Singular asymptotic expansion of the exact control for the perturbed wave equation

1. Introduction

Let Ω = ( 0 , 1 ) , Γ = { 1 } . For any T > 0 , we note Q T : = Ω × ( 0 , T ) , Γ T : = Γ × ( 0 , T ) and Σ T : = ∂ Ω × ( 0 , T ) . This work is concerned with the null controllability property with respect to the parameter ε > 0 of the following linear equation of Petrowsky type (1) y t t ε + ε y x x x x ε − y x x ε = 0 , in Q T , y ε ( 0 , · ) = y ε ( 1 , · ) = y x ε ( 0 , · ) = 0 , y x ε ( 1 , · ) = v ε , in ( 0 , T ) , ( y ε ( · , 0 ) , y t ε ( · , 0 ) ) = ( y 0 , y 1 ) , in Ω . Here, v ε is a control function in L 2 ( 0 , T ) . This system can be interpreted either as a fourth order singular perturbation of the wave equation or as a simplified model for the dynamic of a linear isotropic beam of length and density equal to one. In this case, ε stands for the product of the Young modulus with the area moment of inertia of the cross section (we refer to [23]). y ε = y ε ( x , t ) is the deflection of the beam at point x ∈ Ω and time t ∈ ( 0 , T ) . y 0 denotes the initial position and y 1 the initial velocity assumed in L 2 ( Ω ) and H − 2 ( Ω ) respectively.

For any ε > 0 , v ε ∈ L 2 ( Ω ) and ( y 0 , y 1 ) ∈ L 2 ( Ω ) × H − 2 ( Ω ) , there exists exactly one solution y ε to (1), with the regularity y ε ∈ C 0 ( [ 0 , T ] , L 2 ( Ω ) ) ∩ C 1 ( [ 0 , T ] , H − 2 ( Ω ) ) (see [17], Chapter 4) and the following estimate: ‖ y ε ‖ L ∞ ( 0 , T ; L 2 ( Ω ) ) + ‖ y t ε ‖ L ∞ ( 0 , T ; H − 2 ( Ω ) ) ⩽ c ε ( ‖ y 0 ‖ L 2 ( Ω ) + ‖ y 1 ‖ H − 2 ( Ω ) + ‖ v ε ‖ L 2 ( 0 , T ) ) for some constant c ε > 0 .

Accordingly, for any final time T > 0 , the associated null controllability problem at time T is the following: for each ( y 0 , y 1 ) ∈ L 2 ( Ω ) × H − 2 ( Ω ) , find a control function v ε ∈ L 2 ( 0 , T ) such that the corresponding solution to (1) satisfies (2) ( y ε ( · , T ) , y t ε ( · , T ) ) = ( 0 , 0 ) in L 2 ( Ω ) × H − 2 ( Ω ) . For any ε > 0 , existence of null controls is proved in [17] assuming that T > 2 based on the following observability inequality for some constant C > 0 independent of ε (3) ‖ φ 0 ε ‖ H 0 1 ( Ω ) 2 + ‖ φ 1 ε ‖ L 2 ( Ω ) 2 + ε ‖ φ 0 , x x ε ‖ L 2 ( Ω ) 2 ⩽ C ∫ 0 T ε | φ x x ε ( 1 , · ) | 2 , ∀ ( φ 0 ε , φ 1 ε ) ∈ H 0 2 ( Ω ) × L 2 ( Ω ) , where φ ε solves the corresponding homogeneous adjoint associated to the initial condition ( φ 0 ε , φ 1 ε ) , see (7). We refer to [6] for a recent exposition of the control theory for partial differential equations. Let us introduce the non-empty set of null controls C ( y 0 , y 1 , T , ε ) : = { ( y , v ) : v ∈ L 2 ( 0 , T ) ; y solves (1) and satisfies (2) } .

Since the perturbation parameter ε is small with respect to one, the issue of the asymptotic behavior of elements of C as ε is smaller and smaller arises naturally. It turns out that the system (1) is not uniformly controllable with respect to ε in the sense that the control of minimal L 2 ( 0 , T ) -norm is not uniformly bounded. The following result is proved in [17, Chapter 3], assuming additional regularity on the initial velocity. We also refer to [16].

