Cost-Effective Robust Stabilization and Bifurcation Suppression

Abstract

A novel technque of robust stabilization and bifurcation suppression is proposed. The proposed method, the centre probe method (CPM), stabilizes an equilibrium point of a delay differential equation at a bifurcation point by introducing an impulsive controller that minimizes a given cost functional. The cost functional can weight certain structural properties of the controller, such as the number of nodes controlled (in the stabilization of a complex network). The method takes advantage of the dimension reduction properties of the centre manifold, which makes the method notably efficient to implement. A numerical example is provided to demonstrate its effectiveness in suppressing a Hopf bifurcation and robustly stabilizing a nonlinear network model with 100 linearly coupled nodes, while simultaneously keeping the number of controlled nodes to a minimum and minimizing a cost function that assigns higher cost to nodes with higher degree. The strengths and weaknesses of the method compared to other impulsive stabilization techniques are discussed.

Full text

COST-EFFECTIVE ROBUST STABILIZATION AND BIFURCATION1
SUPPRESSION2
KEVIN E.M. CHURCH∗AND XINZHI LIU†
3
Abstract. A novel technque of robust stabilization and bifurcation suppression is proposed.4
The proposed method, the centre probe method (CPM), stabilizes an equilibrium point of a delay5
differential equation at a bifurcation point by introducing an impulsive controller that minimizes a6
given cost functional. The cost functional can weight certain structural properties of the controller,7
such as the number of nodes controlled (in the stabilization of a complex network). The method takes8
advantage of the dimension reduction properties of the centre manifold, which makes the method9
notably efficient to implement. A numerical example is provided to demonstrate its effectiveness in10
suppressing a Hopf bifurcation and robustly stabilizing a nonlinear network model with 100 linearly11
coupled nodes, while simultaneously keeping the number of controlled nodes to a minimum and12
minimizing a cost function that assigns higher cost to nodes with higher degree. The strengths and13
weaknesses of the method compared to other impulsive stabilization techniques are discussed.14
Key words. Impulsive stabilization, bifurcation suppression, centre probe method, complex15
network16
AMS subject classifications. 34K20, 93D0917
1. Introduction. Stabilization of complex networks and dynamical systems18
both large-scale and small play an important role in science and industry. Since19
the introduction of Lyapunov’s direct method and its various generalizations, the20
technique has seen much application in the development of sufficient conditions for21
the stability of steady states, which can themselves be used to derive controllers22
guaranteeing robust stability and synchronization. The recent survey paper [19] cat-23
alogues recent developments in the stability analaysis of linear time-delay systems by24
Lyapunov-based methods, and one may consult the references therein for background.25
For a short list of specifically nonlinear results, one may consult [11,25,26,27].26
Suppose one has an autonomous n-dimensional retarded functional differential27
equation (RFDE) depending on a parameter ∈Rp,28
˙x=f(xt, ),(1.1)29
30
where xt∈C:= C([−r, 0],Rn) is the solution history with r > 0 finite, and it is
known that x∗= 0 is an equilibrium point for all ∈N⊂Rp, a neighbourhood of the
origin. Systems of this type include linear and nonlinear discrete time-delay systems
with or without parameters:
˙x=Ax(t) + Bx(t−r)) + f(x(t), x(t−r)),
as well as systems with multiple discrete delays and distributed delays. If fin (1.1)31
is Fr´echet differentiable in its first variable, the linearization at x∗= 0 is the linear32
system33
˙y=Df (0, )yt,(1.2)34
35
where Df (0, ) : C→Rnis the Fr´echet derivative of f(with respect to the first36
variable) at the parameter . The dynamics of the linearization determine the local37
∗Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
(k5church@uwaterloo.ca).
†Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
(xzliu@uwaterloo.ca).
1
This manuscript is for review purposes only.

2K.E.M. CHURCH AND X. LIU
behaviour near x= 0 for the nonlinear system (1.1). If one defines the characteristic38
matrix39
∆(λ;) = λI −Df (0, )[eλ(·)I](1.3)40
41
where Ithe n×nidentity matrix, then the eigenvalues (sometimes called characteristic42
roots) are the solutions of the generally transcedental characteristic equation43
det(∆(λ;)) = 0.(1.4)44
45
x∗will be exponentially stable if and only if all eigenvalues have strictly negative real46
part; see [8,18] for background.47
A transition from stability to instability occurs when one or more eigenvalues48
cross the imaginary axis. Such a transition is an example of a bifurcation, and when49
such a bifurcation occurs, the dynamics of the nonlinear system (1.1) near x∗= 050
can quickly become unpredictable. One may consult the survey [6] or the textbooks51
[4,7,10] for background on bifurcation theory. In practical applications, it may be52
that due to some external influence or component failure, a system parameter enters53
a regime where stability is lost and a bifurcation occurs. The goal then shifts to54
stabilization.55
One stabilization methodology that has seen a fair bit of attention in recent years56
is impulsive stabilization [9,12,13,15,16,21,23,28]. These are typically proven by57
means of Lyapunov functionals and are stated in terms of the existence of matrices58
satisfying linear matrix inequalities. They often provide global stability. However, the59
assumptions can be somewhat strong: global Lipschitz conditions are typically needed60
to guarantee convergence and finding matrices satisfying the necessary inequalities can61
be difficult especially for large interconnected systems.62
A related problem in terms of the implementation of impulsive stabilization is63
that, naturally, some controls may be more difficult to implement than others. In64
other words, there may be an explicit cost in implementing an impulsive controller.65
Guaranteed cost impulsive control from has been considered in [14,24,29] among66
others, where the goal is to design the impulsive controller so that a running cost is67
minimized. To contrast, we are interested in average costs associated to impulsive68
controllers, where the cost may be dictated by such factors as the control gain or its69
structure. The latter encompasses such factors as the amount of coupling induced by70
the controller, the amount of diffusivity or lack thereof, or a penalty for accessing or71
modifying certain system states.72
Simultaneously, there may be hard constraints to the types of impulsive controllers73
that are permitted. In pinning control, for example, the jump functionals do not74
induce any additional coupling between nodes – see the aforementiond references. In75
an input-output setting, if one only has access to system outputs, then one might76
want the controller to depend only on the measurements. See later an example in77
Section 2.1.2. We would like to incorporate such hard constraints into our stabilization78
methodology.79
It is our goal to provide an alternative impulsive stabilization approach based on80
centre manifold theory. Our novel method does not require global Lipschitzian con-81
straints and provides an algorithmic way to find an impulsive controller that achieves82
stabilization while simultaneously guaranteeing a prescribed local convergence rate83
and minimizing a cost functional. The jump functionals that lead to stabilization can84
be chosen from a set of admissible functionals that can be set by the control designer,85
thereby incorporating a wide class of hard constraints as described in the previous86
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 3
paragraph. Moreover, bifurcations are suppressed in the sense that any nonlinear87
structures such as periodic orbits that could result from parameter variation will be88
unstable. In other words, the dynamics near the equilibrium will be robust under89
parameter variation.90
With this in mind, our setup is as follows. We assume that at = 0, the charac-91
teristic equation has some number c > 0 of eigenvalues on the imaginary axis, while92
all others have strictly negative real part. This situation corresponds to one where93
stability of x∗= 0 could potentially be either gained or lost by perturbing the pa-94
rameter away from zero, and a bifurcation could therefore occur. Also, we assume95
f:C×N→Rnis C2in a neighbourhood of (0,0) so that it admits a Taylor expansion96
of the form97
f(φ, ) = L0φ+L()φ+O(||φ||2)98
99
near (φ, ) = 0. Specifically, if Df (0, ) denotes the Fr´echet derivative of φ7→ f(φ, )
for fixed, then L0=Df (0,0) and L() = Df (0, )−L0. For simplicity, we will
assume further that L0can be expressed in the form
L0φ=A0φ(0) +
m
X
k=1
Ckφ(−rk) + Z0
−r
C(s)φ(s)ds
for n×nmatrices A0and Ckfor j= 1, . . . , m, some discrete delays rk∈(0, r] and100
integrable C: [−r, 0] →Rn×n. The Riesz representation theorem implies that the101
functional L0could be more general than this, but to keep things simple we will102
assume this form.103
For each ∈N, we consider the problem of finding a linear jump functional B
104
such that, for the impulsive retarded functional differential equation (IRFDE)105
˙x=f(xt, ), t /∈1
hZ(1.5)106
∆x=B,h(t)xt−, t ∈1
hZ,(1.6)107
108
the following conditions are satisfied for || ≤ δ, for some positive δ.109
U.1 The equilibrium x∗= 0 is locally asymptotically stable with local convergence110
rate O(e−γt ).111
U.2 B(t) is optimal in the sense that it minimizes an admissible cost function.112
The frequency of impulse effect, h, is chosen beforehand. The constant γ > 0 is a113
chosen rate parameter. The functional Bis typically the action of a matrix on a114
vector of state observations (possibly delayed) or a sum thereof. The notion of a local115
convergence rate and admissible cost functions will be defined later when we formalize116
the problem more precisely. Condition U.1 guarantees that the equilibrium is stabilized117
and any bifurcations that could lead to a loss of stability of the equilibrium are118
suppressed in the parameter regime || ≤ δ, while specifying a worst-case convergence119
rate. The second condition U.2 ensures that the impulsive control (1.6) is one that120
minimizes an associated cost.121
One might also be interested in conditions under which one can find a jump122
functional Bthat satisfies conditions U.1 and U.2, but is independent of . This123
situation corresponds to a uniform robust cost-effective stabilization, and may be124
desirable when the dimension of the parameter space is very high. We will consider125
this problem as well.126
This manuscript is for review purposes only.

