Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs
We detail in this article the necessity of a change of paradigm for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.
In this paper, we highlight an important shortcoming of some control designs for systems of two heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs). More precisely, we show that imposing finite-time convergence by completely canceling the proximal reflection (i.e the reflection at the actuated boundary) yields, in some cases, zero robustness margins to arbitrarily small delays in the actuation path. In particular, the control laws in recent contributions (see for instance [9, 17, 18, 26, 31]) can have very poor to no robustness to delays due to the cancellation of the proximal reflection. To overcome this problem, we propose some changes in the design of target system to preserver a small amount of this reflection and ensure delay-robustness.
Most physical systems involving a transport phenomenon can be modeled using hyperbolic partial differential equations (PDEs): heat exchangers , open channel flow , , multiphase flow ,  or power systems . The backstepping approach [18, 31] has enabled the design of stabilizing full-state feedback laws for these systems. The generalization of these stabilization results for a large number of systems has been a focus point in the recent literature (details in [9, 17, 18, 31]). The main objective of these controllers is to ensure convergence in the minimum achievable time (as defined in ), thereby neglecting the robustness aspects that are essential for practical applications. Some of these questions have been the purpose of recent investigations: in the presence of uncertainties in the system, the design of adaptive control laws using filter or swapping design is the purpose of [5, 6]. A different approach, towards an engineering use of backstepping, consists in deriving sufficient conditions guaranteeing the exponential stability of the controlled system in presence of uncertainties . The issues of noise sensitivity and performance trade-off are considered in  where a method enabling the tuning of observer and controller feedback aggressiveness is proposed. However, the impact on stability of small delays in the feedback loop, has not been studied yet in this context. It has been observed (see [21, 37]) that for many feedback systems, the introduction of arbitrarily small time delays in the loop may cause instability for any feedback. In particular, in , a systematic frequency domain treatment of this phenomenon for distributed parameter systems is presented. Here, we use these results to cast a new light on feedback control design for linear hyperbolic systems.
The main contribution of this article is a set of necessary and sufficient conditions for classical controllers for linear hyperbolic systems to be robust to small delays. We prove that finite-time stabilization by completely canceling the proximal reflection, often yields vanishing delay margins, making it an impractical control objective. Indeed, the controllers derived in [9, 18, 31] can be unstable in presence of a small delay in the loop due to the cancellation of the proximal reflection. Furthermore, some systems (for which the product of the proximal and distal reflection gains is greater than one) cannot be delay-robustly stabilized, irrespective of the method. Specifically, we show that, for a system of two heterodirectional linear hyperbolic equations with anti-diagonal source terms111For systems with source terms on the diagonal, a transform is first employed that changes the reflection coefficients., if the product of the proximal and distal reflections is:
Greater than one, the system cannot be stabilized robustly to delays.
Smaller than one but greater than one-half, the system cannot be finite time stabilized robustly to delays.
Smaller than one-half the system can be finite-time stabilized robustly to delays.
Our approach is the following: considering the control law proposed in  and using a backstepping approach, the controlled system is mapped to a distributed delay equation. Then, using the Laplace transform we derive the closed-loop transfer function . It is shown to be potentially unstable in presence of small delays. To ensure delay-robust stabilization, we propose some adjustments in the control law proposed in  by means of an additional degree of freedom enabling a trade-off between convergence rate and delay robustness. More precisely, if the plant has some proximal reflection terms, the target system should preserve a small amount of this reflection.
An important by-product of this analysis, detailed in Section III-B is the reformulation of any system of two coupled linear hyperbolic equations as a zero-order neutral system with distributed delay. This result is obtained via a backstepping change of coordinates and yields a new tool for the study of hyperbolic systems.
The paper is organized as follows. In Section II we illustrate the necessity of this change of paradigm with a well-known system of two transport equations. These results are extended in Section III to coupled systems composed of two hyperbolic PDEs. A new control method is then derived in Section IV. The corresponding feedback system is proved to be stable to small delays. Finally some simulation results are given in Section V.
Ii A tutorial example: transport equations
In this Section, we consider the tutorial example of two pure transport equations coupled at the boundaries. We first recall historical results on the delay-robust stabilizability of such systems [30, 37]. Then, we study the behaviour of various control laws in the presence of small delays in the actuation path.
Ii-a Description of the system
We consider the following linear hyperbolic system of two transport equations
evolving in , with the following linear boundary conditions
We will use the term proximal reflection to denote : the reflection at the actuated boundary, and distal reflection to denote : the reflection on the unactuated boundary. The boundary reflections and , and the velocities and are assumed to be constant. The control law is denoted . Moreover, we assume that
The initial conditions denoted and are assumed to belong to . In the following we define the characteristic time of the system as
The product , product of the proximal and distal reflections, is called open-loop gain of the system. We recall the following definition from .
