Delayrobust control design for two heterodirectional linear coupled hyperbolic PDEs
Abstract
We detail in this article the necessity of a change of paradigm for the delayrobust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical tradeoff between convergence rate and delayrobustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delayrobustness. Indeed, for such systems, using a backsteppingcontroller, 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.
I Introduction
In this paper, we highlight an important shortcoming of some control designs for systems of two heterodirectional linear firstorder hyperbolic Partial Differential Equations (PDEs). More precisely, we show that imposing finitetime 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 delayrobustness.
Most physical systems involving a transport phenomenon can be modeled using hyperbolic partial differential equations (PDEs): heat exchangers [42], open channel flow [16], [22], multiphase flow [25], [27] or power systems [41]. The backstepping approach [18, 31] has enabled the design of stabilizing fullstate 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 [36]), 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 [7]. The issues of noise sensitivity and performance tradeoff are considered in [8] 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 [37], 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 finitetime 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 delayrobustly stabilized, irrespective of the method. Specifically, we show that, for a system of two heterodirectional linear hyperbolic equations with antidiagonal source terms^{1}^{1}1For 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 onehalf, the system cannot be finite time stabilized robustly to delays.

Smaller than onehalf the system can be finitetime stabilized robustly to delays.
Our approach is the following: considering the control law proposed in [18] and using a backstepping approach, the controlled system is mapped to a distributed delay equation. Then, using the Laplace transform we derive the closedloop transfer function [19]. It is shown to be potentially unstable in presence of small delays. To ensure delayrobust stabilization, we propose some adjustments in the control law proposed in [18] by means of an additional degree of freedom enabling a tradeoff 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 byproduct of this analysis, detailed in Section IIIB is the reformulation of any system of two coupled linear hyperbolic equations as a zeroorder 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 wellknown 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 delayrobust 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.
Iia Description of the system
We consider the following linear hyperbolic system of two transport equations
(1)  
(2) 
evolving in , with the following linear boundary conditions
(3) 
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
(4) 
The initial conditions denoted and are assumed to belong to . In the following we define the characteristic time of the system as
(5) 
The product , product of the proximal and distal reflections, is called openloop gain of the system. We recall the following definition from [37].
Definition 1
Delayrobust stabilization [37].
The controller where is an operator, delayrobustly stabilizes the system (1)(3) in the sense of [37] if the resulting feedback system stabilizes the system (1)(3) in the sense of the norm and is delayrobustly stable with respect to small delays in the loop. A system is said to be delayrobustly stabilizable if and only if there exists such a .
IiB Openloop transfer function
In this section, we consider as the output of the system (1)(3). Using the method of characteristics, one can easily prove that satisfies the following delay equation.
(6) 
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
(7) 
The transfer function is the openloop transfer function of the system. Depending on the value of the openloop 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
(8) 
where is an arbitrary integer. Consequently using [37, Theorem 1.2] we have the following theorem
IiC Feedback control for an openloop gain smaller than one
IiC1 Finitetime stabilization
In this section, we focus on a system of transport equations for which the openloop gain satisfies . Although in this case the system (1)(3) is already exponentially stable in openloop, one could want to increase the convergence rate or to have finitetime 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 ([4]), 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 semiinfinite system that converges to zero in finite time. The corresponding control law is then defined by
(9) 
Consider now that there is a small delay in the actuation. The output is then solution of the following delay equation
(10) 
which is a zeroorder scalar neutral system. Using classical results on such systems [30], we get that, a necessary condition to have equation (10) stable for any delay is
(11) 
This means that for an openloop 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 [34], the authors design a controller preventing stickslip oscillations of a drillstring (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.
IiC2 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
(12) 
In presence of a delay in the actuation, we get, for the closedloop system, the following delay equation:
(13) 
which is now exponentially stable ([30]) for all if satisfies the following equation:
(14) 
Note that such a exists since (as we assumed ).
IiC3 Second solution: filtering the control law
A second approach to provide a delayrobust stabilization consists in filtering the control law. Let us consider e.g the control law defined by its Laplace transformation as
(15) 
where and are some coefficients that have still to be defined. In presence of a delay in the actuation, we get, for the closedloop system, the following delay equation:
(16) 
which is exponentially stable (see [29]) for all if
(17) 
IiC4 Concluding remarks
Throughout the analysis of a simple system of two transport equations, we have proved (using the results from [37]) 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 , finitetime convergence is not a reasonable objective since the corresponding controller is not always robust to delays. This means that to have delayrobustness one may have to give up finitetime 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 delayrobust stabilization) still holds in the case where indomain couplings exist. To do so, using a classical backstepping transformation (see [18]), the original system is mapped to a distributeddelay 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.
