Control of the 1-d wave equation from an interior moving point

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–7) Control of the 1-d wave equation from an interior moving point C. Castro1 1Dpto. Matem´atica e Inform´atica, ETSI Caminos, Canales y Puertos, Universidad Polit´ecnica de Madrid, E-28040 Madrid. E-mail: [email protected]. Palabras clave: Exact controllability, wave equation, pointwise control Resumen We consider the one-dimensional linear wave equation with Dirichlet boundary conditions in a bounded interval, and with a control acting on a single point which moves following a regular trajectory in time. We analyze the exact controllability problem. 1. Introduction We consider the one-dimensional linear wave equation, on a finite interval domain (0, L), with an interior control fwhich acts on a single moving (in time) point x=γ(t),    utt −uxx =f(t)δγ(t)(x),in 0 < x < L, 0< t < T, u(0, t) = u(L, t) = 0,in 0 < t < T, u(x, 0) = u0(x), ut(x, 0) = u1(x) in 0 < x < L. (1) Here, δγ(t)represents the Dirac measure on x=γ(t) and the function γdescribes the trajectory in time of the location of the control. We assume that the function γ: [0, T]→ (0, L) belongs to the class γ∈C1([0, T ]). We are interested in the following exact controllability problem: Given T > 0, some initial data (u0, u1)and final data (v0, v1),find a control fsuch that the solution uof (1) satisfies u(x, T) = v0(x), ut(x, T ) = v1(x),∀x∈(0, L).(2) Let us briefly describe some related results and the main motivation of this problem. When the control acts in an interior open set ωor one of the extremes of the domain (0, L), it is well known that the corresponding exact controllability property of the wave 1 C. Castro equation holds, for some sufficiently large time T(see [9]). On the other hand, in most practical situations, the support of the control is required to be very small compared to the total size of the domain (0, L) and therefore it is very natural to consider a limit situation where the subinterval ωis reduced to a single point γ∈(0, L). It turns out that the controllability property of system (1) depends on the location of γ. Indeed, it can be shown that this property holds if and only if the only eigenfunction of the Laplacian with homogeneous Dirichlet boundary conditions and vanishing on x=γis the identically zero one (see [10], [11], [1] or the more recent reference [5], for example). In the sequel, the points γfor which this spectral property is satisfied will be referred to as strategic points. The property of γbeing strategic is difficult to establish in practice since it is extremely unstable. In fact γ∈(0, L) is strategic if and only if it is irrational with respect to the length of the interval L. Consequently, controllability properties over points are hard to use in practice. To overcome this difficulty one may consider controls supported on moving points {γ(t)}0≤t≤T, as suggested in [11]. The main advantage of moving controls is that it is easy to construct trajectories {γ(t)}0≤t≤Tfor which the strategic property holds for γ(t)∈(0, L) a.e. in t∈[0, T]. For example, this is the case when we assume that the control is located at a point that moves in time with constant velocity. In this case, γ(t) is irrational, and therefore strategic, a.e. in t∈[0, T ]. Therefore, the exact controllability is likely to hold for such moving controls. The aim of this work is to show that this is indeed the