The aim of this paper is to investigate the use of Pyragas control on the stability of stationary, localized coherent structures in a general class of twocomponent, singularly perturbed, reactiondiffusion systems. We use noninvasive Pyragaslike proportional feedback control to stabilize a singular pulse solution to a twocomponent, singularly perturbed reactiondiffusion system. We show that in a significant region of parameter space, the control can be adjusted to stabilize an otherwise unstable pulse.
Singularly perturbed pulses in twocomponent reactiondiffusion equations are generally unstable in large regions of parameter space. To address this instability, we utilize Pyragas control, which was originally developed for periodic solutions of ordinary differential equations, to create a noninvasive feedback control for singularly perturbed pulses in reactiondiffusion systems. We prove the effectiveness of this control technique by analyzing the spectral stability of the controlled singular pulse in a toy model, using Evans function techniques.
I. INTRODUCTION
Reactiondiffusion systems are mathematical models based on semilinear parabolic partial differential equations.^{1} One of the most fascinating aspects of reactiondiffusion equations is the plethora of patterns that can emerge from their solutions. These patterns include such diverse phenomena spiral waves in a chemical oscillator,^{2} the different animal coat patterns,^{3} Faraday waves,^{4} or various patterns in Rayleigh–Bénard convection,^{5} geology,^{6} or fluid dynamics.^{7}
These patterns, of which some can be classified as Turing patterns,^{8} exhibit a variety of behaviors, including the formation of traveling waves and wavelike phenomena, as well as selforganized structures such as stripes, hexagons, and dissipative solitons.^{9} The study of these patterns has significant implications for understanding natural phenomena and has led to new insights in various fields.^{10}
The focus of this article is on the analysis and control of a specific spatially localized coherent structure in a twocomponent reactiondiffusion equation: a symmetric singular pulse.^{11–13,27,29} An illustration of this pulse is presented in Fig. 1. It is characterized by a noticeable scale separation, which is evident in the difference in pulse width between the two components.
It has been shown^{11,12} that such twocomponent pulses can only be stable when the nonlinear interaction between the two components is sufficiently strong. Even when this is the case, singular pulses are unstable for large regions in parameter space, see Ref. 12 (Lemmas 5.11, 5.12, and 5.14). As such, singular pulses are infrequently observed in nature or experiments. Often only the progression toward a stable steady state is observable, while unstable steady states remain largely imperceptible.
It is, therefore, our goal to introduce a Pyragaslike control term to make unstable pulses visible. Pyragas control^{14,15} is advantageous as it is noninvasive on the pulses, i.e., the control term vanishes and does not change the pulse itself. However, it changes the nearby solutions and thereby the stability properties of the pulse. Another benefit of this control approach is its modelindependence and low implementation cost, as it does not require expensive calculations. Although originally designed for periodic solutions of ordinary differential equations, we adapt it for controlling singularly perturbed pulses in reactiondiffusion systems.
