Periodic Solution

Notice that (1.4) can be written in the form

where we have set

We will first prove that (2.1) admits at least one solution u ∈H1，2(Q) with the help of the Leray-Schauder’s principle (see [16， Theorem 6.A]).

Thus， let us consider the auxiliary problem

for each λ ∈[0，1].Denote the space Y := {u ∈L6(Q)∩C([0，T];L2(Ω));u(0) = u(T)}， which is equipped with the norm‖u‖Y=‖u‖L6(Q)+‖u‖C([0，T];L2(Ω)).We also introduce the mapping Λ:Y ×[0，1]→Y with u=Λ(w，λ) if and only if u is the unique solution to

We shall prove the following results.

Lemma 2.1The mappingΛ:Y ×[0，1]→Yis well-defined， continuous and compact.

Lemma 2.2All functionsusuch thatu=Λ(u，λ)for someλ ∈[0，1]are uniformly bounded inY.

In view of the Leray-Schauder’s principle， these will suffice to affirm that (1.2) admits at least one solution.

Proof of Lemma 2.1Step 1 (Well-posedness of (2.5)) Let u(0)=u0∈L2(Ω) be given.Define J :L2(Ω)→L2(Ω) by J(u0)=u(T)， where u(·) is the solution to

We claim firstly that J is well-defined， which suffices to prove that the above equation admits a weak solution u ∈L2(0，T;(Ω))∩C([0，T];L2(Ω)).Indeed， notice that

We can infer that

Multiplying the first equation of (2.6) by u in the sense of L2(Ω)， and integrating on (0，t)， we can obtain by the above identity that， for any given ε ＞0， there exists Cε＞0， such that

Since |?w|2≥c1|w|2for some c1＞0， we can take ε =， and by the above energy estimate，we can obtain that

Together with the classical Galerkin’s arguments，we can infer that(2.6)admits a weak solution，and it is unique.This subsequently implies the above claim.

Now， we show that the map J is contraction.Letbe two different initial data， u1，u2be the corresponding solutions.Then=u1-u2is the solution to

Similarly as the above estimate， multiplying the first equation of system (2.7) by w， we get that

Since |?w|2≥c1|w|2for some c1＞0， we obtain

where c0=min{c1，γ}.This implies that

Since e-c0T＜1，we infer that the map J is contraction.This implies that(2.5)admits a unique periodic solution in u ∈L2(0，T;(Ω))∩C([0，T];L2(Ω)).

Step 2 (The map Λ is well-defined and compact.) (2.5) can be equivalently written as

Multiplying the first equation (resp.the second equation) of system (2.9) by u (resp.) in the sense of L2(Ω)， and summing these two equations， we can obtain that

This implies that

Integrating the above inequality on [0，T]， and notice that u(0)=u(T)，v(0)=0， we infer that

where C is a constant depending on Ω，T and ψi，i=1，2，3.This implies that

Integrating (2.10) on [0，t]， using (2.11)， we can obtain that

Multiplying the first equation of system (2.9) by -t△u， we obtain that

This implies that

Integrating on [0，T]， we infer from (2.12)–(2.13)that

Notice that u(0)=u(T)， hence

Finally， multiplying the first equation of system(2.9)by-△u，and integrating on [0，t]， we can infer by (2.12) and (2.14) that

By Aubin-Lions lemma， we know that H1，2(Q) is compact imbedded in Y.This implies that Λ is well-defined and compact.

Step 3 (The map Λ is continuous) Let (wn，λn) be a sequence in Y ×[0，1]， and wn→w strongly in Y，λn→λ as n →∞.Let

By the result of Step 2， we see that

Hence， there exists at least a subsequence of unwhich will be denoted by itself， such that

Since F(w) is continuous in w， we see that

Then， we can pass to limit in the equation satisfied by wnand unto get that

Hence， we proved that Λ(wn，λn)→Λ(w，λ) strongly in Y.

