Internal Controllability of Systems of Semilinear Coupled One-Dimensional Wave Equations with One Control 

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From Volterra to Fredholm transformations

Volterra transformations of the second kind are one of the ingredients that made the new backstepping method so powerful on PDEs. Interestingly, the use of a Volterra transformation of the second kind also appears in the study of spectral assignability, which is the innite-dimensional analog of poleshifting. All in all, Volterra transformations of the second kind seem to work well with control systems. But on the other hand, as mentioned previously these transformations have a specic triangular structure. For some stabilization problems, this might be too constraining, and one could consider looking for a larger class of invertible transformations, along with suitable target systems, to use the backstepping method. And indeed, more recently, general kernel operators, also known as Fredholm transformations, have been considered: f(t; x) 7! Z L 0 k(x; y)f(t; y)dy: (1.183).
Even though they require more work, as one has to prove their invertibility from scratch, they have proven to be more suitable regarding the position of the control (see [66, 65] for example), and have also allowed to nd stabilizing boundary feedbacks for hyperbolic systems (see [62] for integro-dierential systems, and in [63] for general balance laws). A striking feature is that the Fredholm transformations in [66, 65] are actually still compact perturbations of the identity, although it is not clear whether they are actually Volterra transformations of the second kind. Let us also note that in [63] the Fredholm operator is sought in the form f(t; x) 7! f(t; x) 􀀀 Z L 0 K(x; y)f(t; y)dy.

From boundary control to distributed scalar inputs

Another signicant evolution is the extension of the backstepping method to systems with a distributed scalar input, which no longer bear any resemblance to nite-dimensional cascade systems: dY dt + AY = u(t)(x); (1.184).
where u is the control. Notwithstanding their diering nature, the backstepping method can still be applied on these systems. For example, by adding some additional assumptions on the controller (x), the authors of [150] and [154] for parabolic systems, and [159] for rst-order hyperbolic systems similar to the ones in Section 1.4.1, are able to apply a Volterra transformation of the second kind to their system, which moves some terms to the boundary but still leaves the input inside the domain.
Then, they apply a second invertible dierential transformation to their simpler target system to move the input to the boundary, which allows them to design an explicit feedback law.
As in the case of boundary control, on some systems the backstepping method seems to work better with Fredholm transformations. For example, in [59], the authors use a Fourier approach to nd a suitable backstepping transformation, in the form of a Fredholm transformation, to achieve rapid stabilization for the bilinear Schrodinger equation. In this case, and also in the results presented in Chapters 3 and 5, we will see that the Fredholm transformations that are found by the backstepping method are not compact perturbations of the identity anymore.

Target systems

As we have mentioned, the choice of a target system is an important ingredient in PDE backstepping, primarily because it ensures the convergence of the backstepping change of variables on the discretization of the system, and essentially because it encodes what destabilizing terms we want to remove, or what stabilizing terms we want to add. For example, in the works on the heat and wave equation ([109, 141, 17, 32]), antidamping terms are removed, and stability is enhanced by adding internal damping:ut 􀀀 u = 1u (1.185) becomes ut 􀀀 u = 􀀀2u (1.186)  where 1; 2 > 0.
In some cases, however, backstepping can achieve more. In [68] and in [14, 13] the authors derive a Volterra transformation of the second kind that moves the internal coupling terms to the boundary. Remarkably, in [68], this allows for a complete cancellation of the boundary input, and yields nitetime stabilization of the linearized system, with a minimal time due to the hyperbolic nature of the system.
Finally, in Chapter 5 we encounter another kind of target system. Indeed, we have found that adding boundary damping rather than internal damping seems to work better to nd a backstepping transformation.

Nonlinear systems

Another advantage of obtaining explicit feedbacks laws is that they can be used to stabilize nonlinear systems. This approach can take dierent forms, depending on the system under consideration. For example, in [39] the authors study a nonlinear KdV equation with a boundary control. Using the backstepping method, they rst compute explicit feedback laws for the linearized system for any given exponential decay rate. Then, they build a solution u to the nonlinear equation step by step, on the intervals [nT; (n + 1)T]. On each interval, they study the image of u by the backstepping transformation, and prove that it decays exponentially. Finally, by patching all the intervals together, they prove that the image of u by the backstepping transformation decays exponentially on R+. Thus, by invertibility of the backstepping transformation, u decays exponentially as well.
Another approach is given in [68]. The authors plug the linear feedback, which stabilizes the linearized system in nite time, into the nonlinear system, and prove the exponential stability of the
nonlinear closed-loop system using Lyapunov functions.
Regarding the systems studied in Chapters 3 and 5, as the feedbacks are explicit as well, one can
hope that the results for linear systems can be extended to nonlinear systems. However a new diculty arises: the feedback laws are not bounded for the state space norm. As a consequence the system obtained by plugging the linear feedback law into the nonlinear system is not that straightforward to study.