We aim to integrate control theory with the theory of pattern existence and stability in singularly perturbed reactiondiffusion systems. Specifically, we aim to control the stability of pulse solutions and develop a novel methodology to regulate the stability of diverse patterns in a broad class of singularly perturbed reactiondiffusion systems.
This paper is organized as follows: In Sec. II, we discuss our model system and introduce the singular pulse as the pattern of interest; we also introduce noninvasive control terms. Next, in Sec. III, we construct an Evans function to determine the spectral stability of the singular pulse. Our main result is presented in Sec. IV, where we demonstrate that noninvasive feedback stabilization can be achieved in a large region of parameter space. We provide a brief summary of the stability proof in this section, with a complete and detailed version of the proof available in the Appendix. We conclude with a short discussion in Sec. V.
II. MODEL, PATTERNS AND CONTROL
A. Model
B. Patterns
C. Control
Note that, consequently, on the linear level, the control is not a multiple of the identity matrix. Therefore, the influence of the control term $\u2113$ is not as straightforward as shifting all eigenvalues to the left. Our goal is to derive conditions on the control function $\u2113$ such that the singular pulse $ ( u p , v p ) $ (5) is a stable solution to (7).
III. PULSE STABILITY AND THE EVANS FUNCTION
The essential spectrum of $ L $ (9) is real, negative, and bounded away from the imaginary axis;^{12} hence, the pulse stability is determined by its discrete spectrum, i.e., the roots of the associated Evans function $ E (\lambda )$. In Ref 12, it is shown that the singularly perturbed structure of the pulse can be used to obtain an explicit characterization of the roots of $ E (\lambda )$ to leading order in $\epsilon $; moreover, these roots perturb regularly in $\epsilon $.
A typical configuration of the spectrum of $ L $ (9) is shown in Fig. 2. The pulse can lose stability when a pair of eigenvalues crosses the imaginary axis (a Hopf bifurcation) or when a real eigenvalue passes through the origin. Note that the pulse spectrum consists of both discrete (point) spectrum and essential (continuous) spectrum, as the spatial domain is unbounded.
IV. PULSE STABILIZATION THROUGH PROPORTIONAL FEEDBACK CONTROL
The main research question that we address in this paper is: Given a singular pulse solution $( u p , v p )$ to (1), can we find control terms K, L such that this singular pulse is a stable solution to (6)?
As the introduction of control terms has a (potentially) significant influence on the stability analysis of the singular pulse, we try to answer the research question formulated above in the context of the toy problem (2), with proportional feedback control in v—Eq. (7). We first present the main outcome of our analysis in Theorem 1 and describe the main ideas of the proof. The full proof of Theorem 1 can be found in the Appendix.
Let $0<\epsilon \u226a1$ be sufficiently small, and assume that $ u \u2217 $ is a nondegenerate solution to (4). Consider the symmetric singular pulse solution $( u p , v p )$ to (2), which is to leading order in $\epsilon $ given by (5), and introduce $\rho :=2 f \u2032 ( u \u2217 ) f ( u \u2217 ) + T o \u2032 ( u \u2217 ) T o ( u \u2217 ) $.

