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Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback
Abstract
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.
... The authors used such an extension of the PDE backstepping method with a Fredholm type transformation, relying mainly on spectral properties of the Schrödinger equation and on a controllability assumption. This was also adapted successfully, despite very different spectral properties, to the linear transport equation in [77]. ...
... Rather, Proposition 3.5 extends the implication "controllability ⇒ stabilizability" to a case where the control operator is unbounded, as is done in [22,60,66,72,77]. ...
... However the equation above is actually purely formal, and the "right way" to formulate it is the weak form in (251), which is not surprising, as, according to Corollary 5.1, T I ν is not defined and thus (252) has no real mathematical meaning. This is already the case for transport equations [76,77]. ...
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.
... As of late, the Fredholm transformation was introduced for the backstepping method as an alternative for certain limitations of the Volterra transformation. In particular, it seems better suited for internal stabilization problems [16,51]. The idea of using transformations remains the same, but proving the existence and invertibility of the transformation is generally more involved. ...
... There are mainly two ways to prove the existence of the transformation, either by direct methods [18,19] or, more commonly, by proving the existence of a Riesz basis. For the latter, we again distinguish two cases: either the Riesz basis is deduced directly by an isomorphism applied on an eigenbasis [17,50,51] or the existence of a Riesz basis follows by controllability assumptions and sufficient growth of the eigenvalues of the spatial operator allowing in particular to prove that the family is quadratically close to the eigenfunctions [16,20,21,27] (see Section 2.2 and Section 4 for a definition). ...
... In infinite dimension, the situation is more complex, but the same philosophy applies. The role of controllability in finding such backstepping transformations has been established for several important PDE models such as the transport equation [51], the KdV equation [20], Kuramoto-Sivashinsky equation [21], the linearized Schrödinger equation [16], the linearized Saint-Venant equation [17]. We highlight that the uniqueness equation T B = B was decisive in [16,17,51] to transform non-local terms emanating from distributed control functions into local terms for the operator equality, but was fundamentally used in an implicit way in [20,21] for boundary controls. ...
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.
... The authors used such an extension of the PDE backstepping method with a Fredholm type transformation, relying mainly on spectral properties of the Schrödinger equation and on a controllability assumption. This was also adapted successfully, despite very different spectral properties, to the linear transport equation in [73]. ...
... Rather, Proposition 3.4 extends the implication "controllability =⇒ stabilizability" to a case where the control operator is unbounded, as is done in [22,58,64,73]. ...
... We can assume that we have chosen γ < γ * without losing the controllability of (56) and (73). Then it is clear thanks to Lemma 5.2 that τ A is an isomorphism of (L 2 ) 2 . ...
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.
... where a is continuous, real-valued,φ is a given real-valued function of space, and at time t, y(t, ·) is the state and u(t) is the control. As in [162], the system can be transformed into so that we focus on systems of the form (4.2) in this article. Hyperbolic systems with an internal control of this form model a variety of physical systems: let us cite the water tank system (introduced in [75] and further studied in [52,126]), which is modelled by Saint-Venant equations with boundary conditions analog to our periodic boundary conditions, and the plug-flow reactor system, where the control is the temperature of the reactor, and there is a given input at the boundary (see [125,127]). ...
... This is a linear transport equation, which we seek to stabilize with an internal, scalar feedback, given by a real-valued feedback law. In [162], we proved the following theorem for system (4.2) when it is controllable: ...
... C) is the largest (resp. smallest) constant such that (4.12) holds, then the constant in (4.13) can be critical in some particular cases (see [162]). Now, for λ > 0, the corresponding feedback law obtained in [162, Section 2.3] using the backstepping method is the linear form F λ−µ ∈ E defined by ...
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.
... where a is continuous, real-valued,φ is a given real-valued function of space, and at time t, y(t, ·) is the state and u(t) is the control. As in [38], the system can be transformed into α t + α x + µα = u(t)ϕ(x), x ∈ [0, L], α(t, 0) = α(t, L), ∀t ≥ 0, so that we focus on systems of the form (2) in this article. ...
... This is a linear transport equation, which we seek to stabilize with an internal, scalar feedback, given by a real-valued feedback law. In [38], we proved the following theorem for system (2): ...
... In [36], the author obtains the same kind of results for the Korteweg-de Vries equation. In this article, we use the explicit feedback laws obtained by the backstepping method in [38] to design an explicit feedback law that achieves finite-time stabilization. ...
