OBLIQUE PROJECTION BASED STABILIZING FEEDBACK FOR NONAUTONOMOUS COUPLED PARABOLIC-ODE SYSTEMS. Karl Kunisch. Sérgio S. Rodrigues

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1 OBLIQUE PROJECTION BASED STABILIZING FEEDBACK FOR NONAUTONOMOUS COUPLED PARABOLIC-ODE SYSTS Karl Kunisch Institute for Mathematics and Scientic Computing, Karl-Franzens-Universität, Heinrichstr. 36, 8010 Graz, Austria Sérgio S. Rodrigues Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstraße 69, A-00 Linz, Austria Abstract. Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given timedependent trajectory of FitzHugh Nagumo type systems. The feedback is explicit and is based on suitable oblique nonorthogonal projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ode systems of the FitzHugh Nagumo type. Simulations are presented showing the stabilizing performance of the feedback control. 1. Introduction. In this work we focus on coupled systems consisting of a parabolic equation with controls, and an ordinary differential equation. This class of systems include well-known models describing electric excitations and the propagation of electric waves in nerve fibers and in heart tissue. In this context the system of equations is referred to as the monodomain equations. Given a bounded domain Ω R d, where d is a positive integer, with smooth boundary Γ = Ω, the controlled monodomain equations read t v = ν v av3 + bv cv M 1 v, w + f + M i=1 u i 1 ωi, v0 = v 0, 1a t w = δw M v, w, w0 = w 0, 1b with Neumann boundary conditions n v Γ = 0, where f = fx, t is an external forcing, M 1 and M are suitable functions coupling the two equations, n is the unit outward normal vector to Γ, and the constants in {ν, δ, a, b, c} are all given and strictly positive. Further, {ω i i {1,,..., M}} is 1c 010 Mathematics Subject Classification. 93D15, 93B5, 93C05, 93C0. Key words and phrases. Exponential stabilization, explicit feedback, monodomain equations, finite-dimensional controller, oblique projections. Corresponding author. 1

2 K. KUNISCH AND S. S. RODRIGUES a family of open sets of Ω and our M actuators {1 ωi = 1 ωi x i {1,,..., M}} L Ω, are the indicator functions of the subsets ω i, { 1 if x ω i, 1 ωi x := 0 if x Ω \ ω i. Finally u = ut, taking values in R M, is a control function at our disposal. In electrophysiology, see [9, Section 1.3.3], the variable v models the transmembrane electric potential of the human heart and w is represents a gating variable. Typical models include the FitzHugh Nagumo, the Rogers McCulloch, and the Aliev Panfilov model. See [5, 15, 18, 1, 6]. The coupling M takes the form Mw, v := [ ] M1 w, v M w, v := [ ] dw + evw γv + ρv where the constants d, e, γ, ρ are also given, and are all strictly positive with the following exceptions defining the model: a e = ρ = 0, for the FitzHugh Nagumo model, [FN]. d = ρ = 0, for the Rogers McCulloch model, [RM]. d = 0, for the Aliev Panfilov model, [AP]. b Assume that a desired heart rhythm is given as the solution of the uncontrolled system 1, corresponding to a suitable initial condition v 0, w 0 L Ω L Ω, t v = ν v av3 + bv cv M 1 w, v + f, v0 = v 0, 3a t w = δw M w, v, w0 = w 0, 3b n v Γ = 0, 3c If v 0, w 0 v 0, w 0, then the behavior of 1 without control can be considerably different from the desired behavior of v, w, even if v 0, w 0 v 0, w 0 is small. Here we look for a feedback control u, so that the solution of 1 goes to the desired heart rhythm v, w, solving 3, provided the difference v 0, w 0 v 0, w 0 is small. We will follow a standard idea: first we find a feedback operator which stabilizes globally the linearization of 1 around v, w, then we use a suitable fixed point argument to conclude that the same feedback operator also stabilizes the nonlinear system locally. More precisely, by direct computations, we can see that the controlled linearization around v, w, satisfies t y + Ay + A ry + Sz M i=1 u i 1 ωi = 0, y0 = y 0 a t z + Dz + Rv = 0, z0 = z 0. b Moreover the difference y, z := v, w v, w, to the targeted solution, solves the system t y + Ay + A ry + Sz M i=1 u i 1 ωi = N 1 y, z, y0 = y 0 5a t z + Dz + Rv = N y, z, z0 = z 0, 5b

3 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 3 with the linear operators y Ay := ν y + y, with Ay, v H 1 Ω,H 1 Ω := ν y, v L Ω + y, v d L Ω, z Dz := δz, y A r y := y 3av y + bvy cy ewy, z Sz := dz + evz, y Ry := γ + ρvy, 6a 6b 6c 6d 6e and the nonlinearities y, z N 1 y, z := ay 3 b + 3avy eyz, y, z N y, z := ρy. 6f 6g Without loss of generality we may suppose that the set of actuators {1 ωi i {1,,..., M}} L Ω is linearly independent. We also set := span{1 ωi i {1,,..., M}}. Let α i, i {1,, 3,... }, be the increasing sequence of repeated Neumann eigenvalues of A = + 1 and let E M := span{e i i {1,,..., M}} be the space spanned by the first eigenfunctions of A in L Ω, 0 < α 1 α α n α n+1 + and Ae n = α n e n. For simplicity let us denote L := L Ω and H 1 := H 1 Ω. We will show that a sufficient condition for global stabilizability of system, and for local stabilizability of system 5, is given by H =, α M+1 > 6 + P E M P S + LL L R 0,LL A r L R 0,LL,H 1 δ 7a R L R 0,LL, 7b where L L, H 1 stands for the space of bounded linear functionals from L into H 1, and L L stands for the space of bounded linear functionals from L into itself. along E M, and P E M Further, P E M : L is the oblique projection in L onto = Id P E M is the complementary oblique projection in L onto E M along. The following regularity condition on the desired trajectory will be used. v, w L R 0 Ω L R 0 Ω. 8 Recall that in [11] it is shown that when S = 0, then 7 is a sufficient stabilizability condition for the parabolic system a. The following Theorems 1.1 and 1. are the main results of this paper. Theorem 1.1 Linearized monodomain equations. Let λ > 0 and 0. If 7 and 8 holds true, then there are constants C 1 and µ > 0, independent of

