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Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach
Abstract
We prove the null controllability of a linearized Korteweg–de Vries equation with a Dirichlet control on the left boundary. Instead of considering classical methods, i.e., Carleman estimates, the moment method, etc., we use a backstepping approach, which is a method usually used to handle stabilization problems.
... People have carried out a lot of research on the KdV equation, see [1][2] and reference therein. For changing the state of KdV for satisfying our purpose, its control problem is introduced, for example, kinds of stability and controllability results for KdV can be found in [3][4][5][6][7] and reference therein. ...
... , | ∈ 0,1 , ∈ , 2 , then, , and satisfies Next, we will prove the existence of kernel function , and . Then, we know , satisfying an integral equation For any ∈ , we define the recursive formula as Next, we will adopt the method in [5] to prove the existence of , . Let ℙ be the space of polynomials of one variable on , we define operator by ...
... We write the operator as the sum of seven linear operators for simplifying the estimate. The definitions of the first six operators are the same as those in [5] as which shows the existence of , . With the same approach in [5], , ∈ is proved. ...
This paper considers the exponential stabilization of coupled ordinary differential equation (ODE)-linearized Korteweg-de Vries (KdV) equation system coupled at right boundary point with left boundary control. Firstly, we transfer the original system into an exponentially stable target system by backstepping transformation. Secondly, we show the existence of the kernels in forward and backward transformation. Finally, we prove the exponential stability of the closed-loop system.
... This was used to prove a host of results on the boundary stabilization of partial dierential equations: let us cite for example [24] and [39] for the wave equation, [46,47] for the Korteweg-de Vries equation, [2]*Chapter 7 for an application to rstorder hyperbolic systems, and also [17], which combines the backstepping method with Lyapunov functions to prove nite-time stabilization in H 2 for a quasilinear 2 × 2 hyperbolic system. ...
... where F = (F n ) n∈Z ∈ E will be characterized below. The regularity comes from the denition of the convolution product, (14), (49) and (46), and one can check, using (47), that k n,λ is a solution of (45). ...
... and nally, by integration by parts in the rst term, (46), and adding α, F λ T λ ϕ (N ) ...
We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \begin{document} (0,L) \end{document}, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than \begin{document} 1 \end{document}, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.
... Because we proved the uniqueness of solutions in the previous section, this constructed solution is the unique solution. The main difficulty is to give estimates on this solution, which is done in the paper [Xia19]. ...
... As the backstepping method has already been used for a rapid stabilization for this KdV system, one may naturally expect the smalltime stabilization. In the recent paper [Xia19], the author used this technique to give a new proof of null controllability of the linearized KdV equation. However, when one considers the stabilization problems, there came a difficulty of lacking regularity on the control (feedback) term y(t, 0). ...
... Here, we are not going to reconstruct the whole theory of transposition solutions, which is already well explained in the book [Cor07a] (one can also see similar cases in [CN17,CRX17]). Based on the method introduced in [CN17] and the estimates given in [Xia19], we are able to stabilize system (4.1.5)-(4.1.8) in small time. ...
This thesis is devoted to the study of stabilization of partial differential equations by nonlinear feedbacks. We are interested in the cases where classical linearization and stationary feedback law do not work for stabilization problems, for example KdV equations and Burgers equations. More precisely, it includes three important cases : stabilization of nonlinear systems whose linearized systems are not asymptotically stabilizable ; small-time local stabilization of linear controllable systems ; small-time global stabilization of nonlinear controllable systems. We find a strategy for the small-time global stabilization of the viscous Burgers equation : small-time global approximate stabilization and small-time local stabilization. Moreover, using a quadratic structure, we prove that the KdV system is exponentially stabilizable even in the case of critical lengths.
... for some constants C 1 , C 2 > 0. Thus, by looking for an invertible transformation under a certain form, degrees of freedom are resolved and the problem of stabilization becomes one of solving the PDE satisfied by the kernel of the Volterra transformation. Since this seminal work, this backstepping strategy has been applied to many different PDE systems, obtaining exponential or rapid stabilization for the wave equation ( [109] and [141]), for the Korteweg-de Vries equation ( [39], more recently [155,156]). A general presentation for an application to first-order hyperbolic systems can be found in [24, chapter 7]. ...