Theorem 1.1 is obtained by using a priori estimates on the solutions and by taking the weak limit in the optimality system characterizing the control of minimal L 2 -norm. This argument holds in higher dimensions too but here we focus on the one-dimensional case for which we can give a detailed asymptotic expansion of the controls.

As mentioned in [17], this controllability problem is singular in the following two meanings: firstly, the convergence result (4) holds for ε v ε and not for v ε ; secondly, the Neumann null type control v ε for (1) degenerates as ε → 0 into a Dirichlet type control v for the limit problem (5). The control of minimal norm being of the form v ε = φ x x ε ( 1 , · ) on ( 0 , T ) , this singularity is related to the boundary layer of length ε which occurs on the adjoint solution φ ε in the neighborhood of the point x = 1 as ε tends to 0.

On the other hand, it is interesting to notice that the spectrum of the underlying operator satisfies a uniform gap property with respect to the parameter ε [24]. Therefore, inequality (3) can be also obtained using the classical approach based on reducing the observability inequality to a moment problem for the associated family of exponentials (we refer for instance to [20] for a recent application in a similar context). The ε term in the right-hand side of (3) comes from the boundary layer that affects to the boundary observability for the eigenfunctions.

Remark also that, without compatibility conditions at ∂ Ω × { 0 } between the initial and boundary conditions, the solution of (1) develops, as ε tends to zero, additional internal boundary layers along the characteristics of (5), namely C l = { ( x , t ) ∈ Q T , x − t = 0 } and C d = { ( x , t ) ∈ Q T , x + t − 1 = 0 } . This case would require a specific asymptotic analysis to capture this behavior and we do not consider it here. Thus we will add compatibility conditions to the initial data when required to avoid this situation

The main reason of this work is to perform an asymptotic analysis of the controls with respect to the parameter ε in order notably to precise the rate of convergence of the pair ( ε v ε , y ε ) . Such analysis is actually mentioned in the open problems section in [17, Chapter 3, Section 5.3]. Precisely, we want to characterize the terms v k , k ⩾ 0 , in the expansion (6) ε 1 / 2 v ε = v 0 + ε 1 / 2 v 1 + ε v 2 + ⋯ . Our study aims therefore to enrich the results obtained in [17, Chapter 3] by giving the precise behavior of the control for the fourth order partial differential equation when the parameter is small. A further motivation is the numerical approximation of v ε , which is a delicate issue when ε is small.

For any fixed function v ε , the asymptotic analysis of solutions of singular systems like (1) can be performed using the matched asymptotic method [10,13]. Precisely, under additional regularity and compatibility assumptions on the data ( y 0 , y 1 ) and on the function v ε , one may construct explicitly strong convergent approximations of y ε . We refer for instance to [1] for an advection–diffusion equation. When the exact controllability issue comes into play, such analysis requires more care, since the regularity/compatibility conditions mentioned above become additional constraints on the control set C . Typically, L 2 regularity for the control is in general not sufficient to get strong convergence results for the corresponding controlled solution, at any order. It is then necessary to enrich the set C and to modify the optimality system accordingly. This is probably the reason for which there exists in the literature only very few asymptotic analysis for controllability problem, a fortiori for singular partial differential equations. We mention [1,19] following [7]. We also mention the book [14] and the review [9] in the close context of optimal control problems.

Hopefully, as noticed in [8,11], a small modification of the set C , still in an L 2 -framework, allows to recover a posteriori smooth controls for smooth data. This property, which does not hold in the context of the advection–diffusion equation discussed in [1,7], is a crucial point for the analysis below. This modification also allows to impose simply appropriate compatibility conditions on ∂ Ω × ( 0 , T ) and therefore avoid the internal layers mentioned above along the characteristics C l and C d .