4 K. KUNISCH AND S. S. RODRIGUES, y 0, z 0, such that the solution of the linear system t y + Ay + A ry + Sz P E M Ay + A r y + Sz λy = 0, y = y 0, 9a satisfies t z + Dz + Rv = 0, z = z 0, 9b yt, zt L L Ce µt s0 y 0, z 0 L L, t. for all y 0, z 0 L L. Theorem 1. Monodomain equations. Let λ > 0 and 0. If 7 and 8 holds true, then there are constants C 1, µ > 0, and ɛ > 0, independent of, y 0, z 0, such that the solution of the system t y + Ay + A ry + Sz P E M Ay + A r y + Sz λy = N 1 y, z, y = y 0, 10a satisfies t z + Dz + Rv = N y, z, z = z 0, 10b yt, zt H 1 L Ce µt s0 y 0, z 0 H 1 L, t, provided that y 0, z 0 H 1 L < ɛ. Remark 1. Observe that 10 is exactly 5, with u given by M u i 1 ωi = P E M Ay + A r y + Sz λy. 11 i=1 Furthermore, by definition E M is an M dimensional space, which together with 35a implies that the ordered family of actuators := 1 ω1, 1 ω,..., 1 ωm is linearly independent with M 1. Thus, denoting the bijection [ ]: R M, with u M i=1 u i 1 ωi, the control u as in 11 is uniquely defined, and we may write u = [ ] 1 P E M Ay + A r y + Sz λy. Remark. Observe that Theorem 1. provides a result on the stabilizability to trajectories, since it asserts that the solution v, w = v, w + y, z of 1 goes exponentially to the targeted trajectory v, w, solving 3, provided 7 holds true, and v 0 v 0, w 0 w 0 H1 L < ɛ. Concerning 7b, it is proven in [17] for 1D domains of the form Ω = 0, L R that for any given r 0, 1 and any given M 1 we can place M actuators 1 ωi = 1 ω M i, with ωi M 0, L, so that the operator norm remains P E M LL bounded as M increases. Furthermore, the total volume covered by the actuators is equal rl, and thus independent of M. The extension of these results to multidimensional rectangles follows from the results in [11, Section.8], see also [11, Remark 3.9]. Note that, once the operator norm of the oblique projection remains bounded remains and α M+1 +, then we can set M large enough so that 7 is satisfied. Thus for an arbitrary coupling quadruple d, e, γ, ρ, we can set M large enough so that system is stable. Recall also that in [] the stabilizability of 1.1 is proven, with a Riccati based feedback, for the case ρ = 0. Comparing the results in this paper to those in [],

5 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 5 we can say that in [] all the actuators are supported in an a-priori given subset of the physical domain Ω and stability of the Riccati based closed-loop system is guaranteed under a condition on the coupling triple d, e, γ, see [, Corollary 3. and Eq. 38]. In this paper stability of the oblique projection based closed-loop system is obtained for an arbitrary coupling quadruple d, e, γ, ρ under a boundedness condition on the operator norm of the oblique projection P E M onto the span of the actuators, and the placement of the actuators is supposed to be at our disposal. Contents. The rest of the paper is organized as follows. In Sect. we derive results on the stability of linear coupled systems in an abstract setting. These results are applied in Sect. 3 to obtain stabilizability conditions for general linear coupled parabolic-ode systems. In Sect. we derive the stabilizability result for the concrete example of the linearized monodomain equations, in particular, it contains the proof of Main Theorem 1.1 in Sect..1 and that of Main Theorem 1. in Sect... Numerical simulations are presented in Section 5, for both the linearized system 9 and the nonlinear system 10, confirming the theoretical results and showing the performance of the explicit oblique projection stabilizing feedback control. Finally the Appendix gathers comments on the existence and uniqueness of weak solutions for the systems involved in the main text. Notation. We write R and N for the sets of real numbers and nonnegative integers, respectively, and we define R r := r, +, for r R, and N 0 := N \ {0}. We denote by Ω R d a bounded open connected subset, with d N 0. For a normed space X, we denote by X the corresponding norm, by X its dual, and by, X,X the duality between X and X. The dual space is endowed with the usual dual norm: f X := sup{ f, x X,X x X and x X = 1}. In case X is a Hilbert space we denote the inner product by, X. For an open interval I R and two Banach spaces X, Y, we write W I, X, Y := {f L I, X t f L I, Y }, where the derivative tf is taken in the sense of distributions. This space is endowed with the natural norm f W I, X, Y := f L I, X + 1/. t f L I, Y In case X = Y, we write H 1 I, X := W I, X, X. The time derivative in the distribution sense of a vector function v taking values in a Banach space X will be denoted by v := d dt v. If the inclusions X Z and Y Z are continuous, where Z is a Hausdorff topological space, then we can define the Banach spaces X Y, X Y, and X + Y, endowed with the norms a, b X Y := a X + b { Y ; a X Y := a, a X Y ; and a X+Y := inf a X, a Y X Y a X, a Y X Y a = a X + a } Y, respectively. We can show that, if X and Y are endowed with a scalar product, then also X Y, X Y, and X + Y are. In case we know that X Y = {0}, we say that X + Y is a direct sum and we write X Y instead. Again, if X and Y are endowed with a scalar product, then also W I, X, Y is. The space of continuous linear mappings from X into Y will be denoted by LX, Y. When X = Y we simply write LX := LX, X. If the inclusion X Y is continuous, we write X Y. We write X d Y, respectively X c Y, if the inclusion is also dense, respectively compact. C [a1,...,a k ] denotes a nonnegative function of nonnegative variables a j that increases in each of its arguments. Finally, C, C i, i = 0, 1,..., stand for unessential positive constants. 1

6 6 K. KUNISCH AND S. S. RODRIGUES. Stability of coupled evolutionary systems. We shall present a stability result for a coupled abstract system in a form which is convenient for our purposes. We introduce the bounded time intervals I :=, s 1, 0 < s 1 < +, 1 where the real numbers, s 1 are given. We shall utilize a a separable Hilbert space H, which we consider as a pivot space, H = H. Further we introduce a subspace G H, and two evolutionary systems v = Av, v = v 0 G, 13 ẇ = Dw, w = w 0 H, 1 in the Hilbert spaces G and H, where for suitable Hilbert spaces V G and W H, we suppose that At LV, V and Dt LW, W are operators in G and H, respectively, with domains DA = DAt G and DD = DDt H, independent of t. Furthermore, we assume that DA d V d G d V d DA and DD d W d H d W d DD. 15 Given linear operators S = St LH, G and R = Rt LG, H, we will consider the coupled system v = Av Sw, v = v 0, 16a ẇ = Dw Rv, w = w 0, 16b whose stability properties are the main focus in this section. Namely, we look for a suitable condition, so that the stability of 16 follows from the stability of each of the systems 13 and 1 separately. Let us set [ ] A S H := G H, V := V W, and M :=. 17 R D Note that, since G H is a closed subspace we may write and we have H = H, G = G, and H = H, M LV, V, DM = DA DD, DM d V d H d V d DM. Assumption 1. For every bounded time interval J R 0, we have the existence and uniqueness of: a G, V-weak solution v for system 13, for any given v 0 G, a H, W-weak solution w for system 1, for any given w 0 H, a H, V-weak solution z = v, w for system 16 for any given z 0 H. Thus, since the existence and uniqueness of solutions is not the focus of this paper, it is simply assumed by us. In the appendix, we briefly recall the standard procedure on the construction of weak solutions as a weak limit of suitable Galerkin approximations. In particular we will see that the above assumption is satisfied for a general class of operators A, D, S, and R. Applications to the linearized monodomain equations will be given in Sect..