Iiia Description of the system
We consider the following linear hyperbolic system which appear in SaintVenant equations, heat exchangers equations and other linear hyperbolic balance laws (see [10]).
(18)  
(19) 
evolving in , with the following linear boundary conditions
(20) 
The insidedomain coupling terms and can be spatiallyvarying, whereas the boundary coupling terms and , and the velocities and are assumed to be constant. Moreover, we assume that
(21) 
The initial conditions denoted and are assumed to belong to . This system is pictured in Figure 1.
IiiB A distributeddelay differential equation
In this section, by means of a classical backstepping transformation, the original system (18)(20) is mapped to a neutral system with distributeddelay.
IiiB1 Volterra transformation: removing insidedomain couplings
We consider the following Volterra change of coordinates defined in [18] by
(22)  
(23) 
where the kernels are defined on by the following set of hyperbolic PDEs:
(24)  
(25)  
(26)  
(27) 
with the following set of boundary conditions:
(28)  
(29) 
Lemma 1 ( [18])
The dynamics of the system in the new coordinates is:
(32)  
(33) 
with the following linear boundary conditions
(34) 
with
(35)  
(36) 
IiiB2 Neutral equation with distributed delay
Using the method of characteristics on equations (32)(33) yields (for all , for all )
(37)  
(38) 
Consequently, combining equations (37)(38) and equation (34), we get:
Thus,
(39) 
where is defined by (5) and where is defined by
(40) 
where for any interval I, is defined by
(41) 
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).
Remark 1
IiiC Openloop analysis
In this section, we consider as the output of the system (32)(33). Taking the Laplace transform of (39), we get
(42) 
We then have the following lemma
Lemma 2
If , then the openloop transfer function has an infinite number of poles with a positive real part.
Proof 1
We assume here that the open loop gain is positive (the case can be treated in a similar way). The poles of the openloop transfer function are the solutions of
(43) 
is differentiable almost everywhere on . Integrating by parts yields
(44) 
where is a bounded function that represents the jumps appearing in the integration by parts. We denote in the following
(45)  
(46) 
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
Proof 2
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 openloop 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 openloop gain is greater than one, one cannot find a controller whose delay margin is nonnull. Consequently, there is a whole class of hyperbolic systems that cannot be delayrobustly stabilized.
Remark 3
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.
IiiD Feedback control for an open loop gain smaller than one
IiiD1 Finitetime stabilization
In this section, we focus on a system of hyperbolic equations for which the openloop gain satisfies . Note that the uncontrolled system can be unstable due to the insidedomain couplings and (see [10]). In [18], using the backstepping approach, a control law that ensures finitetime stabilization of the original system was derived. This control law is defined by
(47) 
where the kernels and are defined by equations (24)(29) and are defined by equations (35)(36).
Consider now that there is a small delay in the actuation. The output is then solution of the following delay equation
(48) 
where is defined by (40). We denote as
(49) 
Taking the Laplace transform of equation (48), we get
(50) 
We can now state the following Theorem:
Theorem 3
Proof 3
This proof uses the same idea as the one used for the proof of Lemma 2. Let us denote
(51) 
and
(52) 
where . Choosing small enough, we have that . Consequently, has an infinite number of roots whose real parts are positive (see [30]). 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 [18] 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 finitetime convergence.
Remark 4
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 insidedomain couplings

the term whose objective is to cancel the proximal reflection and to ensure finitetime 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 IIC2 for the case of transport equations, we proposed a similar adjustment for the control law (47) when .
Iva Control law
Let us consider the following control law:
(53) 
where , , and are defined by equations (24)(29) and where, similarly to (12), the coefficient is chosen such that
(54) 
The objective of such a control law is preserve a small amount of proximal reflection in the target system to ensure delayrobustness, while eliminating insidedomain couplings.