Proof of Lemma 2.2(2.4) can be written as

Multiplying the first equation (resp.the second equation) of system (2.16) by u (resp.)，integrating on Ω， and adding the resulting identities， we obtain that

Notice that， for any ε ＞0， there exists Cε＞0， such that

It follows that

This implies that

Integrating on [0，T]， we can infer that

This in turn implies that

Now multiply the first equation of (2.16) by tut， and integrating on Ω， we get that

Notice that t ≤T and λ ≤1，we can check by Cauchy-Schwaz inequality and Young’s inequality that， for any ε ＞0， there exists Cε＞0， such that

This together with (2.18)–(2.19)imply that

Taking ε small enough， and integrating on [0，T]， we get that

This in turn implies that

It follows that

By Aubin-Lions lemma， we know that H1，2(Q)is compact imbedded in Y.Hence， all functions u such that u = Λ(u，λ) for some λ ∈[0，1] are uniformly bounded in Y.This completes the proof.

Proof of Theorem 1.1By Leray-Schauder’s principle， we can see from Lemmas 2.1–2.2 that (1.1) admits at least one periodic solution.The regularity properties (1.2)–(1.3) follows from the proof of Lemma 2.2.

3 Time Periodic Optimal Control Problems

We prove the existence of optimal solution firstly.

Proof of Theorem 1.2(Part I: The existence of optimal solution.) Let (un，gn) be a minimizing sequence in problem (P)， i.e.，

By the definition of J， {gn} is bounded in L2(Q)， and therefore， on a subsequence， again denoted by n， we have

By Theorem 1.1， we see that

Selecting further subsequences， if necessary， we have

Moreover，it is not difficult to see that

Then letting n goes to ∞in (3.1)， we infer thatsatisfies the system (1.2) andinf(P).

Now， given an optimal solution (u，g)， to prove the second part of Theorem 1.2， that is， the first order necessary optimality condition， we need to firstly consider the approximate control problem as follows:

where X ={u ∈H1，2(Q);u(0)=u(T)} and

Remark 3.1Although we have shown that for each control g ∈L2(Q)， there exists at leat one periodic solution to the state equation， there are still other problems to apply variation directly for the original optimal control problem.The essential problem is that the solution map might be multi-valued and might not be Fr′echet differentiable.Indeed， the linearized equation of the nonlinear equation (1.4) may do not have periodic solution.Therefore， we define an optimization problem to approximate the original control problem.We view the state and control as two independent variables， and view the state equation as constraint.The last term in (3.4) is defined to penalize this constraint.Notice that the optimal solution to the control problem is not necessarily unique， the second and third terms in the right-hand side of (3.4)are introduced to make sure that the optimal solutions for the approximate control problem converge to the specified optimal solution (u，g).

Similar to the first part of the proof of Theorem 1.2， we have the following result.

Lemma 3.1For eachε ＞0， problem(Pε)has at least one solution.

Moreover， for the relation between the optimal solution for (Pε) and the optimal solutionfor the original optimal control problem， we have the following lemma.

Lemma 3.2Let(uε，gε)∈X×L2(Q)be optimal for problem(Pε).Then，

ProofDenote vε=?tuε-△εu+G(uε)+F(uε)-χωgε.Sinceit follows that

By the proof of Theorem 1.1， we see that

Hence， on a subsequence， again denoted by ε， we have

In the space L2(Q)， we introduce the operators

and

It is readily seen that

The operators A and A*are defined by the same formulas (3.10) and (3.11)， where uε=.

To obtain the first order optimality condition， we need to use some properties of operators Aε，，A and A*， which can be stated as follows.(Similar properties for the 2-D and 3-D Navier-Stokes equations has been obtained in [3] and [15] respectively， and here we shall apply the same method to prove these properties for the FitzHugh-Nagumo equation.)

Lemma 3.3The operatorsAε，，AandA*are closed， densely defined， and have closed ranges inL2(Q).Moreover，dim N(Aε) ≤n0， independent onε， and the following estimates hold:

Similarly， the operatorsAandA*are mutually adjoint and estimates(3.12)remain true forAandA*.Here we use the symbolsNandRto denote the null space and the range of the corresponding operators.