7 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 7 Assumption. The operators S and R are essentially bounded: S L R s0, LH, G and R L R s0, LG, H. Assumption 3. There are pairs C A, µ A and C D, µ D, both in [1, + 0, +, so that U A t,s v 0 0 C A e µ At v 0 G and U D t,s w G 0 0 C D e µ Dt w 0 H H 18 for all t 0. Theorem.1. Under Assumptions 3 and, and if the inequality ξ := C AC D S R µ A µ D < 1 19 holds true, then for all ε sufficiently small there exists a constant D c 1 such that the weak solution of system 16 satisfies independently of v 0, w 0. vt, wt G H D c e εt s0 v 0, w 0 G H 0 The proof is given below. Before we derive some auxiliary results, where Assumptions and 3 are assumed to hold true. Lemma.. The weak solution of system 16 satisfies the estimate v L 1 R s0, G C A µ A v 0 G + C AC D S µ A µ D w 0 H + C AC D S R µ A µ D v L 1 R s0, G. 1 where S := S L R s0,lh,g and R := R L R s0,lg,h. Proof. Using Duhamel formula, we integrate the equation in 16b and obtain, for t, t wt = U D t,s w 0 0 Rsvs ds, and U D t,s wt H C D e µ Dt w 0 H + C D t Therefore, for t, we obtain t vt G = U A t,s v 0 0 U A t,s Sswsds G C A e µ At v 0 G + C A S t C A e µ At v 0 G + C A C D S t + C A C D S R e µ At s e µ Dt s Rsvs H ds, e µ At s ws H ds 3 s Let us consider first the case µ A µ D. In this case we obtain t e µ At s e µ Ds w 0 H ds e µ Ds τ vτ G dτ ds.

8 8 K. KUNISCH AND S. S. RODRIGUES and t e µ At s e µ Ds w 0 H ds = w 0 H e µ At+µ D = w 0 H µ A µ D e µ At+µ D e µ A µ D t e µ A µ D t e µ A µ D s ds = w 0 H e µ Dt e µ At 5 µ A µ D t s e µ At s t = e µ At+µ D τ vτ G 1 t = µ A µ D 1 = µ A µ D e µ Ds τ vτ G dτ ds t τ e µ At+µ D τ t e µ A µ D s ds dτ Now, from, 5, and 6, it follows v L 1 R s0, G C A v 0 µ G + C AC D S 1 1 A µ A µ D µ D + C AC D S R µ A µ D e µ A µ D t e µ A µ D τ vτ G dτ e µ Dt τ e µ At τ vτ G dτ 6 + t = C A v 0 µ G + C AC D S w 0 A µ A µ H D + C AC D S R µ A µ D + that is, in the case µ A µ D we have vτ G µ A w 0 H e µ Dt τ e µ At τ vτ G dτ dt. + τ e µ Dt τ e µ At τ dt dτ, v L 1 R s0, G C A µa v 0 G + C AC D S µ A µ D w 0 H + C AC D S R µ A µ D v L 1 R s0, G. 7 It remains to prove that the last inequality also holds in the case µ A = µ D. Thus, let µ A = µ D and notice that we have U D C D e µ D ρt w 0 H for H t, w 0 any ρ 0, µ D, t 0. We know, from 7, that v L1 R s0, G = lim ρ 0 v L 1 R s0, G lim CA ρ 0 µa v 0 G + C AC D S µ A µ D ρ w 0 H + C AC D S R µ A µ D ρ v L1 R s0, G from which we conclude that 7 also holds in the case µ A = µ D. Corollary 1. If inequality 19 holds true, then the weak solution of system 16 satisfies v, w L R s0, G H D c v 0, w 0 G H 8 for a suitable constant D c 1, independent of v 0, w 0.,

9 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 9 Proof. From 1 and 19 we obtain Then, from, we arrive at 1 ξ v L1 R s0, G < C A µ A v 0 G + C AC D S µ A µ D w 0 H. w L R s0, H C D w 0 H + C D R v L 1 R s0, G C D w 0 H + 1 ξ 1 CA C D R v 0 µ G + C AC D S w 0 A µ A µ H D Finally, from 3, we derive which ends the proof. v L R s0, G C A v 0 G + C A S w L R s0, H µ A, Proof of Theorem.1. If v, w solves 16, then vt, wt := e εt s0 vt, wt solves v = A εv Sw, v = v 0, 9a ẇ = D εw Rv, w = w 0. 9b We also have that z solves ż = Az, satisfies z = z 0 if, and only if, zt := e εt s0 zt ż = A εz, z = z 0, and z G C A e µ A εt z 0 G. Similarly u solves u = Du, u = u 0 if, and only if, ut := e εt s0 ut satisfies u = D εu, u = u 0, and u H C D e µ D εt u 0 H. Now if 19 holds true, then it also holds ξ ε := C AC D S R µ A εµ D ε < 1 for a small enough ε 0, min{µ A, µ D }. From Corollary 1 it follows that w, v L R s0, G H D c v 0, w 0 G H 30 for a suitable constant D c 1 independent of v 0, w 0, from which we can conclude 0. Let us now consider a perturbation of system 16 as follows, where η, η L R 0, G H, v = Av Sw + η, v = v 0, 31a ẇ = Dw Rv + η, w = w 0. 31b Corollary. Under the assumptions of Theorem.1 the weak solution of system 31 satisfies vt, wt G H D c e v εt s0 0, w 0 G H + ε 1 η, η, independently of, v 0, w 0. L R 0,G H 3

10 10 K. KUNISCH AND S. S. RODRIGUES Proof. With M as in 17, by Duhamel formula we find that vt, wt G H Ut,s M v t 0 0, w 0 + G H Ut,s M η, η s ds and, by Theorem.1, vt, wt G H D c e εt s0 v 0, w 0 G H + D c t e εt s η, η s ds G H G H e D c e εt s0 εt 1 t v 0, w 0 G H + D c η, η s 1 ε ds, G H from which we obtain Stability of the coupled parabolic-ode closed-loop system. Here we show that the explicit oblique projections based feedback control proposed in [11] for stabilization of nonautonomous linear parabolic equations is also able to stabilize a general class of nonautonomous linear coupled parabolic-ode systems, where the control acts only in the parabolic component. We will prove that the coupled system ẏt + Ayt + A r tyt + Stwt K UM tyt, wt = 0, y = y 0, 33a ẇt + Dtwt + Rtyt = 0, w = w 0, 33b with the explicit feedback y, w K UM y, w := P E M Ay + A r y λy + Sw, 33c is stable, under suitable assumptions on the linear span of the actuators = span{1 ω1, 1 ω,..., 1 ωm } and on the operators in Assumptions. We look at system 33 as an evolutionary system in H H, where H is our pivot separable Hilbert space, H = H. First we present our assumptions on the operators in 33. They will guarantee that the assumptions in the abstract setting of Section are fulfilled. Most of the assumptions are standard, and concern the existence and uniqueness of weak solutions for auxiliary systems. The only nonstandard assumption is the main stabilizability condition given below in Assumption 10. It can be seen as a generalization of the condition presented in [11] for the case of parabolic equations. Let V be another Hilbert space with V H. Assumption. A LV V is symmetric and y, z Ay, z V, V complete scalar product in V. is a From now we will suppose that V is endowed with the scalar product y, z V := Ay, z V, V, which still makes V a Hilbert space. Necessarily, A: V V is an isometry. Assumption 5. The inclusion V H is dense, continuous, and compact.