case under suitable conditions on the function γ(t). It is worth noting that in the context of parabolic equations a similar situation appears. We refer to [3], [7], [1], [11] and the references therein for a detailed analysis of this related problem. The rest of this paper is divided as follows: in section 2 we state the main results, namely the existence of solutions for system (1) in suitable functional spaces and the exact controllability property. Both results can be reduced, by classical duality arguments, to some suitable regularity and observability estimates for the uncontrolled wave equation respectively. In section 3 we give the proof of these estimates. 2. Main results We assume that the function γ: [0, T]→(0, L) belongs to C1and satisfies the following hypothesis: There exist constants c1, c2such that 0< c1<|γ0(t)| ≤ c2<1 for all t∈(0, T ).(3) The control f(t) in (1) is assumed to belong to H−1(0, T ) and the initial data (u0, u1) in the class (u0, u1)∈L2×H−1(0, L). We define the weak solutions of system (1) by transposition (see [8]). To do that let ψ∈L1(0, T;L2(0, L)) be a function and consider the non-homogeneous adjoint wave equation    ϕtt −ϕxx =ψ(x, t) in 0 < x < L, 0 < t < T, ϕ(0, t) = ϕ(L, t) = 0 in 0 < t < T, ϕ(x, T) = ϕt(x, T ) = 0,in 0 < x < L, 0 < t < T. (4) 2 Pointwise control of the 1-d wave equation It is well known that system (4) admits a unique solution ϕof (4) in the class ϕ∈C([0, T]; H1 0(0, L)) ∩C1([0, T]; L2(0, L)).(5) Multiplying the equations in (1) by ϕand integrating by parts we easily obtain, at least formally, the following identity: ZL 0 < u1(x), ϕ(x, 0) >1dx −ZL 0 u0(x)ϕt(x, 0) dx+< f, ϕ(γ(t), t)>t 1dt =ZT 0ZL 0 ψ(x, t)u(x, t)dx dt, for all ψ∈L1(0, T ;L2(0, L)),(6) where <·,·>1and <·,·>t 1denote the duality products between H1 0(0, L) and its dual, and between H1 0(0, T ) and its dual respectively. We adopt identity (6) as the definition of solutions of (1), in the sense of transposition. The following result establishes the existence of solutions for system (1). Theorem 2.1 Assume that γ: [0, T ]→(0, L)is in the class γ∈C1([0, T ]) and satisfies the hypothesis (3). Given any initial data (u0, u1)∈L2×H−1(0, L)and f∈H−1(0, T ), there exists an unique solution uof (1), in the sense of transposition, in the class u∈C([0, T]; L2(0, L)) ∩C1([0, T ]; H−1(0, L)). Moreover, there exists a one-to-one correspondence between the data and the solution in the given spaces. Concerning the exact controllability problem of system (1) the following result holds: Theorem 2.2 Let T > 2Land γ: [0, T]→(0, L)be a function in the class γ∈C1([0, T ]) satisfying the hypothesis (3). Then, system (1) is exactly controllable, i.e. for any initial data (u0, u1)∈L2×H−1(0, L)and final data (v0, v1)∈L2×H−1(0, L),there exists a control f∈H−1(0, T )such that the solution uof (1) satisfies (2). The proof of the existence result above (Theorem 2.1) can be obtained from a suitable regularity property stated below (estimate (8)) by a straightforward duality argument. We refer to [8] for a general description, and [6] or [2] where this is done for very similar problems. The proof of the exact controllability property (Theorem 2.2) is also a straightforward consequence of the observability inequality (9) below and the Hilbert Uniqueness Method introduced by J.-L. Lions in [9]. We also refer to [6] and [2] where this method is applied for similar problems. 