... Aside of the seemingly different approach of hyperbolic systems [15,16,17,47,48], the proof of Step 1 and 2 relied heavily in the literature on the quadratically close criterion. Roughly speaking, it amounts to show, after some computations, that n∈N p∈N\{n} 1 |λ n − λ p + λ| 2 < +∞ which holds if the eigenvalues λ n of the operator A scales as n α with α > 3/2 but fails as soon as α ⩽ 3/2. ...
... We have so far excluded from our discussion the case α = 1 as it seems to be a very specific case with techniques of its own. Indeed, the rapid stabilization for hyperbolic systems was established in [16,17] through direct methods or by identifying the isomorphism applied to the eigenbasis leading to the Riesz basis [15,47,48]. The other results found in the literature were concerned with operators such that α ⩾ 2, and in these cases the Riesz basis properties was proved through the quadratically close criterion, thanks to the sufficient growth of the eigenvalues. ...
... In [17], the authors shown that the hyperbolic equation (8) is finite time stabilizable if and only if it is exactly controllable. Internal rapid stabilization has been established by means of scalar feedback law in the paper [52] for transport equation in a periodic Sobolev spaces with the assumption of the controllability of the studied system. Both the papers explored the backstepping method to study the stabilization issues. ...
... We will prove that such a function ρ 0 does not exist by showing that KerF T = {0}, ∀ T > 1. This gives a contradiction of the assumptions (51) and (52). As a consequence the observability inequality (29) holds. ...
In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval (0,1) by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time . We explore the method of moments to conclude the exact controllability at the critical time T=1. For the case of small time, that is for , we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law.
... Conclude on the rapid stabilization using the operator equality. Aside of the seemingly different approach of hyperbolic systems [13,14,15,41,42], the proof Step 1 and 2 relied heavily in the literature on the quadratically close criterion. Roughly speaking it amounts to show after some computations that n∈N p∈N\{n} 1 |λ n − λ p + λ| 2 < +∞ which holds if the eigenvalues λ n of the operator A scales as n α with α > 3/2 but fails as soon as α ≤ 3/2. ...
... We have so far excluded from our discussion the case α = 1 as it seems to be a very specific case with techniques on its own. Indeed, the rapid stabilisation for hyperbolic systems was established in [14,15] through direct methods or by identifying the isomorphism applied to the eigenbasis leading to the Riesz basis [13,41,42]. The other results found in the literature were concerned with operators such that α ≥ 2, and in these case the Riesz basis properties was proved through the quadratically close criterion, thanks to the sufficient growth of the eigenvalues. ...
Fredholm-type backstepping transformation, introduced by Coron and L\"u, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form for . We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying , a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for . The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order .
... The backstepping method, first introduced by Krstic and his collaborators [26], corresponds to moving the spectrum with the help of some feedback laws. It has been improved in [10,12] so that can be adapted to more one dimensional models [11,42,43]. From a spectrum point of view, this method is different from any other stabilizing techniques concentrating on finite dimensional low frequency terms, as a result it can be applied to hyperbolic systems. ...
... Or equivalently, lim t→T − y(t) ∈ L 2 (Ω) can be proved by the Cauchy sequence argument suggested in [13, page 1018 for (4. 42)]. Therefore, the flow Φ(s, t; y) is well-defined on ∆ × L 2 (Ω). ...
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.
... Rapid exponential stabilization by feedback for the linearized Schrodinger equation has been studied in [63] by using interior control. In the paper [64], the author explored backstepping approach for the one dimensional transport equation. However all these papers have studied the rapid exponential stabilization that is exponential stabilization by feedback with any decay rate for a single PDE. ...
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.
... This method, first introduced by Krstic and his collaborators [28], corresponds to moving the spectrum with the help of some feedback laws. It has been generalized in [29,30] by Fredholm transformation, and has turned out to be efficient for various one-dimensional models [31][32][33][34][35]. However, it is still a challenging open problem to introduce the backstepping method on general multidimensional models. ...
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].
... The usual backstepping approach for PDE, presented in [40], searches for isomorphisms under the form of a Volterra transform of the second kind (see (4.3)), which are conveniently always invertible, among other advantages. Some attempt to introduce a generalized backstepping approach which does not necessarily rely on Volterra transforms have also been introduced in [20,17,27,49,50,18,26]. The Volterra approach has been used in many areas and for many systems in the last decades including parabolic equations (see for instance [5,21,24]), hyperbolic system (see for instance [39,46,3,2,29,30,19]), etc. ...
We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.