11 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 11 Necessarily, we have that y, z V, V = y, z H, for all y, z H V, and also that the operator A is densely defined in H, with domain DA satisfying DA d, c V d, c H d, c V d, c DA. Further, A has a compact inverse A 1 : H DA, and we can find a nondecreasing system of repeated eigenvalues α n n N0 and a corresponding complete basis of eigenfunctions e n n N0 : 0 < α 1 α α n α n+1 + and Ae n = α n e n. We can define, for every β R, the fractional powers A β, of A, by + A β y n e n := n=1 + n=1 α β ny n e n, and the corresponding domains DA β := {y H A β y H}, and DA β := DA β. We have that DA β d, c DA β1, for all β > β 1, and we can see that DA 0 = H, DA 1 = DA, DA 1 = V. For the time-dependent operators we assume the following: Assumption 6. For almost every t > 0 we have A r t LH, V, and we have a uniform bound, that is, A r L R 0, LH, V. Assumption 7. We have W d H d W, and D LW, W as in 15. For any 0 and w 0 H there is one, and only one, H, W-weak solution for ẇt + Dtwt = 0, w = w 0, and there are constants C D 1 and µ D > 0, independent of, w 0, such that the solution w satisfies wt H C D e µ Dt s ws H, for all t s 0. Assumption 8. The coupling operators S and R are both in L R 0, LH. Assumption 9. There exists one, and only one, solution v, w W loc R s0, V W, V W for the system v + P E M Av + P A r v + P Sw = 0, v = v 0, 3a ẇ + Dw + Rv = 0, w = w 0 H, 3b Note that in case K UM t = 0, we have that 33 is in the form of system 16, with G = H and V = V. Note that 33 is in the form of system 16, with G = = P E H and V = M E M V. Let us denote cf. Lemma. P S := P S and R := R L L R 0,LH, R 0,L,H

12 1 K. KUNISCH AND S. S. RODRIGUES Assumption 10. The linear span of the actuators satisfies H =, α M+1 > where inf γ R 0, γ 1 γ >0 γ1 1 Ξ 1 + γ 1 + P E M γ 1 γ A r ty, Y V Ξ 1 := sup,v t,y R 0 V Y H Y V Ξ := P A r P E M L R 0,LH,V. R LH Ξ P E M A r P E M 35a C D P S R + γ 1 γ µ D, L R 0,LH,V Remark 3. Note that 7b is a particular case of 35b, with γ 1 = γ = 1. 35b, 36a 36b Remark. Recall that from [11] by taking y 0 in [11, Theorem 3.6] we know that when S = 0, then the system v +P Av +P A r v = 0 is stable provided 35 holds true, with R = 0. In particular we observe that Assumptions 1 and, and estimates 18 hold true for system 16 with A = A + P A r and S = P S, that is, they hold true for system The stability result. The main result of this section, which will lead to main Theorems 1.1 and 1., is the following. Theorem 3.1. Under Assumptions 10, the coupled system 33 is stable: there are constants C c 1 and µ c > 0 such that yt, wt H H C c e µct s0 y 0, w 0 H H, t. with C c, µ c independent of, y 0, w 0. The proof is presented below, in section Auxiliary results. Here we derive some auxiliary results we will use in the proof of Theorem 3.1. We suppose that the Assumptions 10 do hold true. Recall also the interval I in 1. Notice that G = is a closed subspace of H with G endowed with the norm inherited from H. Let us set V = G V, which we endow with the norm inherited from V. Observe that V is a closed subspace of V, and A maps V onto V. Lemma 3.. There exists a unique, V E M -weak solution for the system Moreover, we have v + P E M Av + P A r v = 0, v = v C D P S R vt H e µt s0 v 0 H, for a suitable µ >. 38 µ D

13 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 13 Proof. The existence and uniqueness is proven in [11, Section 3.1]. Note that we have P Av = AP v = Av for any v E M. Now, proceeding as in [11, Section 3.1], by multiplying the equation by v, we obtain d dt v H = v V P E M A r v, v V,V = v V P E M A rv, v V,V P E M P A r v, v V,V and for any given positive constants γ 1 and γ, d dt v H γ 1 γ v V + γ 1 1 Ξ 1 v H + γ 1 with Ξ 1 and Ξ as in 36. Now from v V takes its values in, we obtain d dt v H γ 1 γ α M+1 γ 1 P P Ξ v LV H α M+1 v H, because the solution v 1 Ξ 1 γ 1 P P E M LV and from 35b, it follows that we can choose γ 1 and γ such that µ := γ 1 γ α M+1 γ1 1 Ξ 1 γ 1 P P which implies 38. E M LV Ξ > Ξ v H, 39 C D P S R, µ D Corollary 3. Given q H 1 R 0, E M and v 0, w 0 H, there exists a unique solution for system v + P E M This solution satisfies Av + q + P A r v + q + P Sw + P q = 0, v = v 0, 0a ẇ + Dw + Rv + q = 0, w = w 0. 0b vt, wt H H D c e εt s0 v 0, w 0 H H + ε L T q, q, 1 R 0,H H with T = max{ T P L R 0,LH T {P E M A, P E M A r, P, R}}, and with suitable constants D c 1 and ε > 0 independent of, v 0, w 0. Furthermore, ε 0, min{µ, µ D }, where µ is as in 38. Proof. When q = 0, the existence and uniqueness of the solution, in the space W loc R s0, V W, E M V W, is given by Assumption 9. Setting A = P A + P A r, from Lemma 3. we have that the solution of v + Av = 0, v = v 0 E M satisfies vt E M C A e µ At v 0 E M with C A, µ A = 1, µ, where µ is as in 38. Then, we observe that ξ = C A C D P S R < 1. µ A µ D