If $f \u2032 ( u \u2217 )=0$, then the singular pulse $( u p , v p )$ is always unstable for any choice of proportional control function $\u2113(v\u2212 v p )$ as implemented in (7).

If $f \u2032 ( u \u2217 )<0$, then it is possible to choose a proportional control function $\u2113(v\u2212 v p )$, as implemented in (7), such that the singular pulse $( u p , v p )$ is stable.

Let $f \u2032 ( u \u2217 )>0$.

If $\rho > 1 u \u2217 $, then the singular pulse $( u p , v p )$ is always unstable for any choice of proportional control function $\u2113(v\u2212 v p )$ as implemented in (7).

If $\rho < 1 u \u2217 $, then it is possible to choose a proportional control function $\u2113(v\u2212 v p )$, as implemented in (7), such that the pulse solution $( u p , v p )$ is stable.

A visual representation of the statement of Theorem 1 is given in Fig. 3. A direct application of Theorem 1 for specific parameter values is shown in Figs. 4 and 5.
It is worthwhile to note that control on one variable only suffices to control both components of the pulse.
Specific conditions that the control function $\u2113$ needs to satisfy to stabilize the pulse can be found in the proof of Lemma 4 in the Appendix. In particular, the essential spectrum is stable if and only if $ \u2113 \u2032 (0)<1$ (A4).
Moreover, note that the controllability of the pulse strongly depends on the quantity $\rho =2(f \u2032 /f)+( T 0 \u2032 / T 0 )$, which corresponds to the logarithmic derivative of $ f 2 (u) T 0 (u)$. This quantity represents the udependent nonlinearity of (2)(a) at $u= u \u2217 $.
While the purpose of this paper is to show the stabilization of pulse that is unstable in the absence of control, it is worthwhile to note that our control scheme can also destabilize an otherwise stable pulse. For example, choosing $ \u2113 \u2032 (0)<1$ destabilizes the pulse through a sideband instability, as the essential spectrum is pushed through the imaginary axis (A4). We do not explore such destabilization scenarios in the current paper; the desirability of pulse destabilization through noninvasive control depends on the model context and application.
V. CONCLUSION AND OUTLOOK
In conclusion, we have demonstrated the possibility of stabilizing singularly perturbed pulses in twocomponent reactiondiffusion equations across significant areas of the parameter plane. Specifically, for the toy model (2), we have identified three regions in the $(\rho ,f \u2032 ( u \u2217 ))$plane: a stable region where control is unnecessary, an unstable region where control is always insufficient, and—our main interest here—a large region where pulse stability can be controlled through our proposed control term.
Several potential avenues for future investigation emerge. First, it may be worthwhile to explore the use of a “true” Pyragas control, in which the control is not proportional but includes a delay term, and as such does not rely on explicit preexisting knowledge of the pulse structure. This poses additional problems in the rigorous stability analysis as difficulties may arise concerning nonlinear stability. For the control scheme investigated in this paper, it was not necessary to differentiate between spectral and nonlinear stability. In the noncontrolled eigenvalue problem (8), spectral stability implies nonlinear stability; this follows from a classical result by Henry^{1} because the operator $ L $ (9) is sectorial. The linear operator associated to the “controlled” eigenvalue problem (12) has the same (sectorial) property. Hence, one can directly infer nonlinear stability from spectral stability. For other control strategies, such as delayed feedback, the sectorial property is generally not retained. In such a case, one needs to be more careful about deducing nonlinear stability; see, e.g., Ref. 25 for generalizations of Henry’s result.
Second, investigating the impact of control on the large scale component u in comparison to the current focus on the small scale component v would provide valuable insights for this twocomponent system. Additionally, exploring the potential of nondiagonal controls would also be of interest.
Third, our current analysis has focused on relatively simple spatially localized coherent structures. It would be interesting to investigate the potential extension of our control scheme to more complex patterns, such as multicircuit configurations^{11} and/or periodic pulse patterns,^{26} and to determine the conditions under which stabilization can be achieved for these more intricate situations.
ACKNOWLEDGMENTS
F.V. was supported by a Humboldt Fellowship. I.S. was supported by the Deutsche Forschungsgemeinschaft, SFB 910, Project A4 “SpatioTemporal Patterns: Control, Delays, and Design.” We would like to thank all the members of SFB 910, in particular Sabine Klapp, Eckehard Schöll, and Bernold Fiedler, for their valuable contributions and continuous support. In addition, we would like to express our gratitude to Nigel Higson for insightful discussions on Weyl’s theorem.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
F. Veerman: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). I. Schneider: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: PROOF OF THEOREM 1
1. Pulse stability and the Evans function
2. An explicit expression for the Evans function
Given $\alpha ,\beta \u2208 R $, $\beta \u22600$ fixed, can we choose the control parameter $ \u2113 \u2032 (0)<1$ such, that the (complex) $ \lambda ^ $solutions to (A18) outside the essential spectrum $ \sigma ^ ess $ (A9) all lie to the left of the line $ Re \lambda ^ =\u2212 \u2113 \u2032 (0)$?
3. Spectral decomposition in the presence of essential spectrum
Let $R( \lambda ^ )= \u27e8 v ^ in ( \u22c5 , \lambda ^ ) , v ^ p \u27e9 2 $ be as in (A29). Then the following statements hold:

$sgn ImR( \lambda ^ )=\u2212 s g n I m \lambda ^ $;

There exists $c>0$ such that, if $ R e \lambda ^ >c$, then $ R e R( \lambda ^ )>0$;

For real $ \lambda ^ > \lambda ^ 0 $, $R( \lambda ^ )$ is positive and strictly monotonically decreasing;

There exist $ d 1 , 2 >0$ such that, if $\u2212 d 1 < R e \lambda ^ < d 2 $, then $ R e R( \lambda ^ )<0$.