... Even when it seems difficult to aim for target systems with finite-time convergence, backstepping can help achieve finite-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 [t n , t n+1 ], the length of which tends to 0, and on which one gets exponential stabilization with decay rates λ n , with λ n → ∞, by applying feedbacks k λn . ...
... Then, for well-chosen t n , λ n , the trajectory thus obtained reaches 0 in time T , with a piecewise H 1 , 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 λ, for the state y and the feedback u := k λ (y): ...
In this thesis we study controllability and stabilization questions for some hyperbolic systems in one space dimension, with an internal control. The first question we study is the indirect internal controllability of a system of two coupled semilinear wave equations, the control being a function of time and space. Using the so-called fictitious control method, we give sufficient conditions for such a system to be locally controllable around 0, and a natural condition linking the minimal control time to the support of the control. Then, we study a particular case where the aforementioned sufficient conditions are not satisfied, applying the return method.The second question in this thesis is the design of explicit scalar feedbacks to stabilize controllable systems. The method we use draws from the backstepping method for PDEs elaborated by Miroslav Krstic, and its most recent developments: thus, the controllability of the system under consideration plays a crucial role. The method yields explicit stationary feedbacks which stabilize the linear periodic transport equation exponentially, and even in finite time.Finally, we implement this method on a more complex system, the so-called water tank system. We prove that the linearized systems around constant acceleration equilibria are controllable if the acceleration is not too strong. Our method then yields explicit feedbacks which, although they remain explicit, are no longer stationary and require the addition of an integrator in the feedback loop.
... This was used to prove a host of results on the boundary stabilization of partial differential equations: let us cite for example [19] and [25] for the wave equation, [31,32] for the Korteweg-de Vries, [2, chapter 7] for an application to first-order hyperbolic systems, and also [15], which combines the backstepping method with Lyapunov functions to prove finite-time stabilization in H 2 for a quasilinear 2 × 2 hyperbolic system. ...
... Proof. It is clear, by definition of H m per , and using (31), that for α ∈ H m per , the expression ...
... The general strategy (as is done in [12], [31]) is to divide the interval [0, T ] in smaller intervals [t n , t n+1 ], the length of which tends to 0, and on which one applies feedbacks to get exponential stabilization with decay rates λ n , with λ n → ∞. Then, for well-chosen t n , λ n , the trajectory thus obtained reaches 0 in time T . ...
We use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate.
... Backstepping based feedback control given by an integral operator, can be computed explicitly and calculated numerically by successive approximation scheme and this approach requires much less computational effort, see [12,15,33]. Moreover, this explicit feedback law is very useful for getting the other results like null-controllability, and finite time stabilization, see [34][35][36]. ...
... In [34], null controllability of the heat equation has been proved by a combination of backstepping method and Lebeau-Robbiano strategy. This approach further gives the null controllability for the KdV equation in [36]. In this context, it will be very interesting to explore this method for studying approximate controllability cases. ...
In this article, we study the boundary feedback stabilization of some one-dimensional nonlinear coupled parabolic-ODE systems, namely Rogers-McCulloch and FitzHugh-Nagumo systems, in the interval (0, 1). Our goal is to construct an explicit linear feedback control law acting only at the right end of the Dirichlet boundary to establish the local exponential stabilizability of these two different nonlinear systems with a decay e^ {-ωt} , where ω ∈ (0, δ] for the FitzHugh-Nagumo system and ω ∈ (0, δ) for the Rogers-McCulloch system and δ is the system parameter that presents in the ODE of both coupled systems. The feedback control law, derived by the backstepping method forces the exponential decay of solution of the closed-loop nonlinear system in both L^2 (0, 1) and H^1 (0, 1) norms, respectively, if the initial data is small enough. We also show that the linearized FitzHugh-Nagumo system is not stabilizable with exponential decay e^ {-ωt} , where ω > δ.
... However, in our case the related eigenfunctions do not form a Riesz basis, due to the fact that the operator is neither self-adjoint nor skew-adjoint. In fact they are not even complete in L 2 (0, L), see [29], which prevents us from directly applying those methods. Due to the existence of the critical length set, it does not seem natural to consider Carleman estimates. ...