4K.E.M. CHURCH AND X. LIU
The structure of the paper is as follows. In Section 2, we precisely formulate127
our cost-effective impulsive stabilization and bifurcation suppression problem. The128
existence of solutions of the problem are considered in Section 3, while Section 4and129
Section 5are devoted to the computation of optimal solutions. The effectiveness of our130
stabilization method is demonstrated in Section 6by way of a numerical simulation.131
A discussion and conclusion follow in Section 7and Section 8respectively. All proofs132
are deferred to the appendix.133
1.1. Notation. Ra×bdenotes the real vector space of a×bmatrices with real134
entries. For A∈Ra×b, the notation Aij denotes the entry in row iand column j.135
If Xis a real vector space, Y⊂Xis a linear subspace and x∈X, we denote
x+Y={x+y:y∈Y} ⊂ X
the affine subspace spanned by Ywith translation x. If Wand Zare two such affine
subspaces, we define for t∈[0,1] the convex combination
tW + (1 −t)Z={tw + (1 −t)z:w∈W, z ∈Z}.
For a vector space X, we will denote Xk=X×X× · · · × Xthe k-fold cartesian136
product of Xwith itself, with kfactors in the product. We will sometimes abuse137
notation and identify elements of Xkwith 1 ×karrays with elements in X.138
For square matrix A∈Rb×b, the symbols ρ(A), det(A) and tr(A) will respectively139
denote the spectral radius, determinant and trace. We will at times suppress the140
parentheses and write simply ρA for the spectral radius of A.141
The symbol || · || will be used for the norm on a relevant real vector space. When142
a specific choice of norm is needed (eg. induced by an inner product), this will be143
stated beforehand.144
RCR denotes the real vector space of functions f: [−r, 0] →Rnthat are contin-145
uous from the right and have limits on the left. In integration theory, such functions146
would be referred to as right-continuous and regulated. RCR contains the continu-147
ous functions C=C([−r, 0],Rn) as a proper subspace. For a function φ:R→Rn
148
and t∈R, we define the segment φt: [−r, 0] →Rnby φt(θ) = φ(t+θ). For a149
matrix-valued function Φ : R→Rn×mwe use the same notation.150
If Xand Yare two normed vector spaces, L(X, Y ) denotes the vector space of151
bounded linear maps from Xto Y.152
For a linear map L:X→Ybetween finite-dimensional vector spaces, we denote153
L+its Moore-Penrose pseudoinverse. If Lis a matrix, then L+is its pseudoinverse.154
In some proofs we might use the symbol L+for other generalized inverses, but this155
will always be done with adequate warning.156
If j∈Zand k∈N+, we denote [j]kthe remainder of jmodulo k. Specifically, if157
we uniquely write j=pk +rfor some r∈ {0, . . . , k −1}and p∈Z, then we define158
[j]k=r.159
For a finite sequence of matrices A0, . . . , Ak−1for k≥1, we define the product
Qk−1
j=0 Ajby iterative composition – that is, multiplication on the left,
k−1
Y
j=0
Aj=Ak−1· · · A0.
For a function f:X→Y, we will use the symbol f(X) for the image of f. For
an element y∈f(X), we denote f−1(y) its preimage:
f−1(y) = {x∈X:f(x) = y}.
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 5
When fis one-to-one, the singleton f−1(y) will be identified with the unique solution160
xof the equation f(x) = y.161
For a complex vector v∈Cn, we denote Re(v) and Im(v) its real and imaginary162
parts, repsectively. That is, Re(v) and Im(v) are the unique elements of Rnsuch that163
v= Re(v) + iIm(v).164
2. Precise problem formulation. In this section we formulate the problem165
outlined in Section 1more precisely. We introduce the linear jump functionals and166
frequencies that will be considered in Section 2.1. The allowable cost functionals are167
introduced in Section 2.2.168
2.1. The space Bof discrete-delay jump functionals. We consider classes of169
linear jump functionals Bdefined by a collection of distinct discrete delays δ1, . . . , δ`∈170
[−d, 0] for some d > 0:171
Bφ =Aφ(0) +
`
X
i=1
Aiφ(δi),(2.1)172
173
Note that given the frequency hof impulse effect, it will need to be assumed that174
the any nonzero delay δi6= 0 satisfies δi/∈1
hZ. This constraint is needed because175
of a technical assumption (the overlap conditon) of the centre manifold theory and176
reduction principle for impulsive delay differential equations, of which our main results177
are based. From the right-hand side of (2.1), we can identify a jump functional with178
an element of the finite-dimensional vector space179
B= (Rn×n)`+1.(2.2)180
181
With this identification, we will abuse notation and write Bφ for B= (A, A1, . . . , A`)∈182
Band φ∈ RCR as the right-hand side of (2.1). Similarly, if Φ = [ φ1· · · φc]∈183
RCRc, then we denote BΦ=[ Bφ1· · · Bφc].184
2.1.1. Pinning stabilization and diagonal jump functionals. In some situ-185
ations, it may not be appropriate to work with the entire space Bof jump functionals.186
For example, suppose the system (1.1) has the structure of a network of Nidentical187
linearly-coupled nodes188
˙x(i)=f(x(i)
t) + c
N
X
j=1
aij ΓH(x(j)(t)), i = 1, . . . , N, x(i)∈Rn
(2.3)189
190
with coupling strength c > 0, Γ = diag(γ1, . . . , γn)>0 an inner coupling matrix,191
H:Rn→Rna nonlinear coupling with H(0) = 0, and A= (aij )∈RN×Na diffusive192
Laplacian matrix representing the coupling configuration of the network, where any193
of these coupling terms may depend on parameters. In impulsive pinning stabilization194
and synchronization, one would choose jump functionals that act only on individual195
nodes – see the references [9,16,23] – and are functionally driven by an error system.196
For simplicity, assume we wish to stabilize the trivial equilibrium x∗= 0, so that197
one does not need to consider a separate error system. An appropriate subspace of Bin198
which one could consider pinning stabilization is the set of block diagonal operators.199
To define these, we note that for system (2.3), we have B= (RnN×nN )`+1 . Each200
element of RnN×nN can be interpreted as an N×Nblock matrix with n×nblocks,201
so that we can write an arbitrary element of Bin the form B= (A, A1, . . . , A`) with202
This manuscript is for review purposes only.