Denoting $ \lambda ^ =a+ib$, we have $ Im k \lambda ^ + m =b k ( a + m ) 2 + b 2 $ for $k,m\u2208 R $. From this observation, it immediately follows that $ sgn Im R c ( \lambda ^ )=\u2212sgn Im \lambda ^ $. For $ R d $, we estimate $ 75 ( a \u2212 5 4 ) 2 + b 2 \u2212 1 ( a + 3 4 ) 2 + b 2 > 75 \u2212 1 b 2 + max ( ( a \u2212 5 4 ) 2 , ( a + 3 4 ) 2 ) >0$, from which it follows that $sgn Im R d ( \lambda ^ )=\u2212sgn Im \lambda ^ $.

Using the same notation as in (I), we have $ Re k \lambda ^ + m = k ( a + m ) ( a + m ) 2 + b 2 $ for $k,m\u2208 R $. From this observation, it immediately follows that $ Re R c ( \lambda ^ )>0$ for all $a>0$. For $ R d $, we find that $ 75 ( a \u2212 5 4 ) ( a \u2212 5 4 ) 2 + b 2 \u2212 a + 3 4 ( a + 3 4 ) 2 + b 2 >0$ for $a> 75 \u22c5 5 4 + 1 \u22c5 3 4 75 \u2212 1 >0$ and any $b\u2208 R $.

If $ \lambda ^ 2 > \lambda ^ 1 >0$, then $ k \lambda 2 ^ + m < k \lambda 1 ^ + m $ for $k,m>0$; hence, it follows that $ R c ( \lambda ^ )$ is strictly monotonically decreasing for real $ \lambda ^ >0$. Moreover, from the proof of (II), we know that $ R c ( \lambda ^ )$ is positive for real $ \lambda ^ >0$. For $ R d ( \lambda ^ )$, we calculate $ d R d d \lambda ^ = 81 \pi 2 8192 ( \u2212 75 ( \lambda ^ \u2212 5 4 ) 2 + 1 ( \lambda ^ + 3 4 ) 2 ) <0$ for $ \lambda ^ > \lambda ^ 0 = 5 4 $. Moreover, $ lim \lambda ^ \u2192 \u221e R d ( \lambda ^ )=0$ and $ R d ( \lambda ^ )\u2192+\u221e$ as $ \lambda ^ \u2193 \lambda ^ 0 = 5 4 $; we conclude that $ R d $ is positive and strictly monotonically decreasing for real $ \lambda ^ > \lambda ^ 0 $.
 For $ R d $, we find that $ 75 ( a \u2212 5 4 ) ( a \u2212 5 4 ) 2 + b 2 \u2212 a + 3 4 ( a + 3 4 ) 2 + b 2 <0$ for $\u2212 3 4 <a< 5 4 $ and any $b\u2208 R $. For $ R c $, we use the estimates $ \kappa 2 + 9 4 > \kappa 2 +1$, $ \kappa 2 + 1 4 > \kappa 2 $ and $ Re 1 \lambda ^ + \kappa 2 + 1 = a + \kappa 2 + 1 ( a + \kappa 2 + 1 ) 2 + b 2 \u2264 1 \kappa 2 + 1 $ when $a\u22650$, to obtain$ Re R c ( \lambda ^ ) \u2264 \u222b 1 \u221e 9 \pi 2 16 \kappa 2 csch 2 \pi \kappa d \kappa = 9 \pi 16 [ 3 + coth \pi \u2212 2 \pi log \u2061 ( \u2212 1 + e 2 \pi ) + 1 \pi 2 Li 2 ( e \u2212 2 \pi ) ] \u2248 9.05 \xd7 10 \u2212 3 . $
The claim follows by continuity of $ Re R= Re R d + Re R c $ in $(a,b)$ for $\u2212 3 4 <a< 5 4 $.
4. Results on proportional feedback control
Using the results obtained in the previous sections, we can formulate the following lemma on solutions to Eq. (A18):
Consider Eq. (A18) for $ \lambda ^ \u2208 C $ with $\alpha ,\beta \u2208 R $ and $ \u2113 \u2032 (0)<1$, where $ v ^ in $ is the unique bounded solution to (A16), and $ v ^ p (\xi )= 3 2 sech 2 \xi 2 $, cf. (5). Then the following statements hold:

If $\beta >0$, then all solutions $ \lambda ^ $ to (A18) lie on the real axis.

If $\beta >0$ and $\alpha \u22640$, then there always exists a real, positive solution $ \lambda ^ > \lambda ^ 0 $ to (A18) for any $ \u2113 \u2032 (0)\u2208 R $.

If $\u2212\alpha \u2265\beta >0$, then for any $ \u2113 \u2032 (0)\u2208 R $, there exists a real, positive solution $ \lambda ^ >\u2212 \u2113 \u2032 (0)$ to (A18).

If $0\u2264\u2212\alpha <\beta $, then there exists a value $ \u2113 \u2032 (0)<\u2212 \lambda ^ 0 $ such that all solutions $ \lambda ^ $ to (A18) obey $ Re \lambda ^ <\u2212\u2113(0)$.

If $\beta >0$ and $\alpha >0$, then there exists a value $ \u2113 \u2032 (0)<\u2212(1+ \lambda ^ 0 )$ such that (A18) has no solutions.

If $\beta <0$, then there exists $ \Lambda ^ <0$ such that, when $ \u2113 \u2032 (0)< \Lambda ^ $, all solutions $ \lambda ^ $ to (A18) lie to the left of the vertical line $ { Re \lambda ^ = c} $, with $c>0$ independent of $\alpha ,\beta $ and $ \u2113 \u2032 (0)$.
1. If $\beta >0$, then all solutions to (A18) lie on the real axis. For the principal complex square root $ z $, which is defined for all z away from the negative real line, it holds that $Re z >0$ and $sgn Im z =sgn Imz$. It follows that, if $\beta >0$, $ImL( \lambda ^ )$ can only equal $ImR( \lambda ^ )$ if both are zero, by Lemma 3 (I).
2. If $\beta >0$ and $\alpha \u22640$, then there always exists a real, positive solution $ \lambda ^ > \lambda ^ 0 $ for any $ \u2113 \u2032 (0)\u2208 R $. By statement 1, we take $ \lambda ^ \u2208 R $. Both $ R d $ and $ R c $ are positive for sufficiently large $ \lambda ^ $ by Lemma 3 (II); moreover, $R( \lambda ^ )\u21920$ as $ \lambda ^ \u2192\u221e$. It follows that $L( \lambda ^ )>R( \lambda ^ )$ for sufficiently large $ \lambda ^ $, since $\beta >0$. Moreover, from the observation that $R( \lambda ^ )\u2192+\u221e$ as $ \lambda ^ \u2193 \lambda ^ 0 = 5 4 $, combined with the fact that $L( \lambda ^ )$ is continuous for $ \lambda ^ \u2265\u22121\u2212 \u2113 \u2032 (0)$ and $L( \lambda ^ )\u2192\alpha \u22640$ as $ \lambda ^ \u2193\u22121\u2212 \u2113 \u2032 (0)$, it follows by continuity of $R( \lambda ^ )$ for $ \lambda ^ > \lambda ^ 0 = 5 4 $ that there exists a real, positive solution to (A18), that lies to the right of the point $ \lambda ^ = \lambda ^ 0 $ and to the right of the point $ \lambda ^ =\u22121\u2212 \u2113 \u2032 (0)$.
3. If $\u2212\alpha \u2265\beta >0$, then for any $ \u2113 \u2032 ( 0 ) \u2208 R $, there exists a real, positive solution $ \lambda ^ >\u2212 \u2113 \u2032 (0)$ to (A18). Since $R( \lambda ^ )$ is positive for real $ \lambda ^ > \lambda ^ 0 $ by Lemma 3 (III), the intersection of the graphs $R( \lambda ^ )$ and $L( \lambda ^ )$ lies above the horizontal $ \lambda ^ $ axis. Since $\beta >0$ and $\alpha <0$, the graph of $L( \lambda ^ )$ is strictly monotonically increasing on its domain and intersects the horizontal axis before intersecting the graph of $R( \lambda ^ )$, as $ \lambda ^ $ increases. Denote the $ \lambda ^ $value for which the two graphs intersect as $ \lambda ^ \u2217 $. From the monotonicity of the graphs, it follows that $ \lambda ^ \u2217 $ is the largest solution to (A18). Moreover, as $L( \lambda ^ )$ if and only if $ \lambda ^ = ( \alpha \beta ) 2 \u22121\u2212 \u2113 \u2032 (0)$, we conclude that $ \lambda ^ \u2217 > ( \alpha \beta ) 2 \u22121\u2212 \u2113 \u2032 (0)>\u2212 \u2113 \u2032 (0)$ since $\u2212\alpha \u2265\beta $.