... Originally introduced to stabilize system exponentially [15,20], recently it is further developed as a tool for null control and small-time stabilization problems [14,17,[28][29][30][31], the so called piecewise backstepping, which shares the advantage that the feedback (control) is well formulated. It consists in stabilizing system with arbitrary exponential decay rate (rapid stabilization) with explicit computable estimates, and splitting the time interval into infinite many parts such that on each part backstepping exponential stabilization is applied to make the energy divide at least by 2. Concerning our KdV case, at least for non-critical cases, rapid stabilization by backstepping is achieved in [13], where they used the controllability of KdV equation with control of the form b(t) = u x (t, 0) − u x (t, L) as an intermediate step. ...
The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.
... However, in our case the related eigenfunctions do not form a Riesz basis, due to the fact that the operator is neither self-adjoint nor skew-adjoint. In fact they are not even complete in L 2 (0, L), see [29], which prevents us from directly applying those methods. Due to the existence of the critical length set, it does not seem natural to consider Carleman estimates. ...
... • Is backstepping another option? Originally introduced to stabilize system exponentially [15,20], recently it is further developed as a tool for null control and small-time stabilization problems [14,29,28,17,30], the so called piecewise backstepping, which shares the advantage that the feedback (control) is well formulated. It consists in stabilizing system with arbitrary exponential decay rate (rapid stabilization) with explicit computable estimates, and splitting the time interval into infinite many parts such that on each part backstepping exponential stabilization is applied to make the energy divide at least by 2. Concerning our KdV case, at least for non-critical cases, rapid stabilization by backstepping is achieved in [13], where they used the controllability of KdV equation with control of the form b(t) = u x (t, 0) − u x (t, L) as an intermediate step. ...
The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.
... This was used to prove a host of results on the boundary stabilization of partial differential equations: let us cite for example [20] and [27] for the wave equation, [35,36] for the Korteweg-de Vries, [2, chapter 7] for an application to first-order hyperbolic systems, and also [16], which combines the backstepping method with Lyapunov functions to prove finite-time stabilization in H 2 for a quasilinear 2 × 2 hyperbolic system. ...
... As we have mentioned in the introduction, one of the advantages of the backstepping method is that it can provide explicit eedback laws for exponential stabilization. This allows the construction of explicit controls for null controllability ( [35,13]) as well as time-varying feedbacks that stabilize the system in finite time T > 0 ( [36,13]). ...
... (Yuhu Wu), bzguo@iss.ac.cn (Bao-Zhu Guo). (Pazoto, 2005;Marx et al., 2017;Kang-Fridman, 2019;Chentouf-Guesmia, 2022), optimal control (Boulanger-Trautmann, 2017), backstepping methods (Xiang, 2019), internal control (Cerpa-Montoya-Zhang, 2020;Komornik-Pignotti, 2020;Parada-Crépeau-Prieur, 2022), localized damping (Perla Menzala-Vasconcellos-Zuazua, 2002;Wang-Zhou, 2023), and others. Among these, the boundary control approach stands out as a highly feasible method due to its ease of implementation in engineering applications, both economically and practically. ...
In this paper, we study the absolute exponential stability of the Korteweg-de Vries-Burgers equation under two distinct types of nonlinear boundary position feedback controls. We propose criteria that adhere to the sector-bounded and parabolic-restricted conditions, thereby encompassing a broad spectrum of nonlinear controllers. For each of these two nonlinear control strategies, we establish the well-posedness of the ensuing closed-loop system through the utilization of the Faedo-Galerkin approximation method. Furthermore, by crafting specific Lyapunov functionals, we demonstrate that the closed-loop systems exhibit global exponential stability in the L 2-space and semi-global exponential stability in the H m spaces for m = 1, 2, 3 respectively. The explicit exponential decay rates of the closed-loop system solutions are determined, depending upon the dissipative and diffusive parameters. Some numerical simulations using a finite difference scheme are presented to illustrate the effectiveness of the proposed controls.
... For example, models of the transmission of electric lines and quantum hydrodynamic models are described by third-order partial differential equations. In papers [1,2], for example, nonlocal boundary value problems for third-order partial differential equations were studied, and in [3][4][5][6][7][8], the properties of solutions to linearized Korteweg-de Vries equations were studied. In [9], the question of the existence and uniqueness of solutions to the Goursat-Dirichlet problem was studied. ...
In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time.