6K.E.M. CHURCH AND X. LIU
Am(i, j)∈Rn×nfor m=∅,1, . . . , ` and i, j = 1, . . . , N . Then, the diagonal subspace203
Bdiag ⊂ B is defined by204
Bdiag ={B= (A, A1, . . . , A`)∈ B :Am(i, j) = 0 ∀i6=j, m =∅,1, . . . , `}.(2.4)205
206
This is indeed a subspace of Band it contains only the linear jump functionals that207
induce no further coupling between different nodes.208
2.1.2. Proportional control. Suppose one wishes to stabilize the system ˙x=209
f(xt, ) using a proportional impulsive control, with a system output y. That is, we210
seek jump functionals B(t) such that 0 ∈Rnof the system211
˙x=f(xt, ), t /∈1
hZ212
y(t) = Hx(t),213
∆x=NB(t)yt−, t ∈1
hZ,214
215
becomes stable, where N∈Rn×mis an input matrix, H∈Rp×nis an output ma-216
trix and y∈Rpis the output. Starting with the space B= (R(n+p)×(n+p))`+1 as217
introduced at the beginning of this section, our porportional control constraint on the218
jump functionals can be imposed by abusing notation and identifying B(t) with an219
element of Bdefined in block form as220
B(t) = 0NB(t)
0HNB(t).(2.5)221
222
We need to verify that z=x y Tsatisfies the equation ∆z=B(t)zt−at times223
t∈1
hZ.224
∆z=∆x
H∆x=NB(t)yt−
HNB(t)yt−=0N B(t)
0HNB(t) xt−
yt−=B(t)zt−,225
226
as desired. The set of elements of Bof the form (2.5) is indeed a proper subspace of227
B, which we denote Bprop. One can complete the transformation to a system of form228
(1.5)–(1.6) by taking the derivative of y. PID controls can be introduced in a similar229
way.230
2.1.3. Cycles of jump functionals. In our IRFDE (1.5)–(1.6), we have left231
the possibility for the jump function to depend on time t. That is, B(t)∈ U for each232
tand some subspace U ⊆ B, while being generally nonconstant. As our method is233
based on centre manifold theory, working with infinite time horizons is difficult so we234
will typically take t7→ B(t) to be periodic. As such, a cycle of jump functionals is an235
element of the product Uk, for natural number k≥1 called the period. We can then236
associate a jump functional in the style of (2.1) by way of the following equivalence.237
If B= (B0, . . . , Bk−1)∈ Uk, we define238
Bj
hφ=B[j]kφ.(2.6)239
240
This definition is sufficient to give meaning to (1.6) since we need only define B(t) at241
the times t=j
hfor j∈Z.242
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 7
2.1.4. Cycles of jump functionals in nonidentical subspaces. In some
applications, it might be that certain controls (quantified by jump functionals) can
only be applied itermittently due to resource limitations. As such, it is worthwhile
considering the case where more generally, we have B(t)∈ U(t), for t∈1
hZand each
subspace U(t) is generally distinct. As in Section 2.1.3 we will assume the periodicity
condition
Uj+k:= Uj+k
h=Uj
h=: Uj
for all j∈Z, and associate t7→ B(t) to a jump functional B= (B0, . . . , Bk−1) in the
product space
U(k):= U0× · ·· × Uk−1.
The construction in the previous Section 2.1.3 corresponds to the special case where243
Ui=U0for all indices i. Regardless, it remains true that U(k)is a linear subspace of244
Bk.245
2.2. Allowable cost functionals. An allowable cost functional will be a func-246
tional C:U → R+satisfying the following properties.247
C.1 Cis continuous, convex and positive-definite.248
C.2 Cis radially unbounded : if Bn∈ B is a sequence with unbounded norm249
||Bn|| → ∞, then C(Bn)→ ∞.250
These properties will eventually be used to show that a formalized version of the251
problem from Section 1admits a solution satisfying the conditions U.1 and U.2. One252
can similarly define allowable cost functions on any linear subspace U ⊆ B using the253
same definition.254
Given an allowable cost functional C, we define the cost of a cycle of jump func-255
tionals of period kin a linear subspace of Bkas follows. For B= (B0, . . . , Bk−1)∈ Bk
256
we set257
C(B) =
k−1
X
j=0
C(Bj).(2.7)258
259
We will somtimes abuse notation and write C:Bk→R+for the associated cost260
functional on the cycles of jump functionals of period k.Bkcan be replaced with any261
linear subspace thereof, so this definition extends naturally to encompass cycles of262
jump functionals in nonidentical subspaces as in Section 2.1.4. Finally, the following263
definition will be useful later.264
Definition 2.1. An allowable cost function C:U → R+is pro jective if there265
exists an inner product h·,·i such that C(B) = hB, B i.266
2.2.1. Weighted matrix norms. A typical allowable cost functional can be267
constructed through the introduction of a weighted matrix norm ||X||W=||W1
2X W −1
2||2
268
for a symmetric positive-definite matrix Wand its principal real square root W1
2, and269
|| · ||2the spectral norm. Indeed, let W0,W1, . . . , W`be positive-definite matrices, let270
w0, w1, . . . , w`∈R+be weight constants, and define a cost function271
C((A, A1, . . . , A`)) = w0||A||W0+
`
X
i=1
wi||Ai||Wi
(2.8)272
273
The weight matrices Witake into account limitations and costs associated to accessing274
and/or modifying the states of the system by impulses. The weights w0, w1, . . . , w`
275
This manuscript is for review purposes only.

8K.E.M. CHURCH AND X. LIU
allow for a weighting of the individual factors that define the control (1.6) relative to276
each other.277
2.3. Problem statement. Having introduced the allowable cost functionals278
and the space Bof linear jump functionals, we can more precisely formulate our279
problem. First, we define local convergence rates.280
Definition 2.2. Let g:R+→R+be a function satisfying limt→∞ g(t)=0.281
The asymptotic O(g(t)) is a local convergence rate of an equilibrium point x∗if there282
exists a neighbourhood Uof x∗and a constant K > 0such for all (s, φ)∈R×U,283
the solution t7→ x(t;s, φ)of (1.5)–(1.6)satisfying the initial condition xs(·;s, φ) = φ284
satisfies the inequality ||x(t;s, φ)−x∗|| ≤ Kg(t−s)for all t≥s.285
Next, we formalize the standing hypothesis that at parameter = 0, our system is286
at a bifurcation point where stability could be either gained or lost by parameter287
variation.288
Spectral gap condition. At parameter = 0, the characteristic equation (1.4)289
has c > 0eigenvalues with zero real part, and all other eigenvalues have real part less290
than some σ < 0. The real number σis the spectral gap.291
Remark 2.3. If the characteristic equation has a candidate bifurcation point at a292
parameter ∗6= 0, one can perform a change of variables to shift the bifurcation point293
to the origin, = 0. As such, no generality is lost by assuming a bifurcation point at294
= 0.295
From this point onward, we assume the spectral gap condition. With these definition296
at hand, the problem whose feasibility we will study and subsequently solve is the297
following.298
Problem A. Let Ube a linear subspace of Bkfor k≥1. For a given rate299
parameter γ > 0and freqency h, determine whether one can, for sufficiently small,300
find an allowable cost functional B∈ U such that the optimality condition301
arg min
Y∈Y(,h;γ)
C(Y) = B
(2.9)302
303
is satisfied, where Y(, h;γ)⊂ U is the set of all linear jump functionals for which304
O(e−γt )is a local convergence rate of the equilibrium point x∗= 0 of (1.5)–(1.6)for305
parameter and frequency h.306
In the formulation of Problem A, we allow the optimal jump functional to depend307
on the parameter . However, in some settings it may be desirable to have a single308
jump functional provide stabilization robustly for all sufficiently small, or it may309
be compuitationally too expensive to generate optimal jump functionals for a large310
sample of parameters. Therefore, the following problem is of interest.311
Problem B. Let Ube a linear subspace of Bkfor some k≥1. For a given rate312
parameter γ > 0and frequency h, find η > 0and single allowable cost functional313
B∈ U such that the optimality condition314
arg min
Y∈Yη(h;γ)
C(Y) = B(2.10)315
316
is satisfied, where Yη(h;γ)⊂ U is the set of all linear jump functionals for which317
O(e−γt )is a local convergence rate of the equilibrium point x∗= 0 of (1.5)–(1.6)for318
parameters || ≤ ηand frequency h.319
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 9
3. Existence of an optimal solution. In this section we state our solutions320
to Problems A and B. The main results are provided in Section 3.1 and Section 3.2.321
All proofs are deferred to the proofs appendix. Illustrative applications of our main322
results of this section as they apply to stabilization and bifurcation suppression are323
postponed until Section 6.324
3.1. Main existence results: Problem A. A first step toward the solution325
of Problem A is provided by the following proposition.326
Proposition 3.1. Write f(φ, )as a Taylor expansion near 0, so that
f(φ, ) = A0φ(0) +
m
X
k=1
Ckφ(−rk) + Z0
−r
C(s)φ(s)ds
| {z }
L0
+L()φ+O(||φ||2),
for L:Rp→ L(C([−r, 0],Rn),Rn)continuous and satisfying L(0) = 0 for some327
discrete delays rk∈(0, r]and a matrix A0∈Rn×n. Let Φ(t)be a real n×cmatrix328
whose columns form a basis of the set of centre generalized eigenfunctions329
E0=[
n∈N(z(t) =
n
X
i=1
ti−1eλtvi,:Re(λ)=0,˙z=L0zt),(3.1)330
331
and compute the c×cmatrix Λsatisfying the identity d
dt Φ(t) = Φ(t)Λ. Introduce the332
transposed operator LT
0,333
LT
0ψ=A0ψ(0) +
j
X
k=1
ψ(rk)Ck+Z0
−r
ψ(−s)C(s)ds,(3.2)334
335
acting on C([0, r],Rn∗). Let Ψ(t)be a real c×nmatrix whose rows form a basis for336
the set of adjoint centre generalized eigenfunctions337
ET
0=[
n∈N(w(t) =
n
X
i=1
ti−1e−λtvi,:Re(λ)=0,˙w=−LT
0wt),(3.3)338
339
and define the invertible matrix Γ∈Rc×cby the equation340
Γ−1= Ψ(0)Φ(0) −
j
X
k=1 Zrk
0
Ψ(s)CkΦ(s−rk)ds · · ·
−Z0
−rZ0
−θ
Ψ(s)C(s)Φ(s+θ)dsdθ.
(3.4)341
342
Introduce the centre monodromy map M,h :Bk→Rc×c,343
M,h(B) =
k−1
Y
j=0
(Ic×c+ ΓΨ(0)BjΦ0) exp 1
h(Λ + ΓΨ(0)L()Φ0),(3.5)344
345
and define the set346
e
Y(, h;γ) = {B∈ U ⊆ Bk:eγ/hρ(M(B)) ≤1}.(3.6)347
348
Let an allowable cost functional Cbe given. Let hbe fixed. The following are true.349
This manuscript is for review purposes only.