Delay-robust stabilization .
The controller where is an operator, delay-robustly stabilizes the system (1)-(3) in the sense of  if the resulting feedback system stabilizes the system (1)-(3) in the sense of the -norm and is delay-robustly stable with respect to small delays in the loop. A system is said to be delay-robustly stabilizable if and only if there exists such a .
Ii-B Open-loop transfer function
where is defined by (5). In the following, we denote the Laplace variable, and use boldface to denote the Laplace transform of a given function, i.e., the Laplace transform of will be denoted . Taking the Laplace transform of (6), we get
The transfer function is the open-loop transfer function of the system. Depending on the value of the open-loop gain this transfer function has either no pole in the Right Half Plane (RHP) or an infinite number of poles in the RHP. More precisely, if , this transfer function has no pole in the RHP whereas if , it has an infinite number of poles whose real parts are positive. They are defined as
where is an arbitrary integer. Consequently using [37, Theorem 1.2] we have the following theorem
Ii-C Feedback control for an open-loop gain smaller than one
Ii-C1 Finite-time stabilization
In this section, we focus on a system of transport equations for which the open-loop gain satisfies . Although in this case the system (1)-(3) is already exponentially stable in open-loop, one could want to increase the convergence rate or to have finite-time convergence. This improvement of the controller performance can be done using impedance matching methods (see [2, 4, 28]). This method is used, for instance, to improve the control performance for the heave rejection problem (), one can match the load impedance (the pressure to flow ration in the frequency domain at the boundary) to the characteristic line impedance (the pressure to flow ratio in the frequency domain in the transmission line). The application of this method in the case of transport equations consists in canceling totally the reflexion term , and get a semi-infinite system that converges to zero in finite time. The corresponding control law is then defined by
Consider now that there is a small delay in the actuation. The output is then solution of the following delay equation
This means that for an open-loop gain such that , one cannot have both robustness to a delay and finite time convergence. This justifies the observations that have been done by industrial practitioners about the limitations of the impedance matching method. For instance, in , the authors design a controller preventing stick-slip oscillations of a drill-string (a dysfunction of rotary drilling, characterized by large cyclic variations of the drive torque and the rotational bit speed). They observed that completely canceling the proximal reflection coefficient could change the dynamics of the string in a way that makes the system unstable. This observation is confirmed by the present analysis.
Ii-C2 First solution: preserving some reflection
Although canceling the proximal reflection to stabilize system (1)-(3) increases the nominal convergence rate, the corresponding feedback system is not robustly stable to delay in the loop when . Thus, it appears necessary to keep some reflection terms. To do so, let us slightly change the control law and replace (9) by
In presence of a delay in the actuation, we get, for the closed-loop system, the following delay equation:
which is now exponentially stable () for all if satisfies the following equation:
Note that such a exists since (as we assumed ).
Ii-C3 Second solution: filtering the control law
A second approach to provide a delay-robust stabilization consists in filtering the control law. Let us consider e.g the control law defined by its Laplace transformation as
where and are some coefficients that have still to be defined. In presence of a delay in the actuation, we get, for the closed-loop system, the following delay equation:
which is exponentially stable (see ) for all if
Ii-C4 Concluding remarks
Throughout the analysis of a simple system of two transport equations, we have proved (using the results from ) that there is a whole class of systems (the ones for which the product is larger than one) that cannot be robustly stabilized in presence of a small delay in the loop. Moreover, it has appear that even for , finite-time convergence is not a reasonable objective since the corresponding controller is not always robust to delays. This means that to have delay-robustness one may have to give up finite-time convergence. In the next section, we show that this change of paradigm still holds for general coupled system of two hyperbolic partial differential equations.
Iii General case of two coupled equations
In this section we consider the general case of two linear coupled hyperbolic PDEs. The main objective is to prove that the results of the previous section (requiring giving up finite time convergence to obtain delay-robust stabilization) still holds in the case where in-domain couplings exist. To do so, using a classical backstepping transformation (see ), the original system is mapped to a distributed-delay neutral system. Deriving the corresponding transfer function, it becomes possible to adjust the results of the previous section to this general case. Interestingly, this approach highlights the potential of backstepping as an analysis tool, rather than just a control design tool.