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Null-controllability and nite-time stabilization

As mentioned earlier, in [68] the structure of the system makes it possible to chose a target system
that converges to 0 in nite time. The same is achieved in [159], where the second transformation
actually maps the system to a hyperbolic system with zero input at the boundary.
Even when it seems dicult to aim for target systems with nite-time convergence, backstepping can help achieve nite-time stabilization. Indeed, a strategy has been developed in [67, 155, 156], using the explicit feedback laws obtained by the backstepping method. The general strategy is to divide the interval [0; T] in smaller intervals [tn; tn+1], the length of which tends to 0, and on which one gets exponential stabilization with decay rates n, with n ! 1, by applying feedbacks kn. Then, for well-chosen tn; n, the trajectory thus obtained reaches 0 in time T, with a piecewise H1, explicit, closed-loop control. For example, in [155] the author derives an explicit feedback law to stabilize a linearized KdV equation exponentially, with a Dirichlet control on the left boundary. This yields the following decay estimates for a given exponential decay rate , for the state y and the feedback u := k(y): ky(t)kL2 e4(1+L)2 p 􀀀tky(0)kL2 .

Backstepping in higher dimension

As far as we know, PDE backstepping has only been applied on 1-D systems, except for some parabolic systems on a parallelepiped, under some assumptions on the diusion and reaction coecients, in [100, 122], or extension to higher dimension of feedback laws elaborated in 1-D ([119]). Indeed the triangular (or spatially causal) structure of Volterra transformations of the second kind makes them dicult to dene on higher-dimensional domains, and geometrical constraints on the system seem necessary as they help reduce the problem to a collection of 1-D problems.
In contrast, our method relies on general kernel operators, and one could consider extending them to higher-dimensional domains, as long as the dierential operators involved have nice spectral properties. This would involve more complicated kernel equations, and the analog of the equations (1.199) would be PDEs instead of ODEs, for which Riesz basis properties could be more challenging to prove.

First case: the linearised system is controllable

As mentioned in section 2.1.3, we build on the method presented in [5]. One of the main ingredients
of this method is the theory of dierential operators, and the notion of algebraic solvability, which we brie y present in the subsection below. The use of algebraic solvability in the study of control systems rst appears in [47], where it was used to prove the stabilisability of nite dimensional systems without drift with time-varying feedbacks. It was rst used in the context of partial dierential equations in [64] for the control of the Navier-Stokes equation.
But rst let us give an informal explanation of our method in the case of a linear system: rst we have to rewrite the control problem using dierential operators. We note D the operator associated with the equation of our control problem. Then, the control problem, given initial and nal conditions, consists in nding (u; v) with those initial and nal conditions, and a control h such that D(u; v; h) = 0.

Table of contents :

1 Introduction 
1.1 A few general notions
1.2 From controllability to stabilization
1.3 Internal controllability of systems of semilinear coupled one-dimensional wave equations in 1-D with a single control
1.4 Stabilization of hyperbolic systems with a distributed scalar input
1.5 Conclusion and prospects
I Internal controllability of coupled wave equations 
2 Internal Controllability of Systems of Semilinear Coupled One-Dimensional Wave Equations with One Control 
2.1 Main results and outline of proof
2.2 First case: the linearised system is controllable
2.3 Second case: an example with an uncontrollable linearized system
2.4 Further questions
II Stabilization of 1-D linear hyperbolic systems with a scalar input 
3 Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback 
3.1 Introduction
3.2 Denition and properties of the transformation
3.3 Well-posedness and stability of the closed-loop system
3.4 Further remarks and questions
4 Finite-time internal stabilization of a linear 1-D transport equation 
4.1 Introduction
4.2 The exponentially stable semigroup
4.3 The limit semigroup
4.4 An explicit example
4.5 Comments and further questions
5 Exponential stabilization of the linearized water tank system 
5.1 Introduction
5.2 Properties of the system and presentation of the method
5.3 Dealing with mass conservation
5.4 Controllability
5.5 A heuristic construction
5.6 Backstepping transformation and feedback law
Appendix 5.A Proof of Proposition 5.2.1
Appendix 5.B Proof of Proposition 5.6.1
Appendix 5.C Expression of the feedback coecients before and after variable changes 


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