4. If $0\u2264\u2212\alpha <\beta $, then there exists a value $ \u2113 \u2032 (0)<\u2212 \lambda ^ 0 $ such that all solutions $ \lambda ^ $ to (A18) obey $ Re \lambda ^ < \u2212 \u2113 ( 0 ) $. We calculate $L(\u22121\u2212 \u2113 \u2032 (0))=\alpha \u22640$ and $L(\u2212 \u2113 \u2032 (0))=\alpha +\beta >0$. Hence, the largest solution $ \lambda ^ \u2217 $ to (A18) is contained in the interval $(\u22121\u2212 \u2113 \u2032 (0),\u2212 \u2113 \u2032 (0))$ if and only if $R(\u2212 \u2113 \u2032 (0))<\alpha +\beta $ by the monotonicity and continuity of $L( \lambda ^ )$ and $R( \lambda ^ )$. From Lemma 3 (III), it follows that $R(\u2212 \u2113 \u2032 (0))<\alpha +\beta $ for sufficiently large $  \u2113 \u2032 (0)  $.
5. If $\beta >0$ and $\alpha >0$, then there exists a value $ \u2113 \u2032 ( 0 ) < \u2212 ( 1 + \lambda ^ 0 ) $ such that (A18) has no solutions. By Lemma 3 (III), there exists $ k ^ > \lambda ^ 0 $ such that $R( \lambda ^ )<\alpha $ for all $ \lambda ^ > k ^ $. Choosing $ \u2113 \u2032 (0)=\u22121\u2212 k ^ $, so that $L( k ^ )=L(\u22121\u2212 \u2113 \u2032 (0))=\alpha $, we see that (A18) cannot have any real solutions, which implies that (A18) does not have any (complex) solutions by statement 1.
6. If $ \beta < 0 $, then there exists $ \Lambda ^ < 0 $ such that, when $ \u2113 \u2032 ( 0 ) < \Lambda ^ $, all solutions to (A18) lie to the left of the vertical line $ { Re \lambda ^ = c} $, with $ c > 0 $ independent of $\alpha ,\beta $, and $ \u2113 \u2032 ( 0 ) $.
Suppose $\alpha \u22640$, and consider $ \lambda ^ \u2208 R $. We can choose $ \u2113 \u2032 (0)<\u2212(1+ \lambda ^ 0 )$, making sure that the domain of $L( \lambda ^ )$ lies entirely to the right of $ \lambda ^ = \lambda ^ 0 $. On this domain, we have $L( \lambda ^ )<0$, while $R( \lambda ^ )>0$, from which we conclude that no real solutions to (A18) exist. The same reasoning can be extended to complex solutions of (A18), under the same assumption that $\alpha \u22640$: because $Re z >0$ for all z away from the negative real line, we have that $ Re L( \lambda ^ )<0$ for all $ \lambda ^ $ on its domain, while there exists a $c>0$ such that $ Re R( \lambda ^ )>0$ for all $ \lambda ^ $ to the right of the line $ { Re \lambda ^ = c} $ by Lemma 3 (II).
Now suppose $\alpha >0$. For $ \lambda ^ \u2208 R $, the same reasoning as in the case $\alpha \u22640$ can be applied, choosing $ \u2113 \u2032 (0)< ( \u2212 \alpha \beta ) 2 \u22121\u2212 \lambda ^ 0 $. The reasoning for complex $ \lambda ^ $ is also analogous to the case $\alpha \u22640$, now with $ \u2113 \u2032 (0)< ( \u2212 \alpha \beta ) 2 \u22121\u2212c$, again using Lemma 3 (II).
5. Finishing the proof of Theorem 1
The results in Lemma 4 can now be used prove Theorem 1, restated here for convenience:
Let $0<\epsilon \u226a1$ be sufficiently small, and assume that $ u \u2217 $ is a nondegenerate solution to (4). Consider the symmetric singular pulse solution $( u p , v p )$ to (2), which is to leading order in $\epsilon $ given by (5), and introduce $\nu :=2 f \u2032 ( u \u2217 ) f ( u \u2217 ) + T o \u2032 ( u \u2217 ) T o ( u \u2217 ) $.