Iii-a Description of the system
We consider the following linear hyperbolic system which appear in Saint-Venant equations, heat exchangers equations and other linear hyperbolic balance laws (see ).
evolving in , with the following linear boundary conditions
The inside-domain coupling terms and can be spatially-varying, whereas the boundary coupling terms and , and the velocities and are assumed to be constant. Moreover, we assume that
The initial conditions denoted and are assumed to belong to . This system is pictured in Figure 1.
Iii-B A distributed-delay differential equation
Iii-B1 Volterra transformation: removing inside-domain couplings
We consider the following Volterra change of coordinates defined in  by
where the kernels are defined on by the following set of hyperbolic PDEs:
with the following set of boundary conditions:
Lemma 1 ( )
The dynamics of the system in the new coordinates is:
with the following linear boundary conditions
Iii-B2 Neutral equation with distributed delay
where is defined by (5) and where is defined by
where for any interval I, is defined by
This invertible coordinate change enable us to rewrite as the solution of a delay equation with distributed delays. Since this transformation is independent of the control law, it means that the class of systems described by (18)-(20), is equivalent to a class of neutral systems with distributed delay, as given by (39).
Iii-C Open-loop analysis
We then have the following lemma
If , then the open-loop transfer function has an infinite number of poles with a positive real part.
We assume here that the open loop gain is positive (the case can be treated in a similar way). The poles of the open-loop transfer function are the solutions of
is differentiable almost everywhere on . Integrating by parts yields
where is a bounded function that represents the jumps appearing in the integration by parts. We denote in the following
Since is bounded, the function converges to 0 for . The function has an infinite number of zeros whose real parts are equal to . The hypothesis of Theorem 5 (see the Appendix) are satisfied and we can then conclude that has an infinite number of zeros whose real parts are strictly positive. This concludes the proof
We can now state the following Theorem
If there exists a controller for system (18)-(20) such that the resulting feedback system is robustly stable to small delays in the loop, it implies that equation (39) is stable (since both system are equivalent). It means that there exists a controller for system (39) such that the resulting feedback system is robustly stable to small delays in the loop. This is impossible (see [37, Theorem 1.2]) since the open-loop transfer function has an infinite number of poles with a positive real part.
We have proved in this section that, similarly to transport equations, if the open-loop gain is greater than one, one cannot find a controller whose delay margin is non-null. Consequently, there is a whole class of hyperbolic systems that cannot be delay-robustly stabilized.
The critical case is not considered here. Indeed one cannot simply adjust the previous proof, since the zeros of are located on the imaginary axis.
Iii-D Feedback control for an open loop gain smaller than one
Iii-D1 Finite-time stabilization
In this section, we focus on a system of hyperbolic equations for which the open-loop gain satisfies . Note that the uncontrolled system can be unstable due to the inside-domain couplings and (see ). In , using the backstepping approach, a control law that ensures finite-time stabilization of the original system was derived. This control law is defined by
Consider now that there is a small delay in the actuation. The output is then solution of the following delay equation
where is defined by (40). We denote as
Taking the Laplace transform of equation (48), we get
We can now state the following Theorem:
This proof uses the same idea as the one used for the proof of Lemma 2. Let us denote
where . Choosing small enough, we have that . Consequently, has an infinite number of roots whose real parts are positive (see ). Moreover, these roots are unbounded. Thus, has an infinite number of roots whose real parts are larger than . Integrating by part we prove that converges to zero for large enough. Using Theorem 5, we have that has at least one root whose real part is strictly positive. This concludes the proof.
The fact that the backstepping controller proposed in  has zero delay margin when means that it cannot be used for practical applications. Specifically, indicates that the feedback systems cannot have both finite time convergence and be robust to delays. Similarly to the case of transport equations, this stability limitation is not due to the backstepping method itself bu is strongly interwoven with the cancellation of the proximal reflection term . To obtain a tractable implementation of a controller for the system (18)-(20), one must have robustness to delays and thereby give up finite-time convergence.
In the next section we propose a different control design by slightly adjusting the control law (47).
Iv A new control paradigm
In this section we slightly modify the control law (47) to overcome the stability limitation exposed above, while maintaining the same structure for the controller. The control law (47) is composed of two parts:
the integral part whose objective is to remove the effect of inside-domain couplings
the term whose objective is to cancel the proximal reflection and to ensure finite-time convergence.
As seen above, the instability of the feedback system in presence of small delay in the loop is mostly due to the term in the control law. It appears consequently necessary to avoid the total cancellation of the proximal reflection (and thereby giving up finite time convergence). Based on the analysis carried out in Section II-C2 for the case of transport equations, we proposed a similar adjustment for the control law (47) when .