Remark 5
The control law can be rewritten as
(55) 
Using the backstepping transformation (22)(23), the system (18)(20) is mapped to
(56)  
(57) 
with the boundary conditions
(58) 
Proof 4
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))
(59) 
since . The other cases can be treated similarly.
Remark 6
The coefficient can be interpreted as a tuning parameter, enabling a tradeoff between performance (convergence rate) and robustness with respect to delays. This parameter has a role similar to the coefficient introduced in [8] in the design of the observer to enable a tradeoff between performance and noise sensitivity.
Remark 7
We need now to prove that the proposed control law is robust with respect to small delays. We have the following theorem.
Theorem 4
Proof 5
Consider a positive delay . Consider the two states and defined by (22)(23). Slightly adjusting the method used to derive (48), we get the following equation satisfied by the output .
(60) 
where is defined by (40). Taking the Laplace transform yields the following characteristic equation
(61) 
where is defined by
(62) 
Let us now consider a complex number such that . We then have
(63) 
Since satisfies (54), there exists such that
(64) 
Let us now focus on the term . We have
The integral on the expression above can be rewritten
(65) 
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
(66) 
Consequently, for , we have
(67) 
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.
Remark 8
For a given value of , the parameter gives a range for admissible delays. However, is not necessarily the maximum admissible delay.
IvB Interpretation of the results and outlook
IvB1 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 KelvinVoigt damping, or other phenomena that would be susceptible of making the delay margins nonnull. However, these results do indicate

that the delayrobustness margins would be very poor for such systems
In this regard, a more quantitative approach to analyzing the performance–delayrobustness tradeoffs 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 tradeoffs has been taken in [7]. In the next sections, we analyze the impact of the results on broader classes of systems.
IvB2 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 predictorlike feedback laws, that focus on stabilizing the (potentially unstable) ODE. This approach, although rigorously correct, is bound to exhibit poor delayrobustness in practice, as detailed in the previous section. To illustrate this point, we consider the following example.
Example 1
Let us consider the following ODEPDE system
(68)  
(69) 
with the boundary conditions
(70)  
(71) 
where satisfies
(72) 
where , and are nonnull constants. Considering as the output of the system and taking the Laplace transform, we get the following characteristic equation
(73) 
This yields
(74) 
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 ODEPDEs systems.
Interestingly, this opens new perspectives for the control of PDEODE 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 closedloop 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 closedloop system is a cascade. In this regard, the stability analysis methods for such systems developed in [20, 38, 39] will be instrumental.
IvB3 Systems with multiple equations
Some physical systems require more than two equations to be properly modeled. For instance, DriftFlux Models described in [1] 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., [3]). To provide finitetime convergence, using a control law such as the one described in [26] or [9] 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 delayrobustness. Applying a transformation similar to the one proposed in section IIIB and writing the corresponding characteristic equation would lead to a matrix neutral equation with distributed delays. One can then use classical method ([30]) 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.
(75) 
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 [18] 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 zeroequilibrium 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 finitetime 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 delayrobust stabilization. This has been done by means of a tuning parameter enabling a tradeoff between convergence rate and delayrobustness. Even if the delayrobustness properties of the outputfeedback 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 controllerdesign for PDEsODEs systems or systems with more than two PDEs. That will be the purpose of future contributions.
Acknowledgment
We thank Laurent Praly, Delphine BreschPietri, 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
(76) 
where is the identity matrix of dimension . For any real number , we denote the open halfplane . We have the following general theorem
Theorem 5
Let us consider an holomorphic function such that
(77) 
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 [12]. For any positive , we denote the set of complex numbers whose distance to the zeros of det is at most :
(78) 
We start proving the following lemma
Lemma 4
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
Proof 6
This proof is similar to the one given in [12]. By continuity of the determinant we have
(79) 
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
(80) 
If there is an unbounded connected component of , then there exists a sequence () of distinct zeros of such that for any , . It yields
(81) 
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
(82) 
The sequence is locally bounded on uniformly in . Consequently, there exists a subsequence 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
(83) 
Because of (79), this sequence can be selected such that converges to . It yields
(84)  
(85) 
Consequently, there is an integer such that has at least one zero in . So, which is a contradiction.
We now prove Theorem 5
Proof 7
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 Lemma4, there exists such that
(86) 
By assumption is such that
(87) 
Since is upperbounded, it yields (developing the determinant)
(88) 
Consequently, for large enough, we get
(89) 
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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