George Avalos † and Irena Lasiecka ∗
1 STATEMENT OF THE PROBLEM AND MAIN RESULTS 1.1 Description of the Problem
Exact Boundary Controllability of a Hybrid PDE System Arising in
Structural Acoustic Modeling
George Avalos
†and Irena Lasiecka
∗†Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, NE
∗Department of Mathematics, University of Virginia, Charlottesville, VA
In this chapter, we consider the active control of a pair of wave equations, each defined on different geometries: one wave equation holds on the interior of a bounded domainΩ; the other wave equation is satisified on a portion of the boundary Γ0 of∂Ω. The respective wave equations are coupled by trace terms on the boundary interfaceΓ0. For this coupled system of equations, we present results of exact controllability in the case that the con-trollers are exerted strictly on the boundary ∂Ω. In particular, we give precise geometric conditions under which control on the “active portion”Γ0only gives exact controllability for the dynamics ofboth wave equations, the interior as well as the boundary wave.
1 STATEMENT OF THE PROBLEM AND MAIN RESULTS 1.1 Description of the Problem
We shall consider here a hybrid partial differential equation (PDE) system that models acoustic pressure in a cavity — taken to be ann-dimensional entity; this cavity in turn is interacting with another environment (manifold), taken to be a wall of dimension n − 1.
The coupling between the structures is accomplished through the back pressure exerted on the wall. These kinds of systems are typical in structural acoustic modeling (see, e.g., [8], [24]). In this chapter, we shall consider a canonical model, in which the hybrid consists of two disparate wave equations that are coupled at the interface. This interface consists of the boundary of the region (cavity). We are thus dealing here with interactions of two hyperbolic PDEs, but other configurations have also been considered in the literature; e.g., a coupling of a parabolic PDE to a hyperbolic PDE, where the hyperbolic component is a wave equation satisfied within the cavity, but with the wall (structure) satisfied by a beam (rather than a wave) equation under the influence of various degrees of structural damping (see [1]).
Our goal here is to studyexact controllability properties, with a (natural) emphasis being on boundary control with respect to initial and terminal data in physically meaningful finit energy states. (However, there is generally no guarantee that during the transit time
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156 G. Avalos and I. Lasiecka
from the initial to the terminal data, the corresponding state will remain in the space of finite energy, that is the underlying control operator for our control operator is not “admissible.”
In part, this is a consequence of the fact that the Lopatinski conditions are not satisfied for the wave equation with Neumann boundary conditions (see [15]).
In order to clearly convey our main ideas and the technical difficulties associated with the exact controllability of PDEs that govern interactive structures, we have chosen to analyze here a relatively simple-looking model; namely, a wave/wave model that displays a variety of the phenonema intrinsic to coupled PDE structures in general. Moreover, this canonical PDE model provides a comfortable setting from which to study the delicate issue of “propagating” controllability properties from one PDE component to the other.
As we shall see below, the geometry of the region plays a critical role in our analy-sis; and it will turn out that some geometries can be particularly exploited for the sake of controllability. Indeed, our results lead to the conclusion that there are favorable geometric configurations that should be kept in mind when one wishes to design acoustic chambers (such as the cabin of a helicopter) that are amenable to active controllers distributed on a portion of the chamber walls.
Other, higher-order equations (of the fourth order on the interface), can and have been considered by the authors as well. See [1], [2], and [5] for an account of these results.
1.2 The PDE Model
LetΩ be a bounded open subset of Rn,n ≥ 2, with Lipschitz boundary ∂Ω = Γ0∪ Γ1, with eachΓinonempty, andΓ0∩ Γ1= ∅. There will eventually be additional assumptions imposed on theΓi(see (A1) and (A2) below). For this geometry in place, we shall study the solutions[z(t, x), v(t, τ)] to the following (controlled) PDE model:
This PDE is coupled to the following on the interfaceΓ0: (1)
As usual, ∂τ∂ denotes the (unit) tangential, and ∂ν∂ the outward normal derivative with respect to Γ. ∂n∂ is here the unit exterior normal derivative with respect to the n − 1 manifold∂Γ0.
To account for the initial data of the problem, we define the spaces,
It can be readily shown thatH is a Hilbert space, with the inner product
With this definition of the inner product, one can then proceed to use the Lumer–Phillips Theorem to show the existence of aC0-group
eAt
t≥0associated with this coupled sys-tem of wave equations. Accordingly, with regard to the PDE (1), a straightforward con-sequence of these dynamics
eAt
t≥0 (see, e.g., [1], [25]) yields the continuity of the mapping
{[ z0, v0] ∈ H, u1= 0, u0= 0} ⇒ [ z, v] ∈ C([0, T ]; H).