3. Observability As we have said, the main results in this paper can be obtained from some inequalities for the uncontrolled wave equation. In this section we prove these inequalities. 3 C. Castro Consider the system    ϕtt −ϕxx = 0,in 0 < x < L, 0< t < T, ϕ(0, t) = ϕ(L, t) = 0,in 0 < t < T, ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x),in 0 < x < L. (7) We assume that (ϕ0, ϕ1)∈H1 0×L2(0, L). The following holds: Proposition 3.1 Assume that γ∈C1satisfies the hypothesis (3). Then, there exists a constant c(γ)>0such that the solution ϕof (7) satisfies ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt ≤c(γ)° °(ϕ0, ϕ1)° ° 2 H1 0×L2.(8) Moreover, if T > 2L, then there exists a constant C(γ)>0such that ° °(ϕ0, ϕ1)° ° 2 H1 0×L2≤C(γ)ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt. (9) Remark 3.1 Estimate (8) is a regularity result for the trace of the solution of the wave equation ϕ(x, t)on the curve defined by the trajectory γ. This result cannot be obtained from classical arguments or semigroup theory. Estimate (9) is an observability inequality which establishes that the total energy of the solutions of the wave equation can be estimated from the value of the solution ϕat γ(t) for a large enough time interval t∈(0, T ). Proof. Note that it is enough to consider smooth solutions since for other solutions we can argue by density. We first prove the estimate (8). We observe that in the one-dimensional wave equation one can change the variables x by tand tby xwithout altering the equation. Thus, D’Alambert formula can be used to obtain the solution ϕ(x, t) in terms of the solution at one extreme, say x= 0,instead of the data at t= 0 as usual. Indeed, we have ϕ(x, t) = 1 2[ϕ(0, t −x) + ϕ(0, t +x)] + 1 2Zt+x t−x ϕx(0, s)ds. (10) If ϕis defined on (x, t)∈[0, L]×[0, T] this formula holds only for those values (x, t) for which 0 ≤t−x≤t+x≤T. However, we can extend the solution of the wave equation ϕto (x, t)∈(0, L)×(−∞,∞) and formula (10) is still valid for the whole domain (x, t)∈(0, L)×(0, T). This is always posible because the wave equation with Cauchy data at t= 0 is well-posed for t≥0 and t≤0. In particular, taking into account the homogeneous Dirichlet boundary conditions in (7) we have ϕ(γ(t), t) = 1 2Zt+γ(t) t−γ(t) ϕx(0, s)ds. Therefore, 2d dtϕ(γ(t), t) = (1 + γ0(t))ϕx(0, t +γ(t)) −(1 −γ0(t))ϕx(0, t −γ(t)),(11) 4 Pointwise control of the 1-d wave equation and 4ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt ≤2ZT 0 (1 + γ0(t))2|ϕx(0, t +γ(t))|2dt + 2 ZT 0 (1 −γ0(t))2|ϕx(0, t −γ(t))|2dt ≤2 sup t∈[0,T ] (1 + γ0(t))2ZT+γ(T) γ(0) |ϕx(0, s)|2ds + 2 sup t∈[0,T ] (1 −γ0(t))2ZT−γ(T) −γ(0) |ϕx(0, s)|2ds ≤4ZT+γ(T) γ(0) |ϕx(0, s)|2ds + 4 ZT−γ(T) −γ(0) |ϕx(0, s)|2ds ≤8ZT+γ(T) −γ(0) |ϕx(0, s)|2ds ≤C° °(ϕ0, ϕ1)° ° 2 H1 0×L2(0,L). Here, the last inequality can be obtained by classical multipliers techniques (see [9]). Now, we prove the estimate (9). We divide the analysis in two cases depending on the sign of γ0. Case A: Assume that −∞ <−c2≤γ0(t)≤ −c1<0 for all t∈(0, T ).From identity (11), we can estimate |ϕx(0, t −γ(t))|as follows: |ϕx(0, t −γ(t))|2 =µ1 + γ0(t) 1−γ0(t)ϕx(0, t +γ(t)) −2 1−γ0(t) d dtϕ(γ(t), t)¶2 =µ1 + γ0(t) 1−γ0(t)¶2 |ϕx(0, t +γ(t))|2+µ2 1−γ0(t)¶2¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 −21 + γ0(t) 1−γ0(t)ϕx(0, t +γ(t)) 2 1−γ0(t) d dtϕ(γ(t), t) ≤(1 + a)µ1 + γ0(t) 1−γ0(t)¶2 |ϕx(0, t +γ(t))|2+µ1 + 1 a¶µ 2 1−γ0(t)¶2¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 for any a > 0 to