Our analysis relies on the matched asymptotic expansion which allows to construct explicitly an expansion of y ε and v ε . The control of minimal L 2 -norm v ε for (1) (precisely, in view of Theorem 1.1, the control which minimizes v → ‖ ε v ε ‖ L 2 ( 0 , T ) over C ( y 0 ; y 1 , ε , T ) ) is given by v ε : = φ x x ε where φ ε solves the homogeneous problem (7) φ t t ε + ε φ x x x x ε − φ x x ε = 0 , in Q T , φ ε = φ x ε = 0 , on Σ T , ( φ ε ( · , 0 ) , φ t ε ( · , 0 ) ) = ( φ 0 ε , φ 1 ε ) , in Ω , associated to the initial condition ( φ 0 ε , φ 1 ε ) , minimizer of the so-called conjugate functional J ⋆ ε : H 0 2 ( Ω ) × L 2 ( Ω ) → R defined by J ⋆ ε ( φ 0 ε , φ 1 ε ) = ε 2 ‖ φ x x ε ( 1 , · ) ‖ L 2 ( 0 , T ) 2 − ( y 0 , φ 1 ε ) L 2 ( Ω ) , L 2 ( Ω ) + ( y 1 , φ 0 ε ) H − 2 ( Ω ) , H 2 ( Ω ) . We clearly see here that the singular character of the control v ε is due to the boundary layer occurring at x = 1 on the quantity φ x x ε ( 1 , · ) . In particular, such singular character does not occur for a distributed control in Ω.

We also emphasize that the asymptotic analysis with respect to ε is also relevant from an approximation viewpoint. Precisely, an approximation of v ε is usually obtained through the minimization of the conjugate functional (see for instance [22] in a similar context). However, as ε goes to zero, the norm ‖ ε φ x x ε ‖ L 2 ( 0 , T ) is not anymore equivalent to ‖ φ 0 ε , φ 1 ε ‖ H 0 2 ( Ω ) × L 2 ( Ω ) so that the minimization of J ⋆ ε becomes ill-conditioned. On the contrary, the characterization of each term in the expansion (6) leads to well-conditioned extremal problems (at the successive orders of ε). This guarantees an accurate approximation of the control of minimal L 2 -norm v ε with respect to the approximation parameter, for arbitrarily small value of ε. This is illustrated below with some numerical experiments.

This document is organized as follows. In Section 2, we set the problem in a suitable functional framework and check that smooth controls can be achieved for smooth data, both for (1) and the limit wave equation. In Section 3, we use the matched asymptotic method and determine expansion of solutions of (1) and of the associated adjoint system (7). In Section 4, we substitute these expansions in the optimality system associated to the control of minimal L 2 -norm and fully characterize each term in the expansion (6). Section 5 justifies these expansions with convergence results with respect to ε. Eventually, we discuss some experiments in Section 6 and provide some perspectives in the concluding section.

2. Minimal L 2 -weighted controls

In this section, following [11], we introduce a class of controls for both system (1) and the limit one (5) which are smooth for smooth data. Note that, in general, this is not the case for controls with minimal L 2 -norm and the asymptotic analysis below cannot be justified for such controls. As described in [11], the problem with the control of minimal L 2 -norm is that there is no way to impose a priori the value of the control at time t = 0 . Therefore, the initial data at the extremity x = 1 and the control at time t = 0 may have different values and this will produce a discontinuity for the controlled solution that can possibly affect to the regularity of the control at larger time. The weight is chosen to ensure that the control is zero in a neighborhood of t = 0 and this allows to match the control with the initial condition at the point x = 1 .

We introduce a weight function with suitable properties that will be used below to define a class of smooth controls. For T > 2 , the minimal time to have controllability of system (1), we consider a positive smooth weight function η ⩾ 0 with compact support in ( 0 , T ) , i.e. η ∈ C 0 ∞ ( 0 , T ) , and such that η ( t ) > η 0 > 0 in a subinterval [ δ , T − δ ] ⊂ ( 0 , T ) with δ such that T − 2 δ > 2 .

We divide the rest of this section in two subsections where we consider separately the results for systems (1) and (5) respectively.