Iv-a Control law
Let us consider the following control law:
The objective of such a control law is preserve a small amount of proximal reflection in the target system to ensure delay-robustness, while eliminating inside-domain couplings.
The control law can be rewritten as
with the boundary conditions
It is sufficient to prove that . To do so, let us consider all the cases depending on the signs of and . If and , we have (using (54))
since . The other cases can be treated similarly.
The coefficient can be interpreted as a tuning parameter, enabling a trade-off between performance (convergence rate) and robustness with respect to delays. This parameter has a role similar to the coefficient introduced in  in the design of the observer to enable a trade-off between performance and noise sensitivity.
We need now to prove that the proposed control law is robust with respect to small delays. We have the following theorem.
where is defined by (40). Taking the Laplace transform yields the following characteristic equation
where is defined by
Let us now consider a complex number such that . We then have
Since satisfies (54), there exists such that
Let us now focus on the term . We have
The integral on the expression above can be rewritten
where we denote the Fourier transform. Since belong to , this yields
Moreover, one can easily prove (Lebesgues) that
We can now choose small enough such that for any , . With this choice of , one can easily check that, , such that
Consequently, for , we have
It means that for , the function does not have any root whose real part is positive. Consequently, equation (60) is asymptotically stable. Thus, using the invertibility of the Volterra transformation (22)-(23), this concludes the proof.
For a given value of , the parameter gives a range for admissible delays. However, is not necessarily the maximum admissible delay.
Iv-B Interpretation of the results and outlook
Iv-B1 Zero delay margins
It is important to stress that the fundamental limitations of, e.g. Theorem 2 would not apply to an actual plant in the strict sense. Models of the form (1) are obviously simplistic and do not capture, e.g., the diffusivity that would stem from Kelvin-Voigt damping, or other phenomena that would be susceptible of making the delay margins non-null. However, these results do indicate
that the delay-robustness margins would be very poor for such systems
In this regard, a more quantitative approach to analyzing the performance–delay-robustness trade-offs made available by the use of backstepping is needed, in particular to assess whether the qualitative approach of the present article remains valid with more realistic models. A first step in analyzing this trade-offs has been taken in . In the next sections, we analyze the impact of the results on broader classes of systems.
Iv-B2 Interconnected systems
An important focus point in the recent literature is the control of interconnected and cascade systems: Ordinary Differential Equations (ODEs) featuring hyperbolic systems in the actuation paths have, in particular, received a lot of attention [11, 15, 24, 40]. A recurrent motivation for studying such systems is the control of mechanical vibrations in drilling, where the hyperbolic PDEs correspond to axial and torsional waves traveling along the drillstring, while the ODE models the Bottom Hole Assembly (BHA) dynamics (see, e.g., [13, 23] for details). The strategy in most approaches consists in transforming the interconnected systems into cascade systems by canceling the reflection at the controlled boundary. This enables the design of predictor-like feedback laws, that focus on stabilizing the (potentially unstable) ODE. This approach, although rigorously correct, is bound to exhibit poor delay-robustness in practice, as detailed in the previous section. To illustrate this point, we consider the following example.
Let us consider the following ODE-PDE system
with the boundary conditions
where , and are non-null constants. Considering as the output of the system and taking the Laplace transform, we get the following characteristic equation
This equation has a similar structure to that of equation (16) in presence of a delay in the actuation. Some of the results described in this paper can then be adjusted for ODE-PDEs systems.
Interestingly, this opens new perspectives for the control of PDE-ODE systems: when the reflection at the controlled boundary is partially or not canceled, the system takes in the general case the form of a neutral equation with distributed delay of the same order as the ODE, similarly to Equation (39). The stability of the target and closed-loop system is then subject to restrictive conditions on the coupling terms that have not been canceled by the controller, contrary to the idealized case where the closed-loop system is a cascade. In this regard, the stability analysis methods for such systems developed in [20, 38, 39] will be instrumental.
Iv-B3 Systems with multiple equations
Some physical systems require more than two equations to be properly modeled. For instance, Drift-Flux Models described in  representing the flow of liquid and gas along the oil wells consist of three distributed equations of conservation. Along with closure relations, this yields a set of three nonlinear transport PDEs with appropriate boundary conditions. The model can be linearized around a given equilibrium profile, which yields a system of the form (18)–(20) with and (see, e.g., ). To provide finite-time convergence, using a control law such as the one described in  or  require to cancel all the reflexion terms. In light of what has been presented in this paper, this does not seem desirable in term of delay-robustness. Applying a transformation similar to the one proposed in section III-B and writing the corresponding characteristic equation would lead to a matrix neutral equation with distributed delays. One can then use classical method () to analyze such equations. However, the results presented here do not straightforwardly extend due to the presence of remaining terms in the target system. This will be the purpose of future contributions.