... • Generally speaking, the Lyapunov functional method aims to find artfully chosen energy and multipliers to characterize the variation of energy from a global point of view. They have been heavily adapted to the study of stabilities and stabilization of systems, including hyperbolic systems of conservation laws [8,38,53], wave equations [44], and parabolic equations [27], among others. The Lyapunov functional in this paper is highly inspired by the one introduced in [52]: the idea is to separate low frequency and high frequency with different weights. ...
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.
... Also, several results (e.g. [15]) were obtained using a backstepping approach, a very powerful method based on a Volterra transformation, developed mainly for PDE in [24], and generalized recently with a Fredholm transformation for hyperbolic systems [13,36,35]. One may look at [21] for a more detailed survey about this method and its use for the Saint-Venant equations. ...
... In the recent years several studies started to look at other more general linear transforms such as Fredholm transforms [52,49,46,154,155,48,75,74]. 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. ...
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.
... Also, several results (e.g. [16]) were obtained using a backstepping approach, a very powerful method based on a Volterra transformation, developed mainly for PDE in [24], and generalized recently with a Fredholm transformation for hyperbolic systems [14,34,35]. One may look at [22] for a more detailed survey about this method and its use for the Saint-Venant equations. ...
We study the exponential stability in the norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.
... Such backstepping approach was also used to derive controllability [25,26] or finite-time stabilization [27,28,29] in both parabolic and hyperbolic settings. Other results using a more general transform were then introduced [30,31]. The main drawback of this method is that it involves controls that are usually using full-state measurements and cannot take the simple form of output feedback controllers (see (3)). ...
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.
... 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.
... Recently, the author has introduced a method to stabilize the multi-dimensional heat equations in finite time [35], which is based on quantitative rapid stabilization relying on spectral estimates and Lyapunov functionals, as well as piecewise feedback laws. Methodologically speaking, the technical spectral estimate is achieved via local Carleman estimates on elliptic operators up to boundaries (as always fulfilling Hörmander's pseudoconvex condition), starting from the seminal paper [26] these results can be regarded as standard, at least compared to wave type operators; the Lyapunov functions [18] aim at finding artfully chosen energy and multiplier to characterize the variation of the energy from a global point of view without knowing any microlocal information, which have been extensively developed in the study of hyperbolic systems of conservation laws as well as scattering theory [6,25,28,36]; the piecewise (in time) feedback law is introduced in [16] to stabilize the one dimensional heat equation in finite time together with the backstepping method, instead of using stationary feedback laws. This method shares several advantages: ...
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.
... • 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.
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 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.
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 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 deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.
Abstract We consider feedback transformations of the backstepping/feedback linearization type that have been prevalent in nite dimensional nonlinear stabilization, and, with the objective of ulti- mately addressing nonlinear PDE’s, generate the rst such transformations for a linear PDE that can have an arbitrary nite number,of open-loop unstable eigenvalues. These transformations have the form of recursive relationships and the fundamental,difculty is that the recursion has an innite number,of iterations. Naive versions of backstepping lead to unbounded,coefcients in those transformations. We show how to design them such that they are sufciently,regu- lar (not continuous but L∞). We then establish closedñloop stability, regularity of control, and regularity of solutions of the PDE.
This article studies a class of control systems in a Hilbert space H given by x ˙(t)=Ax(t)+bu(t), where A generates a holomorphic semigroup on H, u(t) is a scalar control, and the control input b is possibly unbounded. Many systems with boundary or point control can be represented in this form. The author considers the question of what eigenvalues {α k } k∈I the closed-loop system can have when u(t) is a feedback control. S.-H. Sun’s condition on {α k } k∈I [SIAM J. Control Optimization 19, 730-743 (1981; Zbl 0482.93035)] is generalized to the case where b is unbounded but satisfies an admissibility criterion; this condition is generalized further when unbounded feedback elements are allowed. These results are applied to a structurally damped elastic beam with a single point actuator. Similar techniques also prove a spectral assignability result for a damped elastic beam with a moment control force at one end, even though the associated input element is not admissible in the appropriate sense.
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.
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.
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 consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by Saint-Venant's equations. We show how to parameterize the trajectories of the linearized system thanks to the horizontal coordinate of a particular point in the system -- the "flat output", see figure 2-- and a periodic function. The motion planning problem of the linearized model is solved in the general case of joining two steady states. Next we provide an algorithm, based on Godunov scheme, with a dedicated way of dealing with boundary conditions, to numerically simulate the evolution of the nonlinear system. Nonlinear simulations provide a way of checking the accuracy of the motion planning based on the tangent linear system. 1
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.