If $f \u2032 ( u \u2217 )=0$, then the singular pulse $( u p , v p )$ is always unstable for any choice of proportional control function $\u2113(v\u2212 v p )$ as implemented in (7).

If $f \u2032 ( u \u2217 )<0$, then it is possible to choose a proportional control function $\u2113(v\u2212 v p )$, as implemented in (7), such that the singular pulse $( u p , v p )$ is stable.

Let $f \u2032 ( u \u2217 )>0$.

If $\rho > 1 u \u2217 $, then the singular pulse $( u p , v p )$ is always unstable for any choice of proportional control function $\u2113(v\u2212 v p )$ as implemented in (7).

If $\rho < 1 u \u2217 $, then it is possible to choose a proportional control function $\u2113(v\u2212 v p )$, as implemented in (7), such that the pulse solution $( u p , v p )$ is stable.

The pulse solution $( u p , v p )$ is spectrally stable if and only if, for the shifted eigenvalue $ \lambda ^ =\lambda \u2212 \u2113 \u2032 (0)$ (A7), the condition $ Re \lambda ^ <\u2212 \u2113 \u2032 (0)$ (A8) is satisfied. Eigenvalues $ \lambda ^ $ correspond to zeroes of the Evans function $ E ( \lambda ^ ,\epsilon )$ (A10). These zeroes are, to leading order in $\epsilon $, given by the solutions to (A17).
Claim 1 follows from the observation that the only solution $ v in $ to (A14) is the trivial solution when $f \u2032 ( u \u2217 )=0$. Hence, the slow transmission function $ t s $ (A15) does not have a pole at $ \lambda ^ = \lambda ^ 0 $; therefore, no zeropole cancellation in the Evans function (A10)—as detailed in Ref. 12—takes place. It follows that the positive real zero near $ \lambda ^ 0 $ of the fast transmission function $ t f $ is also a zero of the full Evans function $ E $, which means the pulse solution $( u p , v p )$ is unstable.
If $f \u2032 ( u \u2217 )\u22600$, then (A17) is of the form (A18), with $\alpha =\u22126\u22123 T o \u2032 ( u \u2217 ) T o ( u \u2217 ) f ( u \u2217 ) f \u2032 ( u \u2217 ) $ and $\beta =3 f ( u \u2217 ) f \u2032 ( u \u2217 ) 1 u \u2217 $.
When $\beta <0$, it follows from Lemma 4 (6) that there is a $c>0$ such that all solutions of (A18) lie to the left of the line $ { Re \lambda ^ = c} $. Since $c>0$ is, in particular, independent of $ \u2113 \u2032 (0)$, and the statement holds for all $ \u2113 \u2032 (0)< \Lambda ^ <0$, we can choose $ \u2113 \u2032 (0)<\u2212c$, which implies that all solutions of (A18) obey the stability criterion $ Re \lambda ^ <\u2212 \u2113 \u2032 (0)$. Since $\beta <0$ if and only if $f \u2032 ( u \u2217 )<0$, this proves claim 2.