ProofConsider the linear equation

We claim that system (3.13) has a unique solution φ = φε(t;φ0，g) ∈ C([0，T];L2(Ω)) ∩L2(0，T;(Ω))， for each φ0∈L2(Ω)，g ∈L2(Q).Indeed， to get the energy estimate， we multiply the first equation of (3.13) by φ， and it follows that

Notice that ‖uε‖H1，2(Q)≤C， and H1，2(Q)?C([0，T];(Ω))， we can infer that

This implies that

Taking η small enough， and integrating (3.14) from 0 to T， we can obtain by (3.15) that φ=φε(t;φ0，g) satisfies the following energy estimate

This indicates the existence and uniqueness of solution to (3.13).Moreover， if φ0∈(Ω)， it is not difficult to show that the solution φε∈H1，2(Q)?C([0，T];Ω)).

Multiplying (3.13)by -t?tφε， integrating on Q， and using (3.17)， we can get by the similar approach applied to obtain (2.20) that φε(T)∈(Ω)， and

We define Gε: L2(Q) →L2(Ω) by Gε(g) = φε(t;0，g)， and define Γε: L2(Ω) →L2(Ω) by Γεφ0=φε(t;φ0，0).It is clear that

and the estimate (3.18) yields that

Similarly as the proof of Theorem 1.1， we can get that (φ，g)∈Aε， i.e.， Aεis closed.

Now， let Γ ∈L(L2(Ω)，L2(Ω)) be defined by Γφ0=φ(T，φ0，0)， where φ is the solution to

As seen earlier，Γ ∈L(L2(Ω)，(Ω))， and so Γ is completely continuous from L2(Ω)into itself.Moreover，it is not difficult to show that

Since dim N(I-Γ)＜∞， (3.23)implies that there exists n0＞0， such that dim N(I-Γε)≤n0for all ε ＞0.Hence dim N(Aε)≤n0for all ε ＞0， as claimed.Moreover， we claim that

Indeed， otherwise， there exist φ0ε∈N(I -Γε)⊥= R((I -Γε)*)， fε∈R(I -Γε) such that(I -Γε)φ0ε= fεand |fε| = 1，|φ0ε| →∞.LetandWe can see from (3.18) thathas a subsequence which converges strongly in L2(Ω).Sincewe infer that there exists a subsequence of， and|φ0|=1.Moreover，we can see that φ0∈R((I-Γ)*)∩N(I-Γ)，which contradicts the fact that R((I-Γ)*)⊕N(I-Γ)=L2(Ω).

By (3.24)， we see that

Finally， we have that

This implies the first inequality of (3.12).The corresponding properties of the operator A*εfollow from the same arguments as above.The corresponding results of the operators of A and A*follow similarly.

Proof of Theorem 1.2(Part II: The necessary condition of optimality) Let (uε，gε) be optimal for(Pε).For any w ∈X，h ∈L2(Q)fixed， we set=uε+ρw and=gε+ρh.Then∈X ×L2(0，T;Q).Then，

By taking h=0 in (3.27)， we get that

Hence， pε∈D()=X and

By taking w =0 in (3.27)， we get that

This yields that

Define the linear bounded operator D :L2(Q)→L2(Q) by Dp=χωp.(3.31) implies that

Now， by Lemma 3.3 and the closed range theorem， we may write

Then， by (3.29) and Lemma 3.3 again， we get that

On the other hand， since the space N()is finite dimensional，we infer that the restriction of D to N()， still denoted by D， has closed range.Then， we may write

Then， by (3.32)， we see that {} is bounded in L2(Q).Sincen0， there exist p3∈L2(Q) and a subsequence ofstill denoted by itself， such that

By (3.29)， we know that there exist p1∈L2(Q) and a subsequence ofstill denoted by itself， such that

which is equivalent to

Passing to the limit for ε →0， we get

This shows that p1+p3∈D(A*)， and

Passing to the limit in (3.31) for ε →0， we get

Let p= p1+p3∈X.Then p ∈X.By (3.39)–(3.40)， we derive (1.5)–(1.6).This competes the proof.

AcknowledgementThe authors want to thank the anonymous referees for their valuable remarks and suggestions on the original version of this paper.