be chosen later. Here we have used Young’s inequality. Multiplying by 1 −γ0(t) and integrating in t∈(0, T ) we obtain, ZT 0 |ϕx(0, t −γ(t))|2¡1−γ0(t)¢dt ≤(1 + a)ZT 0 1 + γ0(t) 1−γ0(t)|ϕx(0, t +γ(t))|2¡1 + γ0(t)¢dt +µ1 + 1 a¶ZT 0 4 1−γ0(t)¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt ≤(1 + a)1−c1 1 + c1ZT+γ(T) γ(0) |ϕx(0, s)|2dt +µ1 + 1 a¶4 1 + c1ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt. (12) 5 C. Castro Therefore, ZT−γ(T) −γ(0) |ϕx(0, t)|2dt −(1 + a)1−c1 1 + c1ZT+γ(T) γ(0) |ϕx(0, t)|2dt ≤µ1 + 1 a¶4 1 + c1ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt. (13) Now we take the constants T0, a such that T0−γ(T0) + γ(0) = 2Land 0 <a<1+c1 1−c1−1 respectively. Then, from the 2L-periodicity of the solutions of the wave equation (7) and the fact that γis decreasing we can estimate the left hand side of (13) with T=T0as follows ZT0−γ(T0) −γ(0) |ϕx(0, t)|2dt −(1 + a)1−c1 1 + c1ZT0+γ(T0) γ(0) |ϕx(0, t)|2dt ≥µ1−(1 + a)1−c1 1 + c1¶Z2L 0 |ϕx(0, t)|2dt. (14) Combining this last inequality with (13) we obtain that there exist constant C > 0 such that ZT0 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt ≥CZ2L 0 |ϕx(0, t)|2dt ≥C0k(ϕ0, ϕ1)kH1 0×L2(0,L), where the last inequality is the classical boundary observability inequality for the onedimensional wave equation (see, for example, [4]). Finally, inequality (9) follows for any T > T0and, in particular, for T > 2L. Case B: Assume now that 0 < c1< γ0(t)< c2<1 for all t∈(0, T ).From identity (11), we estimate now |ϕx(0, t +γ(t))|as follows, |ϕx(0, t +γ(t))|2 =µ1−γ0(t) 1 + γ0(t)ϕx(0, t −γ(t)) −2 1 + γ0(t) d dtϕ(γ(t), t)¶2 =µ1−γ0(t) 1 + γ0(t)¶2 |ϕx(0, t −γ(t))|2+µ2 1 + γ0(t)¶2¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 −21−γ0(t) 1 + γ0(t)ϕx(0, t −γ(t)) 2 1 + γ0(t) d dtϕ(γ(t), t) ≤(1 + a)µ1−γ0(t) 1 + γ0(t)¶2 |ϕx(0, t −γ(t))|2+µ1 + 1 a¶µ 2 1 + γ0(t)¶2¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 for any a > 0 to be chosen later. Multiplying by 1 + γ0(t) and integrating in t∈(0, T ) we obtain, ZT 0 |ϕx(0, t +γ(t))|2¡1 + γ0(t)¢dt ≤(1 + a)ZT 0 1−γ0(t) 1 + γ0(t)|ϕx(0, t −γ(t))|2¡1−γ0(t)¢dt +µ1 + 1 a¶ZT 0 4 1 + γ0(t)¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt. (15) 6 Pointwise control of the 1-d wave equation Therefore, ZT+γ(T) γ(0) |ϕx(0, t)|2dt −(1 + a)1−c1 1 + c1ZT−γ(T) −γ(0) |ϕx(0, t)|2dt ≤µ1 + 1 a¶4 1 + c1ZT 0¯¯¯¯ d dtϕ(γ(t), t)¯¯¯¯ 2 dt. Now we take the constants T0, a such that T0−γ(T0) + γ(0) = 2Land 0 <a<1+c1 1−c1−1 respectively. Then, from the 2L-periodicity of the solutions of the wave equation (7) and the fact that γis increasing we can estimate the left hand side of (13) with T=T0as in (14). Then, we can argue as in the previous case. This concludes the proof. Agradecimientos Supported by the Grant MTM2005-00714 of the MEC (Spain), the DOMINO Project CIT-370200-2005-10 in the PROFIT program of the MEC, the SIMUMAT project of the CAM. Referencias [1] S. Avdonin and S. Ivanov, Families of exponentials: The method of moments in controllability problems for distributed paramenter systems, Cambridge University Press, 1995. [2] C. Castro, Boundary controllability of the one-dimensional wave equation with rapidly oscillating density. Asymptotic Analysis, 20, 317-350, 1999. [3] C. Castro y E. Zuazua, Unique continuation and control for the heat equation from a lower dimensional manifold, SIAM J. Cont. Optim, 42 (4), 1400-1434. [4] S. Cox and E. 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