10 K.E.M. CHURCH AND X. LIU
1. e
Y(, h;γ)is closed and if the spectral gap satisfies γ < σ, there exists δ=350
δ(γ)>0such that for || ≤ δ,351
Bδ(0) ∩e
Y(, h;γ) = Bδ(0) ∩ Y(;γ).(3.7)352
353
2. For each each ∈N, there exists Bsuch that C:e
Y(;γ)→Rattains its354
minimum at B, assuming the domain is nonempty.355
Remark 3.2. To compute a basis for E0, it is sufficient to compute a canonical356
system of Jordan chains for ∆(λ; 0) for all eigenvalues λwith zero real part. See357
[Theorem 4.2, Chapter 7, [8]] for the relevant result. To compute a basis for ET
0, one358
can use the connection between the transpose system ˙w=−LT
0wtand the adjoint,359
with the result being that a basis can be computed using a canonical system of Jordan360
chains for the transpose ∆(λ; 0)T– see [Theorem 5.1, Chapter 7, [8]] for the relevant361
theorem. Alternatively, one could exploit the characterization Ψ(t) = e−ΛtΨ(0) for362
Ψ(0) ∈Rc×nand solve for the unknown coefficients of Ψ(0) by imposing the equality363
Γ = Iin equation (3.4), as suggested in [1].364
Remark 3.3. In the case of a Hopf bifurcation, the basis calculation is much sim-
pler. To obtain the basis matrix for E0, one calculates a nontrivial v∈Cnsatisfying
∆(iω; 0)v= 0 for the critical eigenvalue λ=iω. Then, the basis matrix is
Φ(t) = Re(veiωt ) Im(veiω t).
For the transpose basis ET
0, one computes a nontivial w∈Cnsatisfying wT∆(iω; 0) =
0, and obtains the basis matrix
Ψ(t) = Re(wTe−iωt)
Im(wTe−iωt).
One must then compute Γ explicitly after.365
Thus, the existence of a small solution to Problem A is equivalent to the set366
e
Y(, h;γ) being nonempty. It is therefore important that we determine conditions367
under which that set is nonempty. Also, it would be prudent to ensure that as368
γ→0, the cost of ensuring the convergence rate O(e−γt) for a fixed pair () should369
become arbitrarily small and that the intersections (3.7) are nonempty. Our sufficient370
condition is the following.371
Theorem 3.4. Let U=Bk.e
Y(, h;γ)is nonempty provided Φ(0) and Ψ(0) are372
of rank c. If this is the case, for any selection γ7→ Bγ
of minimizing jump functionals373
for rate parameter γ, one has limγ→0+Bγ
= 0. In particular, if γ > 0is sufficiently374
small, the sets in (3.7)are nonempty.375
The subspace and rank condition of Theorem 3.4 allows us to quickly exclude376
memoryless systems and systems at fold bifurcation points. We have the following377
corollary.378
Corollary 3.5. If c= 1 or L0φ=C φ(0) for an n×nmatrix C, then e
Y(;γ)379
is nonempty.380
We can derive a more general sufficient condition that is implied by the subspace381
and rank condition of Theorem 3.4. It is captured by the following corollary whose382
proof is omitted since it is similar to that of the previous theorem.383
Corollary 3.6. The subspace condition U=Bkand the rank condition on Φ(0)384
and Ψ(0) in Theorem 3.4 can be replaced with the condition M(U) = Rc×c, and the385
conclusions of the theorem hold.386
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 11
3.2. Main existence results: Problem B. As it turns out, the proofs of387
Proposition 3.1 and Theorem 3.4 work with minimal modifications to solve Problem388
B. We have the following analogues and the appropriate variant of Corollary 3.5.389
Proposition 3.7. With the same notation as in Proposition 3.1, define the set390
e
Yη(h;γ) = {B∈ U ⊆ Bk:∀|| ≤ η, eγ/h ρ(M,h(B)) ≤1}.(3.8)391
392
Let hbe fixed. The following are true.393
1. e
Yη(h;γ)is closed and if the spectral gap satisfies γ < σ, there exists δ > 0394
such that for η≤δ,395
Bδ(0) ∩e
Yη(h;γ) = Bδ(0) ∩ Yη(h;γ).(3.9)396
397
2. e
Yη(h;γ)can be written as the intersection398
e
Yη(h;γ) = \
||≤ηe
Y(;γ).(3.10)399
400
3. There exists Bsuch that C:e
Yη(h;γ)→Rattains its minimum at B, assum-401
ing the domain is nonempty.402
Theorem 3.8. Let U=Bk.e
Yη(h;γ)is nonempty provided Φ(0) and Ψ(0) are403
of rank c. If this is the case, then for any selection γ7→ Bγof minimizing jump404
functionals for rate parameter γ, one has limγ→0+Bγ= 0. In particular, if γ > 0is405
sufficiently small, the sets in (3.9)are nonempty.406
Corollary 3.9. If c= 1 or L0φ=Cφ(0) for an n×nmatrix C, then e
Yη(h;γ)407
is nonempty.408
Corollary 3.10. The subspace condition U=Bkand the rank condition on409
Φ(0) and Ψ(0) in Theorem 3.8 can be replaced with the condition M(U) = Rc×cfor410
|| ≤ η, and the conclusions of the theorem hold.411
Remark 3.11. Since 7→ Mis continuous, the conclusion of the above corollary412
is guaranteed to hold for some η > 0, provided M0,h(U) = Rc×c. This can be seen413
by vectorizing the monodromy map and recalling that rank function X7→ rank(X) is414
lower semicontinuous.415
4. Computation of optimal solutions for Problem A: the centre probe416
method. Solving the optimization problem417
minimize C(Y),
subject to Y∈e
Y(;γ),
(Y)418
419
directly appears to be very difficult. The feasible set, introduced in Proposition 3.1, is420
characterized as a sublevel set of B7→ ρ(M,h(B)). While B7→ M,h(B) is smooth,421
the spectral radius is notably irregular and nonconvex. If one wishes to guarantee422
feasibility it is generally necessary to optimize in the space Bk, whose dimension is423
kn2(`+ 1) and can be quite large even for small networks. It is therefore critical424
that we reduce the dimension of the problem before even considering performing an425
optimization task.426
From this point onward, we will assume that a rate parameter γ, system param-427
eter and frequency hhave been chosen. We will suppress all dependence on these428
variables unless necessary. Also, we make the following simplifying assumption on our429
chosen space of cycles of jump functionals of period k≥1.430
This manuscript is for review purposes only.

12 K.E.M. CHURCH AND X. LIU
Assumption. The subspace U(k)⊆ Bkof cycles of jump functionals is given by431
kcopies of a single subspace U ⊆ B. That is, U(k)=Uk.432
This assumption is not strictly needed. All constructions (eg. probe space) can be433
appropriately generalized to allow the individual subspaces making up the product434
U(k)to be distinct, and the major theorems (Theorem 4.11 and Theorem 4.12) have435
appropriate and similarly strong analogues. However, the notation can make the436
presentation difficult to follow. For this reason, we will specialize to this particular437
case.438
To avoid ambiguity later, we define M1:U → Rc×cby
M1(B) = (Ic×c+ ΓΨ(0)BΦ0) exp 1
h(Λ + ΓΨ(0)L()Φ0),
and reserve the symbol Mfor the centre monodromy operator M:Uk→Rc×c. Note439
that this implies the factorization M(B) = Qk−1
j=0 M1(Bj).440
4.1. The probe space P.The map M1:U → Rc×cis affine, and we can
decompose it as
M1=M0+Z, Z = exp 1
h(Λ + ΓΨ(0)L()Φ0), M0(B) = ΓΨ(0)BΦ0Z,
where M0:U → Rc×cis linear.441
Definition 4.1. The probe space P ⊂ Rc×cis the image of M1. That is, P=442
im(M1). Elements P∈ P are called probe elements. The k-probe space is the kth443
cartesian power Pk, and its elements are k-probe elements.444
The following lemmas describe the geometry of the probe space, k-probe space445
and preimages of probe elements under the centre monodromy operator. The proofs446
are omitted.447
Lemma 4.2. The probe space is an affine space of dimension at most c2. It can448
be written in the form P=Z+ im(M0).449
Remark 4.3. If the subspace/rank condition of Theorem 3.4 / Theorem 3.8 is450
satisfied, then Pis a vector subspace of Rc×cand it is precisely P=im(M0).451
Lemma 4.4. The k-probe space Pkis convex.452
Lemma 4.5. Let P∈ P. The preimage of Punder the centre monodromy map453
M1is an affine space, and can be written454
M−1
1(P) = M+
0(P−Z) + ker(M0).(4.1)455
456
The correspondence M−1
1:P⇒Uexhibits a concavity-like property that will be457
essential in the next section. It is summarized by the following lemma.458
Lemma 4.6. Let X, Y ∈ P. For all t∈[0,1],459
tM−1
1(X) + (1 −t)M−1
1(Y)⊆ M−1
1(tX + (1 −t)Y).(4.2)460
461
At this stage we should define the precise link between the k-probe space and the
image of M:Uk→Rc×c. Define the product function G: (Rc×c)k→Rc×cby
G(X0, . . . , Xk−1) = Xk−1· · · X0.
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 13
In terms of the product function, the image of M:Uk→Rc×ccan be written462
equivalently as the image of G:Pk→Rc×c. From this equivalence, we obtain the463
following lemma.464
Lemma 4.7. Let P∈ Pk. For all B∈ M−1(G(P)), we have ρ(M(B)) =465
ρ(G(P)).466
The above lemma provides one way to pull k-probe elements back to cycles of
jump functionals of period k. Namely, compute the ordered product and take the
preimage under M:Uk→Rc×c. However, the preimage under M:Uk→Rc×cof
a given c×cmatrix is not necessarily convex. To remedy this, we can alternatively
define a map Mk:Uk→(Rc×c)kby
Mk(B0, . . . , Bk−1)=(M1(B0),...,M1(Bk−1)).
Then, an analogue of Lemma 4.7 is as follows, strengthened by the convexity of467
M−1
k(P) for any P∈ Pkdue to the componentwise convexity afforded by Lemma468
4.5.469
Lemma 4.8. Let P∈ Pk. For all B∈ M−1
k(P), we have ρ(M(B)) = ρ(G(P)).
Also, M−1
k(P)is convex and given P= (P0, . . . , Pk−1), it can be written
M−1
k(P) = {(B0, . . . , Bk−1) : Bi∈ M−1
1(Pi), i = 0, . . . , k −1}.
4.2. Compatible probe cost. Broadly speaking, our idea for solving the non-470
linear program (Y) is to first solve a related optimization problem in the probe space471
and obtain an optimal solution P∗, pull this optimal solution into the convex space472
M−1
k(P∗) and find a minimizer YP∗of the cost functional. Since every element of473
M−1(P∗) will be feasible provided ρ(P∗)≤e−γ/h – see Lemma 4.7 – the spectral474
constraint does not need to be checked at this final optimization stage. The following475
definition allows us to define the appropriate optimization problem in Pk.476
Definition 4.9. A continuous, radially unbounded convex function ˜
C:Pk→R+
477
is a:478
•local probe-compatible cost (LPCC) for the nonlinear program (Y)if one of479
the following conditions hold:480
1. ˜
C◦ Mk(B)≤ C(B)for all B∈ U , with equality if481
B∈arg min
X∈M−1(M(B))
C(X).(4.3)482
483
In this case we say ˜
Cis a type 1 LPCC.484
2. ˜
C◦ Mk(X)≤˜
C◦ Mk(Y)implies C(X)≤ C(Y). In this case we say ˜
C485
is a type 2 LPCC.486
•global probe-compatible cost (GPCC) if for any global optimum B∗of the487
nonlinear program (Y)and any other feasible solution B, one has ˜
C◦Mk(B∗)≤
488 ˜
C◦ Mk(B), with equality holding if and only if Bis also a global optimum.489
•uniform probe-compatible cost (UPCC) if it is both a LPCC and a GPCC.490
The existence of GPCC/LPCCs will be addressed in Theorem 4.12. Given a491
continuous, radially unbounded convex function ˜
C:Pk→R+, we define a new492
nonlinear program in the probe space, which we call the probe program:493
minimize ˜
C(X),
subject to X∈ Pk,
ρ◦G(X)≤e−γ/h .
(P˜
C)494
495
This manuscript is for review purposes only.