14 1 K. KUNISCH AND S. S. RODRIGUES Observe also that system 0 is in the form of system 31, by setting η = Rq and η = P Aq + P A r q + P q. Then, from Corollary, it follows vt, wt E M H D c e v εt s0 0, w 0 E M H + ε 1 η, η, L R 0, H which implies Proof of Theorem 3.1. Let us set q 0 := P y 0, v 0 := P E M y 0, and qt := e λt s0 q 0. Then we also set the solution v, w for system 0, given by Corollary 3, with the initial condition v, w = v 0, w 0. We start by showing that y, w := q + v, w solves 33. Indeed, with y = q + v, we can rewrite 0a as that is, 0 = ẏ q + P E M Ay + P A r y + P Sw + P q 0 = ẏ + Ay + A r y + Sw P E M Ay P E M A r y P E M = ẏ + Ay + A r y + Sw P E M Ay + A r y + Sw λy E Sw P M q, because q = λq and P E M q = P E M P y = P E M y. Therefore, we conclude that y, w solves 33, with K as in 33c. Finally, we notice that vt, wt is orthogonal to qt, 0 in H H, because vt is orthogonal to qt E M in H. Therefore yt, wt H H = vt, wt H H + qt, 0 H H and, using 1 in Corollary 3, we arrive at yt, wt H H D c e εt s0 v 0, w 0 H H + ε L T q, λq + e λt s0 q 0 H R 0,H D c e εt s0 v 0, w 0 H H + D c e εt s0 3+λ λε 1 T + e λt q 0 H D c e µct s0 y 0, w 0 H H with µ c = min{ε, λ} and D c = max{d c, 1 + D c 3+λ λε 1 T }. The proof of Theorem 3.1 is finished.. Applications. Stabilization of the monodomain equations. Assume that the desired trajectory v, w is given as the solution of the uncontrolled system 1. As mentioned in the Introduction, the difference y, z := v, w v, w satisfies system 10, with the operators defined as in 6. In this section we prove Theorems 1.1 and 1. which concern the stabilization of systems 9 and 10, respectively..1. Proof of main Theorem 1.1. Theorem 1.1 will follow from Theorem 3.1. For this purpose we discuss Assumptions 10, for the choice H = L Ω, V = H 1 Ω, and DA = { h H Ω n h Γ = 0}. Assumptions and 5 are satisfied and due to 8 Assumptions 6 and 8 hold true. It is easy to see that Assumption 7 holds with W = H and C D, µ D = 1, δ. Proposition 1. If 8 and Assumption 10 are satisfied, then Assumption 9 holds.

15 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 15 The proof follows from standard arguments. A few details are given in the Appendix. Thus, under condition 8 and Assumption 10, all the assumptions in Section 3.1 are satisfied. By Theorem 3.1 this implies the following Corollary, which in turn implies Theorem 1.1. From Remark 3 we recall that condition 7, which is assumed in Theorem 1.1, implies that Assumption 10 is satisfied. Corollary. Let λ > 0 and 0. If 8 and Assumption 10 hold true, then there are constants C 1 and µ > 0, such that the solution of the linear system 9 satisfies yt, zt H H Ce µt s0 y 0, z 0 H H, t. for all y 0, z 0 H H. Here the constants C and µ are independent of the triple, y 0, z 0... Proof of main Theorem 1.. To deal with the nonlinear systems, we will need strong solutions. We can prove that such solutions exist due to the fact that the reaction operator A r defined as in 6 satisfies if 8 holds. A r L R 0, LV, H, Proposition. Let 8 and Assumption 10 be satisfied, let µ be as in Corollary, and y 0, z 0 V H, with 0. Then the solution for 9 is strong, that is, y, z W loc R s0, DA H, H H. Moreover, we have e µ s sup 0 y, z C y 0, z 0 V H. s W s,s+1,da H,H H The proof, which relies in part on known arguments, is sketched in the Appendix. The next lemma gathers estimates on our nonlinearity N = N y = N t, y. Let us write, for simplicity, H = H H, V = V H, and DA = DA H. Lemma.1. There exists a constant Ĉ1 0 such that for all pairs p = y, z and p = ỹ, z in DA H, N p N p H Ĉ1 p p V 1 + p ε1 V + p ε V p DA + p DA + Ĉ1 p p DA p ε3 V + p ε V. Further for any given ς > 0 there exists Ĉ 0 such that with N p N p, p p H ς p p V + Ĉ1 + p ε5 V + p ε6 V 1 + p DA + p DA p p H, 3a 3b {ε 1, ε } [0, + and {ε 3, ε, ε 5, ε 6 } [, +. 3c Proof. Recall that from 6, we have N y, z = ay 3 + b 3avy eyz, ρy, which we write as { G1 y, z := ay N = G j, with 3, 0, G y, z := b 3avy, 0, G 3 y, z := 0, ρy, G y, z := eyz, 0. j=1 We will argue that 3 holds with N replaced by G j for j = 1,...,. This implies that 3 holds for N = j=1 G j. From [16, Section 5, Examples 1 and ], we can

16 16 K. KUNISCH AND S. S. RODRIGUES conclude that 3 is satisfied with N replaced by G 1 and G involving only the pde component y of the solution and the pde coordinate of the nonlinearity. Recall that the component y of the solution lives in DA, for almost every time t 0. This regularity is not necessarily satisfied by the ode component z, so we must check the details with the monomial term yz in G. On the other hand G 3 involves the ode component of the nonlinearity, so also in this case we will check the details. Concerning G we proceed as follows: estimate 3 is trivially satisfied for e = 0, thus we consider only the case e 0, then with ζ y, ζ z := p p = y ỹ, z z DA H, we find 1 e G p G p H = yz ỹ z H ζ y z H + ỹζ z H ζ y L z H + ζ z H ỹ L 1 e G p G p, p p H = yz ỹ z, y ỹ H = ζ y z + ỹζ z, ζ y H ζ y L z L + ζ y L ζ z L ỹ L. Next we consider the case d = 3, that is, Ω R 3. An analogous argument can be followed for d {1, }. From suitable Sobolev embeddings, the Agmon inequality, and suitable interpolation inequalities, we obtain 1 1 e G p G p H C 1 ζ y DA z H + ζ z H ỹ DA C 1 p p DA p H + p p H p DA C p p DA p V + p p V p DA e G p G p, p p H C 1 ζ y 1 H ζ y 3 V z H + C 1 p p H ỹ DA C 3 ζ y H z H + ς ζ y V + 1 C 1 p p H 1 + ỹ DA for any given ς > 0, and for suitable constants C 1 > 0, C > 0 and C 3 = C 3 ς > 0. Therefore we can conclude that the inequalities in 3 also hold true for G. Finally, concerning G 3 we proceed as follows: First G 3 p G 3 p H y ỹ H y + ỹ L C 1 p p H p DA + p DA, for C 1 > 0 independent of p and p in DA H, second we find that 1 ρ G 3p G 3 p, p p H = y ỹ, z z H = ζ yy + ỹ, ζ z H Thus 3 holds true also for N replaced by G 3., y + ỹ L ζ y L ζ z L C 1 y + ỹ DA p p H C y DA + ỹ DA p p H. Lemma.. If 8 holds, then the feedback operator in 33c is bounded: K UM L R 0, LH H, H. Proof. Indeed, proceeding as in [11, Proof of Theorem 3.7] we find K UM L R 0,LH H,H P E M LV P α 1 M + A r L,H R 0,LH,V + S + λα 1 L R 0,LH,V 1