We study the exponential stability for the C¹ norm of general 2 × 2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the propagation speeds of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic C¹ Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the C¹ norm. We show that, under a simple condition on the source term, the existence of a basic C¹ (or Cp , for any p ≥ 1) Lyapunov function is equivalent to the existence of a basic H² (or Hq , for any q ≥ 2) Lyapunov function, its analogue for the H² norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the C¹ norm using static boundary feedbacks depending only on measurements at the boundaries, which has a large practical interest in hydraulic and engineering applications.
We address the question of the exponential stability for the norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the norm. We show that the existence of a basic Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the norm and we show that the interior condition is also necessary to the existence of a basic Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the norm, for any .
This paper focuses on the (local) small-time stabilization of a Korteweg–de Vries equation on bounded interval, thanks to a time-varying Dirichlet feedback law on the left boundary. Recently, backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system, instead of Carleman inequalities. We use the “adding an integrator” technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small time.
This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems. Compared with the first edition, more emphasis is put on explicit constructions with attractive properties. Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field.
Key features include:
*Elementary introduction to frame theory in finite-dimensional spaces
* Basic results presented in an accessible way for both pure and applied mathematicians
* Extensive exercises make the work suitable as a textbook for use in graduate courses
* Full proofs includ
ed in introductory chapters; only basic knowledge of functional analysis required
* Explicit constructions of frames and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems
* Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory
* Selected research topics presented with recommendations for more advanced topics and further readin
g
* Open problems to stimulate further research
An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.
Review of the first edition:
"Ole Christensen’s An Introduction to Frames and Riesz Bases is a first-rate introduction to the field … . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research."
— Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005
This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.
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.
This paper deals with the saturated control problem of a class of distributed systems which can be modelled by first-order hyperbolic partial differential equations (PDE). The objective is designing a distributed-parameter state feedback with guaranteed performance for this class of systems, using the Lyapunov stability theory and polynomial sum-of-squares (SOS) programming. For this, a polynomial parameter varying (PPV) model is employed to exactly represent the nonlinear PDE system in a local region of the state space and then, based on it, a PPV state-feedback law is designed guaranteeing exponential stability and actuator saturation in such region. The approach is illustrated here through the standard example of a nonisothermal plug-flow reactor.
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.
In the present article we study the stabilization of first-order linear
integro-differential hyperbolic equations. For such equations we prove that the
stabilization in finite time is equivalent to the exact controllability
property. The proof relies on a Fredholm transformation that maps the original
system into a finite-time stable target system. The controllability assumption
is used to prove the invertibility of such a transformation. Finally, using the
method of moments, we show in a particular case that the controllability is
reduced to the criterion of Fattorini.
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.
Consider the following optimal control problem: Given the dynamical system (1.1) minimize the quadratic functional (1.2) over all u ∈ L2(0, ∞, U), with y solution of (1.1) due to u.
We propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u x (0,t),u x (1,t),u xx (0,t) and u xx (1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H 3 stable and well posed.
The backstepping method is a systematic design tool for boundary control of various types of partial differential equations (PDEs). There has been no attempt to apply it to PDEs whose input is not at the boundary. In this paper, we consider a problem of feedback stabilization of 1-dimensional parabolic (unstable) PDEs with internal actuation based on the backstepping method. Since such a PDE can not be converted to a stable PDE by state feedback and the state transformation used in backstepping, an additional transformation is introduced. Under a certain condition, the newly proposed transformation moves the input from the interior of the domain to the boundary. This enables us to cancel the residual term that causes the open-loop instability by using the input. Furthermore, this transformation is continuously invertible. Therefore, a stabilizing state feedback for the original PDE is derived through the inverse transformation. The results are demonstrated by a numerical simulation.
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.
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.
Preface.- 1. Lp Spaces and Interpolation.- 2. Maximal Functions, Fourier Transform, and Distributions.- 3. Fourier Series.- 4. Topics on Fourier Series.- 5. Singular Integrals of Convolution Type.- 6. Littlewood-Paley Theory and Multipliers.- 7. Weighted Inequalities.- A. Gamma and Beta Functions.- B. Bessel Functions.- C. Rademacher Functions.- D. Spherical Coordinates.- E. Some Trigonometric Identities and Inequalities.- F. Summation by Parts.- G. Basic Functional Analysis.- H. The Minimax Lemma.- I. Taylor's and Mean Value Theorem in Several Variables.- J. The Whitney Decomposition of Open Sets in Rn.- Glossary.- References.- Index.