14 K.E.M. CHURCH AND X. LIU
The program (P˜
C) posesses a global optimum so long as e
Yis nonempty. We also496
define a family of convex programs indexed by P∈ Pkwith feasible set given by the497
preimage M−1
k(P). We call this the inverse probe program:498
minimize C(B),
subject to B∈ M−1
k(P).
(YP)499
500
Remark 4.10. As M−1
k(P) is an external direct sum of kaffine subspaces of B–501
see Lemma 4.5 and Lemma 4.8 – the inverse probe program is actually unconstrained502
after an appropriate affine linear change of variables.503
Our first theorem of this section relates the programs (P˜
C) and (YP) to solutions504
of (Y) under the assumption that ˜
Cis a LPCC or a GPCC. The second theorem505
establishes the existence of at least one UPCC.506
Theorem 4.11 (centre probe method). Assume e
Yis nonempty. Let ˜
C:Pk→507
R+be a global (resp. local) PCC for the nonlinear program (Y). Let P∈ Pkbe a508
global (resp. local) optimum for the program (P˜
C). Let B∗∈ M−1
k(P)be a local509
optimum for the convex program (YP). Then, B∗is a global (resp. local) optimum for510
the program (Y).511
We will refer to the procedure of determining a solution of the program (Y) using512
the probe program in conjunction with the inverse probe program collectively as the513
centre probe method (CPM).514
Theorem 4.12. Define ˜
C:Pk→R+by515
˜
C(P) = min
X∈M−1
k(P)
C(X).(4.4)516
517
This function is continuous, radially unbounded, convex, and a UPCC. We will refer518
to it as the trivializing UPCC.519
The trivializing UPCC is so named because it makes the final step of running the520
nonlinear program (YP) unnecessary. Indeed, once one has computed an optimal521
solution Pfor (P˜
C), the optimal cost for the program (Y) is precisely ˜
C(P). In other522
words, running the program (YP) is equivalent to computing ˜
C(P).523
4.3. Explicit formula for trivializing UPCC under projective cost. Sup-524
pose the cost functional C:U → R+is projective. Recall this means that C(B) =525
hB, B ifor an inner product h·,·i on U(note, on Ukone defines the cost additively526
according to (2.7)). See Section 4.3.1 for examples. Because of Lemma 4.5, we can527
calculate ˜
C(P) for the trivializing UPCC according to528
˜
C(P) =
k−1
X
i=0
min hX, X i:X∈M+
0(Z−Pi) + ker(M0).(4.5)529
530
˜
C(P) and the associated minimizer in M−1(P) in fact has an explicit solution, due531
to the computation of ˜
C(P) being equivalent to squared distance minimization to an532
affine subspace of a Hilbert space. The proof is elemenetary and is omitted.533
Proposition 4.13. Let {m1, . . . , mb}be an orthonormal basis for ker(M0). For534
a given P∈ P, define X(P)∈ U by535
X(P) = M+
0(P−Z)−
b
X
j=1
hM+
0(P−Z), mjimj.(4.6)536
537
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 15
If ˜
Cis the trivializing UPCC and the cost functional is projective, then for any P=
(P0, . . . , Pk−1)∈ Pk, if we define X(P) = (X(P0), . . . , X(Pk−1)), then
˜
C(P) = C(X(P)).
Moreover, M+
0(P−Z)in equation (4.6)can be replaced by any solution Yof the
linear equation M0Y+Z=P, and these are
Y=M+
0(P−Z)+(I−M+
0M0)w
for any w∈ U.538
4.3.1. Example of projective cost functionals. A projective cost functional539
modeled on (2.8) and suitable for implementation (ie. vectorized) is540
C((A, A1, . . . , A`)) = w0~
ATW0~
A+
`
X
i=1
wi~
AT
iWi~
Ai,(4.7)541
542
where w0, w1, . . . , w`>0, W0, W1, . . . , W`∈Rn2×n2are symmetric positive-definite543
matrices and for X∈Rn×n, we define ~
X∈Rn2to be its standard vectorization,544
obtained by stacking the columns of Xon top of one another from left to right. The545
cost (4.7) is projective with the inner product546
h(A, A1, . . . , A`),(B, B1, . . . , B`)i=w0~
ATW0~
B+
`
X
i=1
wi~
AT
i~
Bi.547
548
5. Computation of optimal solutions: Problem B. The centre probe method549
of Section 4can be adapted to find solutions of Problem B. Recall that our proposed550
solution to Problem B derived from Proposition 3.7 requires us to find a constant551
η > 0 such that there exists a solution of the nonlinear program552
minimize C(Y),
subject to Y∈e
Yη(h;γ).
(Yη)553
554
It is important to remark that a solution of Problem B includes the robustness pa-555
rameter η. That is to say, ηis not chosen at the outset. This is crucial, and in general556
one cannot take ηas arbitrary.557
Suppose for the sake of argument that one could choose η > 0 at the outset. One558
may recall from Proposition 3.7 that a solution Bηof (Yη) can only be guaranteed to559
solve the nonlinear program (2.10) if ||Bη|| < δ(η), otherwise there may be secondary560
bifurcations involving eigenvalues crossing the imaginary axis from the left. As δ(η)561
is generally decreasing with respect to η– see the proof of the aforementioned propo-562
sition – one can only guarantee stabilization with an increasingly trivial linear jump563
functional. However, unless the equilibrium x∗= 0 is already exponentially stable564
with rate O(e−γt ), a trivial jump functional should not be able to provide stabiliza-565
tion. Therefore, generally, it is not possible to choose the robustness parameter; a566
controller that stabilizes the equilibrium at the parameter = 0 will generally fail to567
stabilize the equilibrium if the parameter is taken too large.568
This manuscript is for review purposes only.