17 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 17 and from 8 and 6 it follows that A r L R 0, LH, V and S L R 0, LH. Therefore we have the following Corollary, which in turn implies Theorem 1., because 35, assumed in Theorem 1., imply Assumption 10. See Remark 3. Remark 5. We are now prepared to provide the proof for Theorem 1., which relies on fixed point arguments. With two modifications, we can rely on the arguments utilized in [, Section 3.] see also [16, Section 3]. First, the feedback operator in these references is obtained as the solution of a suitable differential Riccati equation, in place of K UM as in 33c. However, thanks to the estimates in Corollary and Proposition we can replace the Riccati-based feedback operator in [, Section 3.] by K UM. Second the conditions on the nonlinearity N in [16, Section 3] are slightly stronger than those available here. In fact, in [16, Section 3], instead of property 3b, the following stronger assumption is assumed: G j p G j p, p p H C 1 + p ε5 V + p ε6 V p DA + p DA 1 p p V p p H + C 1 + p ε5 V + p ε6 V 1 + p DA + p DA p p H. It is clear that that the last estimate implies 3b, because p p V p p H β 1 p p H + β p p V, for arbitrary β > 0. Since Assumption 10 is implied by 7, Theorem 1. follows from the following. Theorem.3. Let 8 and Assumption 10 hold true. Then there are constants C 1 and ɛ > 0, independent of, y 0, z 0, such that the solution of the system 10 satisfies yt, zt V Ce µt s0 y 0, z 0 V, t, 5 provided that y 0, z 0 V < ɛ, where µ > 0 is as in Corollary. Proof. We follow the main argument sketched in [16, Section 3] and [, Section ]. Let us consider the system ẏ = Ay A r y Sz + K UM y, z + g 1, y = y 0 V, 6a ż = δz Ry + g, z = z 0 H. 6b [ ] A 0 with g L, + T, H. Multiplying the dynamics by Lz, with L = 0 δ and DL = DA, gives us y, z L,+T,V + y, z L,+T,DA D 3 T y, z V + g L,+T,H, 7 with the constant D 3 T depending on A r L R 0,LV,H, K L R 0,LH H,H, and T, and independent of and y, z. Here we use Lemma.. For simplicity we next denote H = H H, V = V H, and DA = DA H, and we write 6 as v + Lv + Cv Fv = g, v = v 0 V 8

18 18 K. KUNISCH AND S. S. RODRIGUES [ y0 ] [ ] [ ] [ ] g1 Arc S F1,1 F with v 0 =, g =, C =, and F = 1,. In particular, z 0 g 0 R 0 0 notice that F L R = K 0,LH L R. Here F 0,LH,H 1,1y := K UM y, 0 and F 1, z := K UM 0, z. We will look for a solution of the nonlinear system in a subset Z ϱ µ Z µ of the Banach space Z µ := { v L loc R s0, H } v Z µ < endowed with the norm v Z µ := sup e µ v W r,r+1,d,h. We also set r Z µ loc := {v L loc R s0, H e µ v For a given constant ϱ > 0 we define the subset Z µ ϱ } <, for all r. W r,r+1,d,h Z µ ϱ := { v Z µ v Z µ ϱ v 0 V}. Further we define the mapping Ψ: Z ϱ µ to the solution v of as follows. Z µ loc, v v, taking a given vector v v + Lv + Cv Fv = N v, v = v 0. 9 s Step [i]: a preliminary estimate. Proceeding as in [, Section.] we can conclude that the solution of the system 8 satisfies, for a suitable constant C, sup r 0 e µ z W r,r+1,d,h C z 0 V + sup k N k+1 k e µs s0 gs H ds s Step [ii]: Existence. The fixed point argument. Proceed as in [16, Section 3], by using 50, with N v in the place of f, and using 3a, we can conclude that Ψ maps Z ϱ µ into itself, if z 0 V ɛ 1, where ɛ is small enough, and Ψ is a contraction, if z 0 V ɛ, where ɛ ɛ 1 is small enough. Therefore, we can conclude that if z 0 V is sufficiently small, z 0 V < ɛ, then there exists a unique fixed point z = Ψ z = z Zϱ ɛ for Ψ. That is,. 50 v + Lv + Cv Fv = N v, v0 = v [ y Necessarily := v solves 10. Further, notice that 5 follows from z Z z] ρ µ. s Step [iii]: Uniqueness. We show now that a solution for 10 in the space Z := L loc R 0, D C[0, +, V Z µ ϱ is unique. Indeed, let z 1 and z be two solutions in Z, for 51. Then e := z 1 z solves 9 with g = N z 1 N z in the place of N z. Using 3b, and following standard arguments, we can obtain d dt e H C 1 + z 1 ε5 V + z ε6 V 1 + z 1 D + z D e H, which implies et H e t Gs s d e H, for all t, with Gs := C 1 + z 1 s ε5 V + z s ε6 V 1 + z 1 s D + z s D. Using e = 0 we find z t z 1 t = et = 0, for t. This ends the proof.

19 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS Numerical simulations. Here we present simulation results for 9 and 10. As actuators we will take suitable piecewise constant indicator functions {1 ωi i {1,,..., M}} H. In this section H := L Ω. In Figure 1 we can see the placement of rectangular actuators regions. Recall also that the possibly repeated eigenvalues α i of the Laplacian, under Neumann boundary conditions increase likely linearly, as we would expect from Weyl asymptotic formula [1]. See also [10, 8] a The triangulation of the domain Ω b First 9 Neumann eigenvalues of. Figure 1. The domain Ω, and the computed first Neumann Laplacian eigenvalues. The following simulations were carried out using a finite elements based spatial approximation of our equations, with a Crank Nicolson time discretization. We will set the parameters in the FitzHugh Nagumo, Rogers McCulloch, and Aliev Panfilov models so that we will have 3 constant steady states v i, ŵ i, i {1,, 3} and so that the first pde components are v 1 = 0, v = 1, and v 3 =. As we will see: for M as in -[FN] or [RM]: v 1, ŵ 1 = 0, 0, v, ŵ = 1, 10, v 3, ŵ 3 =, 0. 5a for M as in -[AP]: v 1, ŵ 1 = 0, 0, v, ŵ = 1, 5, v 3, ŵ 3 =, 0. 5b We will argue that for all 3 models: the linearizations around the steady state v 1, ŵ 1 and around the steady state v 3, ŵ 3 are stable, the linearization around the steady states v, ŵ is unstable. We will consider the systems 9 and 10 with two choices as reference trajectories. The first one is the unstable steady state vt, wt = v, ŵ, 53a the second one is spatially and temporally dependent and is defined as follows: we consider the time-dependent vector function T = T 1, T : R 0 R, { v + t t T 1 t = v 3 v if 0 t t 1, v 3 + t t 1 v v 3 if 1 < t t, { ŵ + t t T t = ŵ 3 ŵ if 0 t t 1, ŵ 3 + t t 1ŵ ŵ 3 if 1 < t t,