Consideration is given to the placement of the spectrum of the closed-loop operator A plus BK resulting from use of a linear feedback control law u equals Kx in the infinite dimensional linear control system x prime equals Ax plus Bu. For a class of systems in Hilbert space with certain assumptions on the spectrum of the operator A, a complete characterization of the achievable spectra is obtained. The proofs are carried out in an operator-theoretic context.
We consider the problem of boundary feedback stabilization of homogeneous equilibria in unstable fluid mixtures that are governed by unstable linear reaction-convection-diffusion equations. We extend boundary feedback control laws designed for the one-dimensional reaction-diffusion equation using the backstepping method to this higher-dimensional case. We show that, under certain mathematical conditions on the velocity field, boundary feedback controls similar to the ones for one-dimensional equations also works for the higher dimensional case and exponentially stabilize the homogeneous equilibrium zero at any given decay rate.
We prove the well-posedness of a linear closed-loop system with an explicit
(already known) feedback leading to arbitrarily large decay rates. We define a
mild solution of the closed-loop problem using a dual equation and we prove
that the original operator perturbed by the feedback is (up to the use of an
extension) the infinitesimal generator of a strongly continuous group. We also
give a justification to the exponential decay of the solutions. Our method is
direct and avoids the use of optimal control theory.
We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.
In this article we examine the effect of linear feedback control in the hyperbolic distributed parameter control system By means of a reduction to canonical form similar to the one already familiar for finite-dimensional systems we show this system to be equivalent to the controlled difference-delay system The theory of nonharmonic Fourier series is then employed to investigate the placement of eigenvalues in the closed loop system. Boundary value control and canonical form for observed systems are also studied.
We study the wellposedness and the main features of a class of feedback control systems. The involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator, so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or Petrowski type. The feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed-loop system. These results are proved under a controllability assumption and the proofs rely on general results about the algebraic Riccati equation associated with the linear quadratic regulator problem.
Incluye bibliografía e índice
In this paper the inequality which is satisfied by the determinant of the solution of the Lyapunov matrix equation A'Q + QA = - D is presented. The result makes possible a lower estimate of product eigenvalues of the matrix Q and dependence from eigenvalues of the matrices A and D . This result corresponds to those presented in [2] and [3], where an estimate of the extremal eigenvalues of the matrix Q is presented. This estimate depends on the eigenvalues of the matrices A and D .
The results in [1] and [2] for the matrix Lyapunov equation are extended to the case of an algebraic matrix Riccati equation. Some errors in [1] and [2] are pointed out by a counter example. The estimations obtained in this note are shown to be exact for certain cases. Similar results are possible for the discrete algebraic matrix Riccati equation.
A constructive proof is given for finding constant feedback gains that stabilize a linear time-invariant controllable system. It is not necessary to transform variables or to specify pole locations.
This paper develops discontinuous control methods for minimum-phase semilinear infinite-dimensional systems driven in a Hilbert space. The control algorithms presented ensure asymptotic stability, global or local accordingly, as state feedback or output feedback is available, as well as robustness of the closed-loop system against external disturbances with the a priori known norm bounds. The theory is applied to stabilization of chemical processes around prespecified steady-state temperature and concentration profiles corresponding to a desired coolant temperature. Two specific cases, a plug flow reactor and an axial dispersion reactor, governed by hyperbolic and parabolic partial differential equations of first and second order, respectively, are under consideration. To achieve a regional temperature feedback stabilization around the desired profiles, with the region of attraction, containing a prescribed set of interest, a component concentration observer is constructed and included into the closed-loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice. Performance issues of the discontinuous feedback design are illustrated in a simulation study of the plug flow reactor.
. We prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates. Our approach is direct and constructive and avoids many of the technical di#culties associated with the usual methods based on Riccati equations. We give several applications for the wave equation and for Petrovsky systems. Key words. observability, controllability, stabilizability by feedback, partial di#erential equation, wave equation, Petrovsky system AMS subject classifications. 35L05, 35Q72, 93B05, 93B07, 93C20, 93D15 PII. S0363012996301609 1. Introduction. Let# be a nonempty bounded open set in R n having a boundary # of class C 2 , and consider the following problem: (1.1) y ## -#y = 0 in# × (0, #), (1.2) y(0) = y 0 and y # (0) = y 1 in# , (1.3) y = u on # × (0, #). Considering u as a control function, a natural problem is to seek stabilizing feedback laws u = F (y, y # ). In order to motivate our work, let us r...
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