... Armed with the Ce C √ λ estimates (2.16)-(2.17), exactly the same procedure proposed in [36,52,53] by using piecewise stabilizing controls leads to the null controllability. In this section we construct similar feedback laws while keeping an extra attention on control costs. ...
The finite time stabilizability of the one dimensional heat equation is proved by Coron–Nguyên [16], while the same question for multidimensional spaces remained open. Inspired by Coron–Trélat [17] we introduce a new method to stabilize multidimensional heat equations quantitatively in finite time and call it Frequency Lyapunov method. This method naturally combines spectral inequality [35] and constructive feedback stabilization. As application this approach also yields a constructive proof for null controllability, which gives sharing optimal cost with explicit controls and works perfectly for related nonlinear models such as Navier–Stokes equations [52].
... Proof. We mimic the proof of the finite-time stabilization of the heat equations [25,52], as relatively standard; see also [28,50,51] for similar results. The proof is followed by five steps: ...
We construct explicit time-varying feedback laws that locally stabilize the two-dimensional internal controlled incompressible Navier–Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as well as quantitative null-controllability with explicit controls having e^{C/T} costs.
... The key idea is to look for a Volterra transformation of the second kind (which has the advantage of being invertible) mapping the original system to a target system for which the stability is easy to prove. These transformations were extensively used in the last decades, for instance for the heat equation [2,11,12], for first order hyperbolic linear then quasilinear systems [28,67], and for many particular cases (see [14,70,71] for the KdV equations, [32] for coupled PDE-ODE systems, [29] for the viscous Burgers equations, or [51] for an overview), the goal of each new study being to show that such a transformation exists. However, considering only a special type of invertible transformations necessarily restricts the cases where this method can be applied. ...
In this article we study the so-called water tank system. In this system, the behavior of water contained in a one dimensional tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.
... These transforms are more general, and therefore potentially more powerful, but they are not always invertible and proving the invertibility of the candidate transform becomes one of the main difficulties. A good overview of the method when using a Volterra transform can be found in [103] or [150], and in [48, Section 2.1] for more general transforms. ...
Hyperbolic systems model the phenomena of propagations at finite speeds. They are present in many fields of science and, consequently, in many human applications. For these applications, the question of stability or stabilization of their stationary state is a major issue. In this paper we present state-of-the-art tools to stabilize 1-D nonlinear hyperbolic systems using boundary controls. We review the power and limits of energy-like Lyapunov functions; the particular case of density–velocity systems; a method to stabilize shock steady-states; an extraction method allowing to use the spectral information of the linearized system in order to stabilize the nonlinear system; and some results on proportional-integral boundary control. We also review open questions and perspectives for this field, which is still largely open.
... Such a description would also be interesting for finite time stabilization problems, namely Ce Cλ β type estimates. So far such estimates have been achieved via different methods, relying on direct energy estimates [22], Bessel functions [25], iterative methods [47], and spectral inequalities [48,49]. However, so far such important property has not yet been discovered for Fredholm type backstepping methods. ...
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.
... Feedback control given by an integral operator in backstepping can be computed explicitly and calculated numerically by successive approximation scheme (see (2.6)), and this approach requires much less computational effort. Moreover, this explicit feedback law is very useful for getting the other results like null controllability and finite time stabilization [16], [20], [65]. ...
... The key idea is to look for a Volterra transformation of the second kind (which has the advantage of being invertible) mapping the original system to a target system for which the stability is easy to prove. These transformations were extensively used in the last decades, for instance for the heat equation [12,11,2], for first order hyperbolic linear then quasilinear systems [65,28], and for many particular cases (see [14,69,68] for the KdV equations, [32] for coupled PDE-ODE systems, [29] for Burgers equations, or [51] for an overview), the goal of each new study being to show that such a transformation exists. However, considering only a special type of invertible transformations necessarily restricts the cases where this method can be applied. ...
In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.
... Although sometimes observers can be designed to tackle this issue [32]. Other methods exist, as for instance the study of stability based on time delay systems introduced in [8] where the authors give 1 see [14,Definition 1.4.3] for a proper definition and [15] for an overview of this method 2 see [22] for more details criteria for exponential stability in the W 2,p norms for any p ≥ 1 (see also [33]). ...
In this paper we study the global exponential stability in the norm of semilinear 1-d hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the norm with respect to both interior and boundary disturbances.