In short, the uncontrolled problem (i.e., the PDE (1) withui= 0) is well posed in the basic space of energyH. Thus, the well-posedness of the system is not an issue here; the goal of our chapter is rather to determine exact controllability properties1of (1) with boundary controlsu1,u0taken in prescribed spaces of controlsU1, U0, say, under the imposition of specific conditions upon the geometryΩ.
1.3 The Main Results
Our first result in Theorem 1 below states thatall finit energy states are controlled exactly with controls located on Γ0alone (with these controls acting only on the v-component).
This result does require, however, that the geometry be “appropriate” to the situation; viz., the domainΩ is convex and the “roof” of the acoustic chamber is not too “deep.” (See As-sumption (A1) and Fig. 1.) In addition, the controlu0must be of the appropriate topological strength; i.e.,u0 ∈ [H1(0, T ; L2(Γ0))]. So for our first result (Theorem 1(a) below), we assume the following:
Assumption (A1) Assume thatΩ is a bounded subset of Rn, with boundaryΓ = Γ0∪ Γ1, Γ0∩ Γ1= ∅, with Γ0being flat. Moreover assume the following:
1The classical definition of exact controllability is intended here. Namely, the PDE (1) is exactly controllable if there is aT∗> 0 such that for terminal time T > T∗, one has the following reachability property: for all initial data[z0, v0] ∈ H and preassigned target data [zT, vT] ∈ H, there exist control functions [u1, u0] ∈ U1× U0
(to be specified), such that at terminal timeT the corresponding solution [z, v] to (1) satisfies [z(T ), v(T )] = [zT, vT].
158 G. Avalos and I. Lasiecka
Figure 1 A triple{Ω, Γ0, Γ1} that satisfies Assumption (A1).
(i)Ω is convex; (ii) there exists a point x0∈ R2such that (x − x0) · ν ≤ 0 for all x ∈ Γ1.
The special vector field that is available, in case that Assumption (A1) holds true — constructed in [19] and denoted below ash in (13) — will be used in the derivation of the observability inequality associated with exact controllability (see (3) below). In particular, this specialh appears in the wave multipliers classically used to estimate the energy of the z-wave equation (see, e.g., [13], [14], [23], [27], [28]). The behavior of h on the inactive portion of the boundary i.e., h · ν|Γ1 = 0, is a key driver in our first result. With control on Γ0 only, and under Assumption (A1), the PDE (1) is exactly controllable onH. A discussion concerning the possible configuration of those triples {Ω, Γ0, Γ1} that satisfy Assumption (A1) is given in Appendix C in [19]. A canonical example of such a triple is given in Fig. 1.
On the other hand, if the geometry of the acoustic chamber is unrestricted (see As-sumption (A2)), then from our previous discussion, we clearly must have additional con-trol onΓ1. In this case, the second part of Theorem 1 states that all finit energy states are controlled exactly with controls u0 ∈ L2(0, T ; H−14(Γ0)) (located on Γ0) and u1 ∈ L2(0, T ; L2(Γ1)) (located on the roof of a chamber of arbitrary geometrical configu a-tion). So in our second result (Theorem 1(b) below), we make the following assumption:
Assumption (A2) Assume thatΩ is either a bounded subset of Rnwith smooth boundary Γ, or else Ω is a parallelepiped. Moreover, assume boundary Γ = Γ0∪ Γ1, whereΓ0 is flat. No assumptions are made onΓ1(see Fig. 2).
If Assumption (A2) holds true, then one has exact controllability of (1) for arbitrary initial data of finite energy, with the control region taken to beΓ0∪Γ1. The point of making the Assumption (A2) is that in this case, one can take a radial vector fieldh to assist in the multiplier method to be employed to estimate the (acoustic) wave energy. However, since Assumption (A2) is much less restrictive than (A1) — in particular, no impositions are made on the hard wallsΓ1 — the correspondingh cannot be expected to help with the high-order terms on Γ1, and hence the need for control on the hard walls. A common feature in both of our main results is the critical use of the “sharp” regularity theory, which has been developed to handle the tangential derivatives (on the boundary) of solutions to wave equations (see [17] and Lemma 4) below).
With these assumptions, we now state our main results concisely.
Figure 2 A triple{Ω, Γ0, Γ1} that satisfies Assumption (A2).
Theorem 1 (a) Let Assumption (A1) stand, and set U1 = {0} ; U0 =
H1
0, T ; L2(Γ0) .
Then for long enough terminal time T , the problem (1) is exactly controllable on H within the class of U1× U0-controls.