20 0 K. KUNISCH AND S. S. RODRIGUES where s N stands for the smallest integer which is smaller than s s < s + 1. Then we define our reference trajectory vt, x, wt, x as vt, x 1, x, wt, x 1, x = T 1 t + T 1 t, x 1, x, T t + T t, x 1, x, with R T 1 t, x 1, x := sinπt cosπt + 0.1x 1 x + x 1 sin0.x 1, T t, x 1, x := 0.1 sinπt sinπt + 0.1x 1 x + 0.1x R 3x. 53b Notice that v, w is periodic in time with period, vt +, x, wt +, x = vt, x, wt, x for all t 0. Furthermore, for any given nonnegative integer m N, vm, x, wm, x = v, ŵ and vm + 1, x, wm + 1, x = v 3, ŵ 3. That is, the trajectory v, w oscillates between the unstable steady state v, ŵ and the stable steady state v 3, ŵ 3. The initial condition in all the simulations, unless otherwise explicitly stated, is the constant v 0, w 0 = 1, 1. 5 In the feedback 33c, we have set the parameter λ = The linearized FitzHugh Nagumo model. Here we consider the linear system 9, with the parameters in 6 set as ν = 1, δ = 0.01, a = 1, b = 3, c = 1, d = 0.1, e = 0, γ = 0.1, ρ = 0. We can see that the vectors as in 5a are constant steady states of the uncontrolled and unforced nonlinear coupled system 1 i.e., with u = 0 and f = 0. Indeed we can see that for a constant vector v, ŵ we have v = 0 and that [ ] [ ] ξ1 v, ŵ a v ξ v, ŵ = := 3 b v + c v + dŵ + eŵ v ξ v, ŵ δŵ γ v + ρ v satisfies, for v {0, 1, }, [ ] [ dŵ a b + c + dŵ + eŵ ξ0, ŵ =, ξ1, ŵ = δŵ δŵ γ + ρ [ 8a b + c + dŵ + eŵ ξ, ŵ = δŵ γ + ρ Therefore with the parameters as in 56 we find that ξ0, 0 = ξ1, 10 = ξ, 0 = 56 ], 57a ]. 57b [ 0 0], which shows that the vectors as in 5a are constant steady states for system 1. Observe also that the linearization is stable around 0, 0 and, 0, and is unstable around 1, 10. Indeed, the linearization around a constant vector v, ŵ reads [ [ [ ] y 0 y t = + z] 0 0] z and we find that with the parameters as in 56: [ 3a v + b v c eŵ d e v γ ρ v δ ] [ y z], 58

21 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 1 [ ] c d For v, ŵ = 0, 0, the eigenvalues of the matrix are characterized γ δ by 1 λ δ λ + dγ = 0, that is, λ δλ + δ + dγ = 0. Therefore, λ = 1 + δ ± 1 + δ δ + dγ. It is clear that both eigenvalues have strictly negative real part, because 1 + δ δ [ + dγ < 1 + δ. ] 1a + b c d For v, ŵ =, 0, the eigenvalues of the matrix are γ δ again characterized by 1 λ δ λ + dγ = 0, that is, they both have strictly negative real part. [ ] 3a + b c d For v, ŵ = 1, 10, the eigenvalues of the matrix γ δ are characterized by λ δ λ + dγ = 0, that is, λ + δ λ δ + dγ = 0. Thus λ = δ ± δ δ + dγ = δ ± + δ + δ + dγ. It is clear that the largest eigenvalue have strictly positive real part, because + δ + δ + dγ > 0 and δ > The case of the constant unstable steady state. Here we consider the timeindependent trajectory v, w = 1, 10, as in 53a, which is the constant unstable steady state for the FitzHugh Nagumo model. In the figures below the annotation dyn=feed refers to the dynamics under the feedback control and dyn=free to the free dynamics without control. The annotation model=... specifies the type of the monodomain model defined by the selection of parameters cf.. In Figure we see the performance of our feedback control, which is able to stabilize system 9. The slope of log y, z H H is approximately 30 5 = 0.0 for time t 5, which means that y, z H H decreases exponentially with rate approximately δ = Recall that the explicit feedback has been constructed to stabilize the pde component, and as a corollary the stability of the ode component follows as well. Consequently we cannot expect to obtain a rate of decay for the ode component which is better smaller than δ. In Figure we can see also the curve illustrating the squared norm y H of the pde component versus the squared norm z H of the ode component. The configuration at initial time t = 0 is the starting point of the curve located at y 0 H, z 0 H, the red dot. The configuration at final time t = 30 in case of Figure is the end point of the curve the green dot. Further notice that the free dynamics is exponentially unstable, the norm is increasing exponentially with rate approximately = The case of a time-dependent trajectory. We consider the time-dependent trajectory v, w as in 53b. In Figure 3 we observe that after time t 5 the norm of the solution of the linear system 9 decreases exponentially with rate approximately δ = The free dynamics is exponentially unstable with norm increasing exponentially with rate approximately The nonlinear FitzHugh Nagumo model. We consider again the timedependent reference trajectory as in 53b, but now for the nonlinear system 10. In Figure we see that the norm of the controlled nonlinear system also decreases exponentially to zero with rate The free dynamics is not stable, the norm of the solution does not decrease as time increases.

22 K. KUNISCH AND S. S. RODRIGUES Figure. Norm of solution. Linearization around the steady state Linearization around the time- Figure 3. Norm of solution. dependent trajectory Figure. Norm of solution. Around the time-dependent trajectory The linearized Rogers McCulloch model. Here, in system 9, we take the parameters ν = 1, δ = 0.01, a = 1, b =, c =, d = 0, e = 0.1, γ = 0.1, ρ = 0. Note that from 57, with the parameters as in 59, we find [ 0 ξ0, 0 = ξ1, 10 = ξ, 0 =, 0] 59