... Proof of Theorem 5.1. We mimic the prove of the finite time stabilization of the heat equations [16,35], as relatively standard, see also [19,34,33] for similar results. The proof is followed by five steps: ...
We provide explicit time-varying feedback laws that locally stabilize the two dimensional internal controlled incompressible Navier-Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as well as quantitative null controllability with explicit controls having costs.
... Armed with the Ce C √ λ estimates (2.17)-(2.18), exactly the same procedure proposed in [13,40,41] by using piecewise stabilizing controls leads to the null controllability. In this section we construct similar feedback laws while keeping an extra attention on control costs. ...
The null controllability of the heat equation is known for decades [19,23,30]. The finite time stabilizability of the one dimensional heat equation was proved by Coron--Nguy\^en [13], while the same question for high dimensional spaces remained widely open. Inspired by Coron--Tr\'elat [14] we find explicit stationary feedback laws that quantitatively exponentially stabilize the heat equation with decay rate and estimates, where Lebeau--Robbiano's spectral inequality [30] is naturally used. Then a piecewise controlling argument leads to null controllability with optimal cost , as well as finite time stabilization.
... Actually under such radial assumption the NLKG can be more or less regarded as an one dimensional wave equation, for which the study of controllability, stability and stabilization is nearly complete as the situation is much more clear than the higher dimension ones (while GCC is automatically fulfilled), for example local controllability and global controllability even with different type of nonlinearities by Zuazua [44,46] (see also [11] for an introductory proof of local controllability), exponential stabilization by Ricatti type strategy Coron-Trélat [16] or by Backstepping [37] (see also [13,43], so far this method only has been applied for one dimensional models). ...
We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in . First we show that the system is linearly unstable near the static solution for any dissipative boundary condition . Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially with rate less than , which is sharp, provided that the radius of the ball L satisfies .
... The presented procedure avoid solving the kernel equation in higher dimension case. Meanwhile, the backstepping method can also be used in finite time stabilization and controllability of PDEs (see [11] and [12]). ...
In this paper, we discuss the stabilization of the heat-like equation in rectangle domain with boundary control. We firstly transfer the heat-like equation in 2D into series of heat equations in 1D via eigenfunction expansion, then, by using the normal backstepping transformation, we obtain the existence and estimate for series of kernel functions in transformations. Finally, combined with the states and controls estimate of series of heat equations, we finally obtain the exponential and finite time stabilization of the heat-like equation in 2D.
In this paper, we consider the small-time local controllability problem for the KdV system on an interval with a Neumann boundary control. In 1997, Rosier discovered that the linearized system is uncontrollable if and only if the length is critical, namely for some integers k and l. Coron and Cr\'epeau (2003) proved that the nonlinear system is small-time locally controllable even if the linearized system is not, provided that is the only solution pair. Later, Cerpa and Crepeau showed that the system is large-time locally controllable for all critical lengths. In 2020, Coron, Koenig, and Nguyen found that the system is not small-time locally controllable if . We demonstrate that if the critical length satisfies with , then the system is not small-time locally controllable. This paper, together with the above results, gives a complete answer to the longstanding open problem on the small-time local controllability of KdV on all critical lengths since the pioneer work by Rosier
As you know, the third order partial differential equation is one of the basic equations of wave theory. For example, in particular, a linearized Korteweg-de Vries type equation with variable coefficients models ion-acoustic waves into plasma and acoustic waves on a crystal lattice. In this paper, the properties of solutions of а class of the third order degenerate partial differential equations with variable coefficients given in a rectangle were studied. Sufficient conditions for the existence and uniqueness of a strong solution have been established. Note that the solution of the degenerate equation does not retain its smoothness, therefore, these difficulties in turn affect the coercive estimates.
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first prove that, under some assumptions on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. Then, we prove that the Fredholm transformation constructed for the Laplacian also leads to the local rapid stability of the viscous Burgers equation.
We construct explicit time-varying feedback laws leading to the global (null) stabilization in small time of the viscous Burgers equation with three scalar controls. Our feedback laws use first the quadratic transport term to achieve the small-time global approximate stabilization and then the linear viscous term to get the small-time local stabilization.
In this paper we study the global exponential stability in the L2 norm of semilinear 1-d hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the L2 norm with respect to both interior and boundary disturbances.
We consider a 1-D linear transport equation on the interval (0,L), with an internal scalar control. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized in finite time, and we give an explicit feedback law.