When $\alpha >0$ and $\beta >0$, it follows from Lemma 4 (5) that (A18) does not have any solutions when $ \u2113 \u2032 (0)$ is negative and $  \u2113 \u2032 (0)  $ is sufficiently large. As $f \u2032 ( u \u2217 )\u22600$, the zeroes of $ t f ( \lambda ^ ,\epsilon )$ near $ \lambda ^ 0 $ and $ \lambda ^ 2 $ are canceled by the poles of $ t s ( \lambda ^ ,\epsilon )$; the remaining zero $ \lambda ^ 1 =0$ is stable for $ \u2113 \u2032 (0)<0$. It follows that all zeroes the Evans function $ E ( \lambda ^ ,\epsilon )$ (A10) obey $ Re \lambda ^ <\u2212 \u2113 \u2032 (0)$, which implies that $( u p , v p )$ is spectrally stable. The condition $\alpha >0$ is equivalent to $ T o \u2032 ( u \u2217 ) T o ( u \u2217 ) f ( u \u2217 ) f \u2032 ( u \u2217 ) >\u22122$, while $\beta >0$ is equivalent to $ f ( u \u2217 ) f \u2032 ( u \u2217 ) 1 u \u2217 >0$. Since both $ u \u2217 >0$ and $f( u \u2217 )>0)$ by assumption, the latter condition is equivalent to $f \u2032 ( u \u2217 )>0$. This allows us to rewrite the $\alpha >0$condition as $ T o \u2032 ( u \u2217 ) T o ( u \u2217 ) <\u22122 f \u2032 ( u \u2217 ) f ( u \u2217 ) $, thereby proving the second part of claim 3 for $2 f \u2032 ( u \u2217 ) f ( u \u2217 ) + T o \u2032 ( u \u2217 ) T o ( u \u2217 ) <0$.
When $0\u2264\u2212\alpha <\beta $, it follows from Lemma 4 (4) that all solutions to (A18) obey the stability criterion $ Re \lambda ^ <\u2212 \u2113 \u2032 (0)$. As before, the condition $\beta >0$ is equivalent to $f \u2032 ( u \u2217 )$, while the condition $\beta >\u2212\alpha \u22650$ can be rewritten as $0\u22642 f \u2032 ( u \u2217 ) f ( u \u2217 ) + T o \u2032 ( u \u2217 ) T o ( u \u2217 ) < 1 u \u2217 $, which finalizes the proof of claim 3 b).
When $\u2212\alpha >\beta >0$, it follows from Lemma 4 (3) that there always exists a solution to (A18) that does not satisfy the stability criterion $ Re \lambda ^ <\u2212 \u2113 \u2032 (0)$, regardless of the value of $ \u2113 \u2032 (0)$. This implies that, under the condition $\u2212\alpha >\beta >0$, the pulse $( u p , v p )$ is unstable for any choice of the control function $\u2113$. The condition $\beta >0$ is equivalent to $f \u2032 ( u \u2217 )>0$, while the condition $\u2212\alpha >\beta $ can be rewritten as $ f \u2032 ( u \u2217 ) f ( u \u2217 ) + T o \u2032 ( u \u2217 ) T o ( u \u2217 ) > 1 u \u2217 $, which proves claim 3 a).