16 K.E.M. CHURCH AND X. LIU
5.1. Uniform CPM. Because of the smoothness of the vector field (1.1), any569
linear jump functional that guarantees the local convergence rate O(e−(γ+s)t) at pa-570
rameter = 0 for some s > 0 will automatically guarantee the local convergence rate571
O(e−γt ) for || ≤ η, for some η > 0 that generally depends on s. This follows from572
hemicontinuity arguments; see the related proof of Proposition 3.1.573
Theorem 5.1 (Uniform CPM). Let B∈ Ukbe a linear jump functional produced574
by the CPM (local or global) at parameter = 0 and convergence rate γ0satisfying575
σ > γ0=γ+s, for s > 0a small safety parameter. There exists η=η(s)>0such576
that Bis a feasible solution of the program (Yη).577
The above theorem does not guarantee optimality of the candidate solution. How-578
ever, for typical problems where the performance (linear-order convergence rate) of579
the candidate Bdeteriorates when ||becomes large, we are guaranteed that the580
solution is optimal for some robustness parameter.581
Corollary 5.2 (Optimality). With the notation from the previous theorem,582
suppose for some 6= 0,ρ(M(B)) > e−γ/h. Then, there exists η∈(0,||)such that583
Bis an optimum (local or global) of the program (Yη).584
6. Stabilization of a neural network near a Hopf point. In this section we585
consider primarily by way of example how our uniform CPM can be used to stabilize586
a complex network near a Hopf point. We incorporate an inhomogeneous weighting587
on the cost of controlling each node and address how one can minimize the number588
of controlled nodes while still taking advantage of the performance improvements589
inherent to the CPM.590
The neural network we will consider in this section is a slight modification of an591
example considered in [16]. For xi∈R2for i= 1,...,100, consider the nonlinear592
network model593
˙xi=−xi(t) + Btanh(xi(t)) + Dtanh(xi(t−1))+
N
X
j=1
aij xj(t)(6.1)594
595
with connection weight matrices
B=2−0.11
−5 3.2, D =−1.6−0.1
−0.18 −2.4
and linear coupling determined by the matrix A= (aij)N×N, which is the negative
of a graph Laplacian associated to a small world network graph on 100 nodes. We
define tanh(y)=[ tanh(y1) tanh(y2)]Tfor y= (y1, y2)∈R2, and ≥0 is a neural
activation strength parameter. It is known [16] that when decoupled, the individual
nodes determined by the dynanical system
˙y=−y+Ctanh(y(t)) + Dtanh(y(t−1))
exhibit chaotic dynamics with a double-scroll-like attractor, and that the origin is596
unstable. Thus, when = 1, the diffusivity condition implies that the nonlinear597
network model (6.1) is chaotic and the origin is unstable. However, when = 0,598
the origin is globally asymptotically stable. There is therefore a bifurcation at some599
critical activation strength ∗∈(0,1).600
Our goal is to robustly stabilize the network (6.1) in a parameter neighbourhood601
of ∗with frequency h= 1 and period k= 1 using our uniform CPM. We will602
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 17
assume decoupled controls, so that we take our candidate jump functionals from the603
diagonal subspace Bdiag. To introduce some heterogeneity, we will assume that some604
nodes are more difficult to control than others, and we will incorporate this into the605
associated cost functional. Specifically, nodes with a higher degree of connectivity will606
be assigned a higher cost. We will also attempt to minimize the number of controlled607
nodes.608
Our methodology is as follows. First, we recall the small-world network topology609
and introduce the cost functional we will be using in Section 6.1. Then, we encode610
the parameter-dependent system (6.1) as the user input to DDE-BIFTOOL and use611
the included GetStability routine to determine the critical parameter ∗where the612
bifurcation occurs, and determine its type. We also use this routine to compute the613
critical eigenvalus λ=±iω on the imaginary axis. We then implement the uniform614
CPM at the critical parameter ∗and determine a locally cost-minimizing controller615
B0and a generating probe element P∗for a target local convergence rate O(e−t
5) –616
that is, we take the target rate parameter to be γ=1
5. These steps are carried out617
in Section 6.2.618
The feasible jump functional B0is cost-minimizing in the affine space M−1(P∗)619
but might not minimize the number of controlled nodes. We provide a solution to620
this secondary minimization problem in Section 6.3.621
In Section 6.4 we assess the performance of the jump functional derived using622
the uniform CPM in conjunction with the node-minimizing process of Section 6.3.623
Specifically, we compare the output of the neural network model with and without624
our pinning control at a range of parameters ∈[∗, ∗
2). The bifurcation is suppressed625
in this regime, but a secondary bifurcation point is identified at the parameter value626
=∗
2<1. The implications of this bifurcation are discussed.627
6.1. Network topology, cost functional and the trivializing UPCC. Small-628
world networks [22] capture a network topology involving both a high degree of clus-629
tering and short average path-lengths. They can be constructed by starting with a630
ring lattice on Nvertices with kedges per vertex (specifically, knearest neighbours)631
and “rewiring” each edge randomly with a specified probability, p.632
We wrote a script in MATLAB R2018a to generate a Watts-Strogatz small-world633
graph Gon N= 100 vertices with parameters k= 8 and p= 0.3. The matrix A634
defining the linear coupling in (6.1) is then obtained by taking the negative graph635
laplacian: A=−laplacian(G). The computed graph that was used in this example636
is displayed in Figure 1.637
The degree of node iis precisely |aii| ≥ 1. Based on this, if we uniquely write
B∈ Bdiag as a tuple B= (B1, . . . , B100) for Bi∈R2×2, we can define the cost
functional C:Bdiag →R,
C(B) = 1
∆(G)
100
X
i=1
|aii|hBi, BiiF,
where h·,·iFis the Frobenius inner product on R2×2. Thus, the cost of controlling
node iis linearly scaled relative to its degree, normalized with respect to the maximum
degree. This cost functional is projective with inner product
hX, Y i=1
∆(G)
100
X
i=1
|aii|hXi, YiiF.
This manuscript is for review purposes only.

18 K.E.M. CHURCH AND X. LIU
5
6
7
8
9
10
11
12
Fig. 1.The small-world graph used in the example of Section 6. Nodes are coloured with
intensity varying based on their degree, while the sizes indicate relative degree. Parameters were
N= 100 nodes with initial connections to k= 8 nearest neighbours and rewiring probability p= 0.3.
As such, the trivializing UPCC has an explicit formula from Proposition 4.13:638
˜
C(P) = M+
0(Z−P)−
b
X
j=1
hM+
0(Z−P), mjimj
2
(6.2)639
640
and || · || =ph·,·i is the norm induced by h·,·i.641
6.2. Precomputation and uniform CPM. DDE-BIFTOOL [5] was used to642
identify parameters where bifurcations in the neural network model (6.1) occur. The643
tool detected a Hopf bifurcation (centre subspace of dimension c= 2) at parameter644
= 0.5621. Further numerical examination revealed that the trivial equilibrium is645
locally stable at parameter ≤0.5620, and unstable at parameter ≥0.5621. For our646
purposes, we chose ∗= 0.5621 to be the approximate bifurcation point. The critical647
eigenvalues were also computed by DDE-BIFTOOL, and it was found that they are648
λ∗= 0.0001 ±0.375i. Note that the real part is positive, which is a consequence of649
our choosing the parameter on the unstable side of the bifurcation point. We allow650
ourselves to be content with this approximation.651
To calculate the matrices Φ(t) and Ψ(t), we numerically computed the right and
left eigenvectors associated to the eigenvalue of the characteristic matrix ∆(λ∗;∗)
with the smallest absolute value. If ∗were the true bifurcation parameter rather
than a numerical approximation and λ∗was the true critical eigenvalue, we would
merely compute the kernel. Then, we calculate the matrices Φ(t) and Ψ(t) following
Remark 3.3. The rank of Φ(0) and Ψ(0) were both verified to be equal to 2. Following
this we defined a three-dimensional cell array U=cell(100,2,2) and populated it
with basis elements for Bdiag according to the assignment
U{i, j, k}= (0,...,0, Ejk ,0,...,0),
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 19
with the nonzero entry in the ith position. An ordered basis {V1, . . . , V400}was then
defined according to the rule
V4(i−1)+2(j−1)+k=U{i, j, k}, i = 1,...,100, j, k ∈ {1,2}.
Then, relative to the aforementioned ordered basis for Bdiag and the standard ordered652
basis {E11, E21 , E12, E22 }for R2×2, the map M0:Bdiag →R2×2was vectorized as653
a 4 ×400 matrix. The result was rank 4, and it follows that our example satisfies654
Corollary 3.10 with η= 0 (after a change of coordinates, treating ∗as zero). Following655
Remark 3.11, we are guaranteed that the uniform CPM posesses at least one feasible656
solution.657
The probe program (P˜
C) was first solved in MATLAB R2018a using the smooth
constrained solver fmincon from the optimization toolbox. The explicit formula for
the trivializing UPCC from (6.2) was used, and we supplied the solver with an explicit
gradient. The solver was initialized at the infeasible matrix (vectorized)
P0=100 100
100 100 
and converged quickly to the feasible matrix (vectorized)
P∗=0.7317 0.2672
−1.4729 0.3783 .
We then initialized the non-smooth patternsearch solver at this feasible point. To658
standard tolerances and minimum mesh size P∗was optimal. Proposition 4.13 pro-659
vides a feasible jump functional B0=X(P∗), but as we mentioned earlier, this660
candidate does not minimize the number of controlled nodes.661
6.3. Minimizing the number of controlled nodes. As discussed in the out-662
line, the jump functional B0minimizes the cost functional, but it does not necessarily663
control a minimal number of nodes. To address this, we recall that because of Lemma664
4.5 and Proposition 4.13, the cost function C:Bdiag →Ris in fact constant on the665
hyperplane666
H(P∗) = 

M+
0(P∗−Z) + Qy −
b
X
j=1
hM+
0(P∗−Z) + Qy, mjimj:y∈ U 

,667
=

B0+Qy −
b
X
j=1
hQy, mjimj:y∈ U 

,668
669
where Q=I−M+
0M0. Thus, if we wish to minimize the number of controlled670
nodes while maintaining the minimal cost, it suffices to solve the following sequence671
of unconstrained nonlinear programs:672
minimize
100
X
i=1
tanh µ||B0
i+πi(Qy||2),
subject to y∈ U,
(Nµ)673
674
where we define the linear operator Q:U → U by
Qy=Qy −
b
X
j=1
hQy, mjimj,
This manuscript is for review purposes only.