In this paper, the authors consider the Gevrey class regularity of a semigroup associated with a nonlinear Korteweg-de Vries (KdV for short) equation. By estimating the resolvent of the corresponding linear operator, the authors conclude that the semigroup generated by the linear operator is not analytic but of Gevrey class δ ∈ (3/2, ∞) for t > 0.
This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the -norm and the decay rate can be tuned to be as large as desired if the initial data is small enough.
Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.
We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equa- tion on bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval, a Neumann nonhomogeneous boundary condition at the right end-point which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.
We consider the one dimensional Schr\"odinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.
This paper presents an output feedback control law for the Korteweg-de Vries equation. The control design is based on the backstepping method and the introduction of an appropriate observer. The local exponential stability of the closed-loop system is proven. Some numerical simulations are shown to illustrate this theoretical result.
Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries
equation posed on a finite interval [0,2π7/3]{[0,2\pi\sqrt{7/3}]}.
The equation comes with a Dirichlet
boundary condition at the left end-point and both the Dirichlet and Neumann
homogeneous boundary conditions at the right end-point. It is known that the
associated linearized equation around the origin is
not asymptotically stable. In this paper, the nonlinear
Korteweg–de Vries equation is proved to be
locally asymptotically stable around the origin through the center
manifold method.
In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. Analyzing the Korteweg–de Vries equation restricted on the local center manifold, we obtain a polynomial decay rate of the solution.
In this work, we are interested in the small time local null controllability
for the viscous Burgers' equation on the line
segment [0,1], with null boundary conditions. The second-hand side is a
scalar control playing a role similar to that of a pressure. In this setting,
the classical Lie bracket necessary condition introduced by
Sussmann fails to conclude. However, using a quadratic expansion of our system,
we exhibit a second order obstruction to small time local null controllability.
This obstruction holds although the information propagation speed is infinite
for the Burgers equation. Our obstruction involves the weak norm of
the control u. The proof requires the careful derivation of an integral
kernel operator and the estimation of residues by means of weakly singular
integral operator estimates.
These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in L 2 -norm.
The exact boundary controllability of linear and nonlinear Korteweg-de
Vries equation on bounded domains with various boundary conditions is
studied. When boundary conditions bear on spatial derivatives up to
order 2, the exact controllability result by Russell-Zhang is directly
proved by means of Hilbert Uniqueness Method. When only the first
spatial derivative at the right endpoint is assumed to be controlled,
a quite different analysis shows that exact controllability holds too.
From this last result we derive the exact boundary controllability for
nonlinear KdV equation on bounded domains, for sufficiently small initial
and final states.
The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s. Both
exact control and stabilization problems have been intensively studied since then and significant progresses have been made
due to many people's hard work and contributions. In this article, the authors intend to give an overall review of the results
obtained so far in the study but with an emphasis on its recent progresses. A list of open problems is also provided for further
investigation.
We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.
The control of the surface of water in a long canal by
means of a wavemaker is investigated. The fluid motion is governed
by the Korteweg-de Vries equation in Lagrangian coordinates.
The null controllability of the elevation of the fluid surface
is obtained thanks to a Carleman estimate and some weighted inequalities.
The global uncontrollability is also established.
The so-called unified method expresses the solution of an initial-boundary
value problem for an evolution PDE in the finite interval in terms of an
integral in the complex Fourier (spectral) plane. Simple initial-boundary value
problems, which will be referred to as problems of type I, can be solved via a
classical transform pair. For example, the Dirichlet problem of the heat
equation can be solved in terms of the transform pair associated with the
Fourier sine series. Such transform pairs can be constructed via the spectral
analysis of the associated spatial operator. For more complicated
initial-boundary value problems, which will be referred to as problems of type
II, there does \emph{not} exist a classical transform pair and the solution
\emph{cannot} be expressed in terms of an infinite series. Here we pose and
answer two related questions: first, does there exist a (non-classical)
transform pair capable of solving a type II problem, and second, can this
transform pair be constructed via spectral analysis? The answer to both of
these questions is positive and this motivates the introduction of a novel
class of spectral entities. We call these spectral entities augmented
eigenfunctions, to distinguish them from the generalised eigenfunctions
introduced in the sixties by Gel'fand and his co-authors.