(b) Let Assumption (A2) stand, and set
U1 = L2(0, T ; L2(Γ1)) U0 = L2(0, T ; H−14(Γ0)).
Then likewise for long enough terminal time T , the problem (1) is exactly controllable on H within the class of U1× U0-controls.
Remark 2 The precise specificatio of the controllability time is T > 2diam(Ω). This is on account of the terminal time needed for Holmgren’s Uniqueness Theorem to hold true. This theorem is needed at the level of a “compactness/uniqueness” argument so as to eliminate lower-order terms that corrupt the associated observability estimate (see (35) and (37)).
Remark 3 The specificatio here of natural boundary conditions for the v-component in (1) is not critical in the derivation of our observability results. In fact, one could obtain a similar exact controllability result for (1), with instead v|Γ0 = 0 on ∂Γ0.
1.4 About the Problem and the Literature
The PDE system (1) and other classes of PDE models seen in the literature and that are associated with the mathematical descriptions of stuctural acoustic flow, are principally characterized as a coupling of distinct dynamics (be it of hyperbolic/hyperbolic or hyper-bolic/parabolic type), with the coupling being accomplished across boundary interfaces (see [1], [2], [6], [8], [9], [20], [24]). In the present case, we have a chamberΩ, with hard, rigid wallsΓ1and its flexible wall portionΓ0. Moreover, the flow within this chamberΩ is assumed to be of the acoustic wave type. Given this assumption of acoustic-type flow,
160 G. Avalos and I. Lasiecka
an interior wave equation (in z) consequently appears in the governing of the PDE sys-tem, this wave equation being under the influence of Neumann forcing datavt. In turn, the acoustic wave is coupled to the boundary wave equation (inv) on Γ0 (this boundary wave is to account for the dynamics involved in the flexibleΓ0). The coupling mechanism between the two dynamics is brought about through the boundary traces of the respective wave velocities.
In accordance with the classical notion of exact controllability for PDEs, we shall concern ourselves here with deriving reachability properties with respect to initial and ter-minal data in thefinit energy space H, as defined in (2). Moreover in line with the applied problem addressed in the literature, a principal objective is the attainment of some state-ment that assures the exact controllability of the PDE system, in the case that control is exerted on the active wallΓ0only (that is,Γ1is to be inactive). However, this objective, which is really an accommodation to the needs of the given physical problem, is at odds with the geometric conditions necessary for exact controllability, and which are prescribed in [7]. In fact, in the structural acoustic control problem, as stated in [6] and [1], the active control region Γ0 comprises one side of the chamber wall only. In consequence, for an arbitrary triple {Ω, Γ0, Γ1}, one will generally not have exact controllability with control implemented solely onΓ0, as the necessary conditions of geometric optics will not be sat-isfied. In particular, it is now known that in order to control thez-wave equation from the boundary, it is necessary that the support of the control region be sufficiently large (see [7]). Thus prescribing control only onΓ0will generally be insufficient, unlessΓ0is large relative toΩ.
Another (topological) complication presented here is a direct consequence of the cou-pling involved between the two dynamics, i.e., the acoustic interaction (1) presents a sit-uation where the component z on Ω, as the solution of the wave equation with forcing boundary data vt, is subject to the “smoothing effects” due to the (L2) regularity of the Neumann boundary datavt(see, e.g., [16], [18]). As such, it will have a certain measure of regularity. Therefore, a sole controlu0— that isu1= 0 in (1) — acting strictly on the wave component v, in the absence of geometric assumptions, will generally not be strong enough to drive the acoustic variable z to an arbitrary state of finite energy.
To the best of our knowledge, Theorem 1(a) and (b) constitutes the first exact control-lability results for structural acoustic interactions in finite energy spaces, and with general spatial domainsΩ. All other results (see [22] and references therein) pertain to controllabil-ity on specified subspaces of finite energy — such as those described by the asymptotic be-havior of Fourier coefficients — and moreover these results are proved for very special ge-ometries only — a 2D rectangle — with very large classes of controls:H−2(0, T ; L2(Γ0)).
From the mathematical point of view, the key ingredients in our proofs are the fol-lowing:
(i) Sharp trace regularity for the wave equation in the absence of Lopatinski conditions (a distinguishing feature of the Neumann case); see Lemma 5.
(ii) Microlocal analytical estimates that allow the absorption of tangential (wave) traces by time derivatives on the boundary; see Lemma 4.
(iii) A recent result in [19] concerning Carleman’s estimates for the wave equation with the controlled Neumann part of the boundary. These estimates lead to the aforemen-tioned special vector fieldh that allows us, in this chapter, to handle the uncontrolled portion of the boundary so as to derive the requisite observability estimates.