20 K.E.M. CHURCH AND X. LIU
1 10 20 30 40 50 60 70 80 90 100
Node index
5
6
7
8
9
10
11
12
Degree
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Absolute gain
Fig. 2.Plot of the degree (curve, solid blue) of nodes i= 1,...,100 versus the absolute gain
(stems, dashed dot orange) of node iby the cleaned jump functional. Fourteen nodes are controlled –
the small gain applied to node 61 is not visible under the scale of the present graph, as the associated
absolute gain was ||B∗
61|| = 0.01. Note that more expensive nodes were assigned a lower gain control,
and vice versa.
and πi(B1, . . . , B100) = Biis the projection onto the ith factor. The objective function675
of (Nµ) precisely counts the number of nodes controlled by the candidate B0+Qy∈676
M−1(P∗) in the limit µ→ ∞. Consequently, if yµis a solution for parameter µand677
yµ→y∗, then678
B∗=B0+Qy∗
(6.3)679
680
is a feasible jump functional having optimal cost that controls a minimal number of681
nodes.682
After apropriate vectorization consistent with Section 6.2, the smooth nonlinear683
program (Nµ) was solved using the unconstrained solver fminunc from MATLAB684
R2018a initially with parameter µ= 1 and initialized at y= 0. Subsequently, the685
program was solved with increments µ7→ µ+1
2and initialized at the previous solution.686
Upon reaching µ= 3, the solver made step sizes smaller than 10−7and we manually687
halted the process.688
The resulting jump functional B∗= (B∗
1, . . . , B∗
100) was cleaned by setting B∗
i7→ 0689
if ||B∗
i||F<10−4, where ||·||Fis the Frobenius norm. The result was a jump functional690
that pinned fourteen nodes. Defining ||B∗
i|| to be the absolute gain of node i, a plot691
of the absolute gain relative to the degree is provided in Figure 2. From this figure,692
it is clear that our algorithm prioritized the pinning of nodes that have a low degree.693
Indeed, all nodes with degree 5 and 6 are pinned, and the only other node to be694
pinned was the degree 9 node at index 61.695
6.4. Performance of the controller and secondary bifurcation. Plots of696
sample trajectories of the system without the pinning controller are given in Figure697
3, while those with pinning are provided in Figure 4. Several illustrative parameters698
are used.699
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 21
(a) = 0.5621 (b) = 0.59
(c) = 0.61 (d) = 0.63
Fig. 3.Sample trajectories (vertical axis) from random constant initial conditions drawn from
the standard normal distribuition, rescaled to the interval [−1,1] and plotted for various represen-
tative time intervals (horizontal axis). The neural activation parameter for the given simulation
is listed below its frame. Notice the transition from a stable periodic orbit to a stable equilibrium in
the parameter interval [0.61,0.63], indicative of another bifurcation point.
By comparing the two figures, it should be clear that our pinning control B∗
700
derived from the uniform CPM with subsequent node minimization succeeds in ex-701
ponentially stabilizing the trivial equilibrium in the activation parameter interval702
[∗,0.61], while when our controller is not present, the equilibrium is unstable. In-703
creased volatility and decreased convergence rates follow as the parameter increases,704
and a secondary bifurcation occurs somewhere in the parameter interval [0.61,0.63].705
This is consistent with our observation that the system without the pinning control706
appears to undergo a bifurcation in the interval [0.61,0.63]; see Figure 3. We sug-707
gested in the preamble to Section 5that such secondary bifurcations of the model708
without impulses may be responsible for poor performance or complete failure of the709
impulsive controller if the parameter is too far away from the bifurcation point. The710
presence of a secondary bifurcation in the interval [0.61,0.63] in both cases – with711
and without the controller – is consistent with this claim.712
7. Discussion. We have proposed a method of stabilizing a candidate equilib-713
rium point of an autonomous delay dynamical system at or near a bifurcation point714
This manuscript is for review purposes only.

22 K.E.M. CHURCH AND X. LIU
(a) = 0.5621 (b) = 0.59
(c) = 0.61 (d) = 0.63
Fig. 4.Sample trajectories (vertical axis) from random constant initial conditions drawn from
the standard normal distribuition, rescaled to the interval [−1,1], with different neural activation
parameters, plotted for various representative time intervals (horizontal axis). (a) Activation pa-
rameter = 0.5621; the trivial equilibrium is quickly stabilized. (b) Activation parameter = 0.59;
the trivial equilibrium is exponentially stabilized but with a slightly smaller rate parameter. (c) Ac-
tivation parameter = 0.61; the trivial equilibrium is stabilized, but the rate parameter is low. (d)
Activation parameter = 0.63; the control no longer stabilizes the trivial equilibrium. A secondary
bifurcation has occured.
using a novel impulsive stabilization approach based on invariant manifold theory. We715
have also introduced cost structure; our method identifies a jump functional (impul-716
sive controller) that minimizes a given cost functional, should certain controllers have717
cost associated to their implementation. The method, which we call the centre probe718
method (CPM), takes advantage of the dimension reduction inherent to the dynamics719
on the centre manifold. If the cost is projective, the CPM can be implemented with720
great efficiency.721
Our method differs from the majority of impulsive stabilization methods in that722
one does not design the controller to satisfy a system of matrix inequalities, but rather723
solves a pair of optimization problems. The most computationally expensive of the724
two – the probe program (P˜
C) – is done in a low-dimensional space determined by725
the number of eigenvalues crossing the imaginary axis at the bifurcation point. The726
second one – the inverse probe program (YP) – is convex, smooth and unconstrained727
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 23
(in an appropriate coordinate system), and can be solved efficiently using out-of-the-728
box nonlinear solvers. This is the greatest strength of our method compared to those729
derived from Lyapunov functionals based on linear matrix inequalities: constraint sat-730
isfaction for the CPM is done in a low-dimensional space, and if the cost is projective731
the low-dimensional probe program is sufficient. To contrast, the matrix inequalities732
of the other methods are checked in the original (potentially) high-dimensional space733
and must also be synthesized therein.734
Being an inherently local method, the CPM does not rely on global Lipschitzian735
condtions on the vector field or any boundedness contraints. The higher-order terms736
are not important and, notably, can grow superlinearly as ||x|| → ∞ without con-737
sequence. This is a strength of the CPM compared to Lyapunov functional-based738
linear matrix inequality (LMI) sufficient conditions for impulsive stabilization. The739
latter universally require Lipschitzian or Lipschitz-like conditions on the vector field740
in order to rigorously ensure stability. An unfortunate counter-point is that, being a741
local method, the CPM cannot guarantee global stability even if the vector field has742
a small Lipschitz constant.743
As mentioned in the introduction, minimization of a cost functional defined on744
a space of admissible jump functionals (impulsive controllers) appears to be unique745
to our method. Our framework contains as a subcase pinning impulsive stabilization,746
so in the stabilization of a complex network one can use our method to minimize the747
number of controlled nodes.748
The CPM was tested on an artificial neural network model with 100 identical749
linearly coupled nodes, in Section 6. A projective cost was chosen that weighted the750
cost of controlling each node based on its degree, with higher degree nodes being more751
expensive. We identified a bifurcation point and implemented the CPM to identify752
a probe element and an associated affine space of jump functionals that minimize753
the cost functional. The number of nodes was minimized by solving a sequence of754
unconstrained problems in this affine space, with the result being a pinning impulsive755
controller that controls fourteen specific nodes.756
We demonstrated by way of numerical simulations that our controller stabilizes757
the trivial equilibrium in a parameter interval near the bifurcation point, but that758
a secondary bifurcation occurs when the parameter becomes too large. A similar,759
secondary bifurcation occurs in the system without the impulsive controller.760
A weakness of our method in its current state is that such secondary bifurcations761
are barriers to further impulsive stabilization. Suppose for example that the uniform762
CPM is implemented at scalar parameter = 0 and a controller B(t) is found, but763
that a secondary bifurcation occurs at the parameter ∗6= 0. That is, the system764
˙x=f(xt, ), t /∈1
hZ765
∆x=B(t)xt−, t ∈1
hZ766
767
undergoes a bifurcation at the parameter =∗. Technically, the CPM cannot768
be applied at this parameter because the system is no longer an autonomous delay769
differential equation. In principle, the CPM could be extended to this case, but the770
computation of basis functions in the centre fiber bundle (the time-varying analogue of771
the centre subspace needed to handle nonautonomous systems in infinite-dimensional772
spaces) becomes much more difficult, and these are absolutely essential to the CPM.773
Such basis functions are inherently discontinuous and, to our knowledge, there are at774
present no rigorous numerical methods available to compute them in general.775
This manuscript is for review purposes only.