V Simulation results
In this section we illustrate our results with simulations on a toy problem. The numerical values of the parameters are as follow.
The (positive) delay in the loop is denoted . The parameters values are chosen such that
Figure pictures the norm of the state using the control law presented in  without any delay ( s) and then in presence of a small delay in the loop ( s). As expected by the theory, with this control law, the system converges in finite time to its zero-equilibrium when there is no delay in the loop but becomes unstable in presence of a small delay.
Figure pictures the norm of the state using the new control law (53) ( is chosen equal to ) for the same situations ( and s) and for a larger delay ( s). As expected by the theory, the system is now robustly stable to delays in the loop. However, this improvement in terms of delay margin comes at the cost of a diminution of the convergence rate. This example illustrates the importance of this change of paradigm.
Vi Concluding remarks
Inspired by the results obtained for a system of two transport equations, we have proved that control laws ensuring finite-time stabilization are often not robust to arbitrarily small delays in the loop due to the complete cancellation of the proximal reflection. Consequently it appeared essential to make a change of paradigm, focusing on delay-robust stabilization. This has been done by means of a tuning parameter enabling a trade-off between convergence rate and delay-robustness. Even if the delay-robustness properties of the output-feedback controller (crucial for any application on an industrial problem) were not studied there, the result presented in this paper is a new step towards a complete analysis of the properties of the backstepping controller. This change of paradigm implies to rethink the controller-design for PDEs-ODEs systems or systems with more than two PDEs. That will be the purpose of future contributions.
We thank Laurent Praly, Delphine Bresch-Pietri, Sebastien Boisgerault and Miroslav Krstic for their valuable comments.
The work of the second author was supported by the Research Council of Norway, ConocoPhillips, Det norske oljeselskap, Lundin, Statoil and Wintershall through the research center DrillWell (203525/O30) at IRIS.
Appendix A Appendix
In this appendix, we prove an important result of complex analysis. Let us consider some strictly positive integers and , a sequence of constant matrices and a sequence of positive constants . We consider the holomorphic function defined for every complex number by
where is the identity matrix of dimension . For any real number , we denote the open half-plane . We have the following general theorem
Let us consider an holomorphic function such that
If the function has an infinite number of zeros on , then the function has an infinite number of zeros whose real parts are strictly positive.
To prove this theorem, we slightly adjust the proof from . For any positive , we denote the set of complex numbers whose distance to the zeros of det is at most :
We start proving the following lemma
-zero clusters and lower bound. Let us consider and . There exists such that any connected component of the set is bounded and such that if . Moreover, there exists such that on
This proof is similar to the one given in . By continuity of the determinant we have
Let us denote the number of zeros of whose modulus is smaller than . The function satisfies the assumptions of [35, Theorem VIII]. Thus, there exists a positive constant such that
If there is an unbounded connected component of , then there exists a sequence () of distinct zeros of such that for any , . It yields
Choosing implies that any connected component of the set is bounded.
Consider now the following complex analysis result. Let us consider a sequence of numbers in such that
The sequence is locally bounded on uniformly in . Consequently, there exists a sub-sequence that converges locally uniformly to an entire function (Montel’s theorem). Due to (79), this function cannot be identically zero. If , we define as the multiplicity of . Otherwise we set . Using Hurwitz’s theorem, we get that for small enough, and for large enough, has precisely zeroes in the open disk . Since , it yields that .
Let us consider a sequence of bounded connected components of , defined for large enough such that and . For any such and for small enough, we can find a real such that has a least zeros in the open disk . This contradicts the result of the previous paragraph for the sequence .
By the contradiction, let us assume that has no strictly positive lower bound on . We can then find a sequence such that
Because of (79), this sequence can be selected such that converges to . It yields
Consequently, there is an integer such that has at least one zero in . So, which is a contradiction.
We now prove Theorem 5
Let us consider , has an infinite number of zeros on . Let be such that any connected component of that contains such a zero is bounded and included in . Since the zeros of are isolated, every contains a finite number of zeros, and the collection of sets is infinite.
Let us consider a sequence of connected component of . Since these components are bounded, we can define as the closed contour of . Due to Lemma-4, there exists such that
By assumption is such that
Since is upper-bounded, it yields (developing the determinant)
Consequently, for large enough, we get
Rouch’s theorem implies that and have the same number of zeros inside of each (). Consequently has an infinite number of zeros on . So has an infinite number of zeros whose real part is strictly positive. This concludes the proof.
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