23 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 3 which shows that the vectors as in 5a are constant steady states for system 1. For the linearization around a constant steady state 58, we find that with the parameters as in 59: [ ] c d For v, ŵ = 0, 0, the eigenvalues of the matrix with d = 0 are γ δ given by { c, δ}. Thus both eigenvalues [ are strictly negative. ] 1a + b c d e For v, ŵ =, 0, the eigenvalues of are γ δ characterized by 0 λ δ λ + eγ = 0, that is, λ + δλ + eγ = 0. Therefore, λ = δ ± δ 8eγ, and we can conclude that both eigenvalues have a strictly negative real part because δ[ 8eγ < δ. ] 3a + b c 1 d e For v, ŵ = 1, 10, the eigenvalues of are γ δ characterized by λ δ λ + eγ = 0, that is, λ + δ λ δ + eγ = 0. Thus, λ = δ ± δ + δ, because from 59 we have eγ = δ. It is clear that the largest eigenvalue has a strictly positive real part The case of the constant unstable steady state. Here the reference trajectory v, w = 1, 10 is as in 53a. In Figure 5 we can see that the explicit feedback is able to stabilize the system 9 exponentially with rate approximately δ = 0.01, while the free dynamics is exponentially unstable with rate approximately Figure 5. Norm of solution. Linearization around the steady state The case of a time-dependent trajectory. We consider the time-dependent trajectory v, w as in 53b. In Figure 6 we can see that the explicit feedback is able to stabilize the system 9 exponentially with rate approximately δ = 0.01, while the free dynamics is exponentially unstable with rate approximately The nonlinear Rogers McCulloch model. Here, we consider the timedependent trajectory v, w as in 53b, and consider the corresponding nonlinear system 10 with the parameters as in 59. In Figure 7 we see that our explicit feedback is able to exponentially stabilize system 10. The norm of the controlled system decreases exponentially with rate approximately δ = Notice also that the free dynamics is unstable.

24 K. KUNISCH AND S. S. RODRIGUES Linearization around the time- Figure 6. Norm of solution. dependent trajectory Figure 7. Norm of solution. Around the time-dependent trajectory The linearized Aliev Panfilov model. Here we consider the linear system 9, with the parameters ν = 1, δ = 0.01, a = 1, b =.5, c = 1, d = 0, e = 0.1, γ = 0.1, ρ = Note that from 57, with the parameters as in 60, we find [ ] 0 ξ0, 0 = ξ1, 5 = ξ, 0 =, 0 which shows that the vectors as in 5b are constant steady states for system 1. For the linearization around a constant steady state 58, we find that with the parameters as in 59: [ ] c d For v, ŵ = 0, 0, the eigenvalues of the matrix are { c, δ}. γ δ They are both strictly negative. [ ] 1a + b c d e For v, ŵ =, 0, the eigenvalues of are characterized by 3 λ δ λ + eγ 0. = 0, that is, λ δλ + δ = 0. γ 0. δ Therefore, λ = 3+δ± 3 + δ δ, and we have that both eigenvalues have a strictly negative real part.

25 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 5 [ ] 3a + b c 1 For v, ŵ = 1, 5, the eigenvalues of d d 0 δ given by λ { δ, 1 }, that is, the largest eigenvalue is strictly positive The case of the constant unstable steady state. Here we consider the timeindependent trajectory vt, wt = 1, 5, is an in 53a. In Figure 8 we see that our explicit feedback is able to stabilize system 9. The norm of the solution decreases exponentially with rate approximately δ = The norm of the free dynamics is exponentially increasing with rate approximately 0.5. are Figure 8. Norm of solution. Linearization around the steady state The case of a time-dependent trajectory. We consider again the linear system 9, with the time-dependent trajectory v, w as in 53b. In Figure 9 we confirm that also in this case the feedback is able to stabilize system 9. The norm decreases exponentially with rate approximately δ = The free dynamics is not stable. The norm of the solution is not converging to zero, and is close to a periodic behavior Linearization around the time- Figure 9. Norm of solution. dependent trajectory The nonlinear Aliev Panfilov model. We take the time-dependent trajectory v, w as in 53b, and the corresponding nonlinear system 10, again with the parameters as in 60. For Figure 10 we ran the simulation for time t [0, 30] as in previous examples. In this situation the results do not allow us to conclude that the feedback is stabilizing. The norm seems to decrease but, we cannot yet clearly see a rate of stability.

26 6 K. KUNISCH AND S. S. RODRIGUES Therefore we ran the simulation for a larger time interval and the results are shown in Figure 11. Again we observe that an exponentially decreasing rate is achieved. In fact, it is approximately δ = 0.01 the slope for time t 100 is approximately 0.0. We also observe that the norm corresponding to the free dynamics is not converging to zero, that is the free dynamics is not stable Figure 10. Norm of solution. Around the time-dependent trajectory Figure 11. Norm of solution. Around the time-dependent trajectory On the sign of the leading coefficient a. In the above examples the linearization based feedback control is able to stabilize the nonlinear system for the initial condition 1, 1, as in 5. Notice that in the models above, the sign of the leading coefficient a = 1 of av 3 b v + cv is positive, and in this case we may say that is has the good sign, in the sense that such sign somehow favors stability of the nonlinear system, for example it may guarantee the existence of global solutions cf. [7, Chapter 5, Remark 5.1], for the uncoupled parabolic equation. The positive sign of a may explain why the feedback still works for the nonlinear system for such a large initial condition. Recall that simulations in [16], for a single uncoupled parabolic equation, show that the feedback is able to stabilize the nonlinear system only if the initial condition is small enough. This is also the statement of Theorem.3, concerning the coupled system. Below we will consider the a model where we take a negative leading coefficient a. Notice that our results are independent of the sign of a. We will consider

27 STABILIZATION FOR COUPLED PARABOLIC-ODE SYSTS 7 the following parameters ν = 1, δ = 0.01, a = 1, b = 0, c = 1, d = 0, d = 0.1, γ = 0.1, ρ = [a ] 61 which roughly can be seen a variant of the Aliev Panfilov model with the negative sign for a and c and with b = 0. In such situation we may expect the uncontrolled solution of the uncoupled corresponding parabolic system to blow up in finite time. See [7, Chapter 5], [1], and references therein. Therefore we may expect the solution of the corresponding coupled system to explode as well The linearized a model. We consider the time-dependent trajectory v, w as in 53b, with 1, 1 in the place of the steady states: that is with v, ŵ := 1, 1 =: v 3, ŵ 3. In Figure 1 we see that the norm of the controlled solution of system 9 decreases exponentially with rate approximately δ = The norm of the free dynamics increases exponentially with rate approximately Linearization around the time- Figure 1. Norm of solution. dependent trajectory The nonlinear a model. Here we consider the nonlinear system 10. We will test with the resized initial conditions v 0, w 0 ε = εv 0, w 0, with ε {1, 0.1, 0.05}. 6 First in Figure 13 we consider the free dynamics for the nonlinear system 10. We observe that the norm of the solution blows up in finite time, for all initial conditions as in 6. Next we consider the corresponding systems under the action of the feedback control. In this case, in Figure 1, we can see that our feedback is not able to stabilize the system for the initial conditions 1, 1 and 0.1, 0.1. But, it is able to stabilize the system if we take the smaller initial condition 0.05, 0.05 which is consistent with our local stabilization result for the nonlinear system, as stated in Theorem Final Remarks.