In this work, we are interested in the small time global null controllability
for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment
[0,1]. The second-hand side is a scalar control playing a role similar to that
of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two
controls (namely the interior one u(t) and the boundary one y(t,0)). In this
setting, we show that small time global null controllability still holds by
taking advantage of both hyperbolic and parabolic behaviors of our system. We
use the Cole-Hopf transform and Fourier series to derive precise estimates for
the creation and the dissipation of a boundary layer.
Studied here is an initial-and boundary-value problem for the Korteweg–de Vries equation posed on a bounded interval with nonhomogeneous boundary conditions. This particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed. It is shown here that this initial-boundary-value problem is globally well-posed in the L 2 -based Sobolev space H s (0, 1) for any s ! 0. In addition, the mapping that associates to appropriate initial-and boundary-data the corresponding solution is shown to be analytic as a function between appropriate Banach spaces.
This paper deals with the stabilization problem for the Korteweg-de Vries equa-tion posed on a bounded interval. The control acts on the left Dirichlet boundary condition. At the right end-point, Dirichlet and Neumann homogeneous boundary conditions are considered. The proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data. Moreover, the decay rate can be tuned to be as large as desired. The feedback control law is designed by using the backstepping method.
In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering
applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop
unstable eigenvalues and is not restricted to a particular type of boundary actuation. Stabilization is achieved through the
design of coordinate transformations that have the form of recursive relationships. The fundamental difficulty of such transformations
is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes
infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how
to design coordinate transformations such that they are sufficiently regular (not continuous but L
∞). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied
by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.
It is known that the linear Korteweg-de Vries (KdV) equation with homogeneous Dirichlet boundary conditions and Neumann boundary control is not controllable for some critical spatial domains. In this paper, we prove in these critical cases, that the nonlinear KdV equation is locally controllable around the origin provided that the time of control is large enough. It is done by performing a power series expansion of the solution and studying the cascade system resulting of this expansion. (c) 2008 Elsevier Masson SAS. All rights reserved.
We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.
We consider the boundary controllability problem for a nonlinear Korteweg-de Vries equation with the Dirichlet boundary condition. We study this problem for a spatial domain with a critical length for which the linearized control system is not controllable. In order to deal with the nonlinearity, we use a power series expansion of second order. We prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough.
This paper deals with the stabilization of a rotating body-beam
system with torque control. The system we consider is the one studied by
Baillieul and Levi (1987). Xu and Baillieul proved (1993) that, for any
constant angular velocity smaller than a critical one, this system can
be stabilized by means of a feedback torque control law if there is
damping. We prove that this result also holds if there is no damping
We address the problem of Dirichlet and Neumann boundary control of the Kuramoto-Sivashinsky equation on the domain [0, 1]. First we note that, while the uncontrolled Dirichlet problem is asymptotically stable when an "anti-di#usion" parameter is small, and unstable when it is large (we determine the critical value of the parameter), the uncontrolled Neumann problem is never asymptotically stable. We develop a Neumann feedback law that guarantees L 2 -global exponential stability and H 2 -global asymptotic stability for small values of the anti-di#usion parameter. The more interesting problem of boundary stabilization when the anti-di#usion parameter is parge remains open. Our proof of global existence and uniqueness of solutions of the closed-loop system involves construction of a Green function and application of the Banach contraction mapping principle. Key Words: Kuramoto-Sivashinsky equation; boundary control; stabilization; spectral analysis. To appear in Nonlinear Analysi...
In this paper, we deal with controllability properties of linear and nonlinear Korteweg–de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain some local exact controllability results for the nonlinear KdV equation.
This paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side. The feedback control law is designed by first using a Volterra transformation of the second kind and then using an invertible Fredholm transformation. Both existence and invertibility of the transformations are easily obtained.
We consider in this paper the problem of the Lagrangian controllability for the Korteweg-de Vries equation. Using the N-solitons solution, we prove that, for any length of the spatial domain and any time , it is possible to choose appropriate boundary controls of KdV equation such that the flow associated to this solution exit the domain in time T.
In this work, we consider the problem of boundary stabilization for a quasilinear 2×2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H2 exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4×4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.
We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and one leftward convecting transport PDE. We design a controller, which requires a single control input applied on the leftward convecting PDE's right boundary, and an observer, which employs a single sensor on the same PDE's left boundary. We prove exponential stability of the origin of the resulting plant-observer-controller system in the spatial L-2-sense.
Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems.
This paper is devoted to the study of the local rapid exponential
stabilization problem for a controlled Kuramoto-Sivashinsky equation on a
bounded interval. We build a feedback control law to force the solution of the
closed-loop system to decay exponentially to zero with arbitrarily prescribed
decay rates, provided that the initial datum is small enough. Our approach uses
a method we introduced for the rapid stabilization of a Korteweg-de Vries
equation. It relies on the construction of a suitable integral transform and
can be applied to many other equations.
This paper is devoted to the study of the rapid exponential stabilization
problem for a controlled Korteweg-de Vries equation on a bounded interval with
homogeneous Dirichlet boundary conditions and Neumann boundary control at the
right endpoint of the interval. For every noncritical length, we build a
feedback control law to force the solution of the closed-loop system to decay
exponentially to zero with arbitrarily prescribed decay rates, provided that
the initial datum is small enough. Our approach relies on the construction of a
suitable integral transform.
We present in detail a linear, constant-coefficient initial/boundary value problem for which the classical method of eigenfunction expansions fails. We note that the new method recently introduced by A. Fokas and B. Pelloni (2005) in [3] can be successfully applied to the same problem.
We study an initial–boundary-value problem of a nonlinear Korteweg–de Vries equation posed on the finite interval where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear Korteweg–de Vries stationary equation is of dimension 1. This is for example the case if .
We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg-de Vries equation in a bounded inter-val. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by L. Rosier and by O. Glass and S. Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be x-independent. Then we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time T . Finally, introducing a fourth control on the first derivative at the right endpoint, we get the global exact controllability, for any positive time T .
This paper is devoted to present some positive and negative controllability results for the viscous Burgers equation. More precisely, in the context of null controllability with distributed controls, we present sharp estimates of the minimal time of controllability T(r) of initial data of L2-norm less or equal to r. In particular, we see that (global) null controllability does not hold in general (unless the control is exerted everywhere). The same results apply to similar boundary controllability systems with one boundary control.
In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.
Résumé
Ce papier concerne l'équation de Burgers visqueuse. On étudie les propriétés de contrôlabilité exacte de cette équation quand on contrôle les deux extrémités de l'intervalle. Dans un premier résultat, on démontre que la contrôlabilité globale à zéro n'est pas vraie pour un temps petit. Enfin, dans un deuxième résultat on démontre que la contrôlabilité exacte n'est pas vraie pour un temps long.
We consider a problem of boundary feedback stabilization of first-order hyperbolic partial differential equations (PDEs). These equations serve as a model for physical phenomena such as traffic flows, chemical reactors, and heat exchangers. We design controllers using a backstepping method, which has been recently developed for parabolic PDEs. With the integral transformation and boundary feedback the unstable PDE is converted into a “delay line” system which converges to zero in finite time. We then apply this procedure to finite-dimensional systems with actuator and sensor delays to recover a well-known infinite-dimensional controller (analog of the Smith predictor for unstable plants). We also show that the proposed method can be used for the boundary control of a Korteweg–de Vries-like third-order PDE. The designs are illustrated with simulations.
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation u t(x, t) = u xx(x, t) + a(x)u(x, t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term u xx] but also the destabilizing heat is generating (mathematically due to the term au with a > 0). We show that for any given continuously differentiable function a and any given positive constant λ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of λ. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022-2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165-176].
In this paper, we deal with both nonviscous and viscous Burgers-type equations on a bounded interval. We study the global exact controllability of these equations when we have three controls: one control is the right member of the equation and is constant with respect to the space variable; the two others are the boundary values. For a first time, we are interested in nonviscous Burgers-type equations and we prove their global exact controllability for every time with such controls thanks to the return method. Then we prove the global exact controllability of the viscous Burgers equation for every time .
In this paper, we deal with the viscous Burgers equation with a small dissipation coe-cient ". We prove the (global) exact controllability property to nonzero constant states, that is to say, the possibility of flnding boundary values such that the solution of the associated Burgers equation is driven to a constant state. The main objective of this paper is to do so with control functions whose norms in an appropriate space are bounded independently of ", which belongs to a suitably small interval. This result is obtained for a su-ciently large time.