24 K.E.M. CHURCH AND X. LIU
Another limitation of our method is that it cannot in general be used to stabilize776
an unstable equilibrium point in a delay differential equation. The method only777
has a chance of success if there is a sufficiently nearby perturbation of the system778
such that the equilibrium satisfies the spectral gap condition. In the absence of779
such a nearby perturbation, traditional impulsive stabilization approaches based on780
matrix inequalities derived from Lyapunov functional-based LMI sufficient conditions781
for stability are more appropriate.782
The nonconvexity of the probe program does pose certain difficulties insofar as
implementation is concerned. These all centre around the characterization of the
feasible set as being the intersection of Pkwith the sublevel set {X∈(Rc×c)k:
ρ◦G(X)≤e−γ/h }.As we recalled previously at the beginning of Section 4, the
spectral radius ρ:Rc×c→Ris non-convex and non-smooth. However, on the set
V={X∈Rc×c: every eigenvalue of Xis simple},
the spectral radius is continuously differentiable and locally Lipschitz continuous. Vis783
also open and dense in Rc×c. As a consequence, methods based on gradient sampling784
[3,17] can be used to solve the program (P˜
C) with guaranteed convergence results.785
8. Conclusion. We have proposed the centre probe method, a novel impulsive786
stabilization method for delay differential equations based on invariant manifold the-787
ory. The method takes advantage of the low-dimensional properties of the centre788
manifold at a bifurcation point of the system to be stabilized, and allows an impul-789
sive controller to be identified that minimizes a cost functional that can take into790
account many structural features of the controller that may affect its implementation791
cost. The method consists of a pair of optimization problems, with the most com-792
putationally heavy constraint satisfaction being done in a typically low-dimensional793
space.794
The CPM has a few weaknesses. The uniform CPM is only guaranteed to provide795
some finte parameter interval where robust stability is guaranteed near a bifurcation796
point. The interval cannot be selected a priori and the controller generated by the797
uniform CPM willl generally fail to stabilize the system if the parameter is taken too798
far away from the bifurcation point. Also, the CPM cannot generally be used to stabi-799
lize an unstable equilibrium point: it is necessary for there to be a sufficiently nearby800
perturbation of the system such that the equilibrium point undergoes a bifurcation.801
Our method has several strengths compared to impulsive stabilization methods802
based on matrix inequalities derived from Lyapunov functionals. It does not rely803
on Lipschitzian (or Lipschitz-like) conditions to rigorously guarantee stability and804
the constraint satisfaction step is done in a low-dimensional space. Also, in complex805
networks of identical coupled oscillators, the dimension of this low-dimensional space806
remains low even while the number of nodes increases, while the number of constraints807
in a matrix inequality-based stability condition increases quadratically with the num-808
ber of nodes. Our method allows one to consider cost structure and optimize the809
controller with respect to this cost, as well as impose hard constraints on the types810
of controllers permitted.811
Acknowledgements. We are grateful to the reviewers for their careful reading812
of the manuscript and their helpful comments. Kevin E.M. Church acknowledges813
the support of the Natural Sciences and Engineering Research Council of Canada814
(NSERC) through the Alexander Graham Bell Canada Graduate Scholarships Pro-815
gram.816
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 25
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works and Learning Systems, 27 (2016), pp. 1486–1501, https://doi.org/10.1109/TNNLS.905
2015.2449898,http://ieeexplore.ieee.org/document/7161350/.906
[27] X. Zhao, P. Shi, Y. Yin, and S. K. Nguang,New Results on Stability of Slowly Switched907
Systems: A Multiple Discontinuous Lyapunov Function Approach, IEEE Transactions on908
Automatic Control, 62 (2017), pp. 3502–3509, https://doi.org/10.1109/TAC.2016.2614911.909
[28] Y. Zhou, C. Li, T. Huang, and X. Wang,Impulsive stabilization and synchronization of910
Hopfield-type neural networks with impulse time window, Neural Computing and Applica-911
tions, 28 (2017), pp. 775–782, https://doi.org/10.1007/s00521-015-2105-7.912
[29] G. Zong, X. Wang, and H. Zhao,Robust finite-time guaranteed cost control for impulsive913
switched systems with time-varying delay, International Journal of Control, Automation914
and Systems, 15 (2017), pp. 1–9, https://doi.org/10.1007/s12555-015-0314-6.915
Proofs.916
8.1. Proof of Proposition 3.1.1. That the set e
Y(, h;γ) is closed follows by917
the continuity of the spectral radius function ρ:Rc×c→R+. To prove the equality918
(3.7), consider the dynamics on the c-dimensional slice of the parameter-deppendent919
centre manifold at the point x= 0, for small parameter (, B)∈Rp× U of the IRFDE920
˙x=L0xt+L()xt+O(||xt||2), t /∈1
hZ(8.1)921
∆x=B(t)xt−, t ∈1
hZ,(8.2)922
923
where we have overloaded the notation and identify Bwith the functional on the
right-hand side of (2.1) and, if k > 1, using (2.6). At the parameter (, B) = 0, the
centre fiber bundle (equivalently, centre subspace) is determined by the homogeneous
linear equation without impulses
˙y=L0yt,
so the claim concerning Φ(t) and Ψ(t) as being basis matrices associated to the
eigenspaces E0and ET
0is true – see [8]. Moreover, we trivially have the Floquet
decomposition
Φt(θ) = Φ(0)eΛθeΛt:= Q(θ)eΛt
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 27
due to the characterization d
dt Φ(t) = Φ(t)Λ. This implies that Qt= Φ0is constant924
in t. In the notation of Section 3.3 from [2], the projection operator Pc(t) is constant925
and we therefore have926
Pc(t)χ0= Φ0hΨ0,Φ0i−1hΨ0, χ0i=QΓΨ(0) = Φte−ΛtΓΨ(0)927
928
Applying [Lemma 3.1, [2]], the dynamics on the c-dimensional slice of the parameter-929
dependent centre manfiold are given to linear order in z∈Rcby930
˙z= Λz+ ΓΨ(0)L()Φ0z+O(||z||2), t /∈1
hZ(8.3)931
∆z= ΓΨ(0)B(t)Φ0z(t−), t ∈1
hZ,(8.4)932
933
for (, B) sufficiently small. The monodromy matrix associated to the equilibrium934
z= 0 is precisely M,h (B), and the dynamics restricted to the centre manifold there-935
fore has O(e−γt ) as a convergence rate if and only if ρM,h(B)≤e−γ/h . Since all936
other Floquet exponents of the linearization of (8.1)–(8.2) have strictly negative real937
parts and the spectrum is upper hemicontinuous with respect to perturbations in938
(, B), the same convergence rate is achieved locally near x= 0 in the original non-939
linear system due to linearized stability principles and the assumption σ < γ. The940
converse follows by the same argument.941
942
2. Since the cost C:U → Ris nonnegative, the set X=C(e
Y(, h;γ)) ⊂Ris943
bounded below, so there exists a sequence Bn∈e
Y(, h;γ) such that C(Bn)→infX944
as n→ ∞. The sequence Bncannot be unbounded because Cis radially unbounded,945
from which we conclude by the Bolzano-Weierstrass theorem that Bnadmits a con-946
vergent subsequence having a limit B,h ∈e
Y(, h;γ), with the latter inclusion justified947
by part 1. As Cis continuous, we conclude that C(B,h) = infX.948
8.2. Proof of Theorem 3.4.To begin, we assume k= 1. If there exists a949
linear jump functional B∗such that M,h(B∗) = e−γ/h I, then the nonemptiness of950
e
Y(, h;γ) result will follow. Thus, it suffices to solve the equation951
(I+ ΓΨ(0)B∗Φ0) exp 1
h(Λ + Ψ(0)L()Φ0)=e−γ/h I.(8.5)952
953
If Φ(0) and Ψ(0) are rank c, there exists a left-inverse Φ+(0) and right-inverse Ψ+(0)
such that Φ+(0)Φ(0) = In×nand Ψ(0)Ψ+(0) = Ic×c. Then, the jump functional B∗
;γ
defined by
B∗
;γξ= Ψ+(0)Γ−1(e−γ/h −1) exp −1
h(Λ + Ψ(0)L()Φ0)Φ+(0)ξ(0)
satisfies (8.5). Since B∗
;γ→0 as γ→0+, it follows by minimality that also C(B∗
;γ)→954
0 as γ→0+for any selection γ7→ B∗
;γof cost-minimizing jump functionals for rate955
parameter γ. As Cis continuous and positive-definite, we conclude that B∗
;γ→0.956
If k > 1, consider the period kcycle of jump functionals B= (B∗
;γ/k , . . . , B∗
;γ/k ).957
Then M,h(B) = e−γ /hIand the argument proceeds as before.958
8.3. Proof of Corollary 3.5.If c= 1, then Φ(t) is a n×1 column vector959
and, as it constitutes a basis for E0, it cannot be identically zero. Moreover, as960
This manuscript is for review purposes only.

28 K.E.M. CHURCH AND X. LIU
Φ(t) = Φ(0)eΛt, we cannot have Φ(0) = 0. Consequently, Φ(0) has rank one. The961
same argument applies to Ψ(0).962
If L0φ=Cφ(0) for an n×nmatrix C, then Φ(t) = Z(t)Mfor some n×cwith963
matrix Mand Z(t) is the fundamental matrix solution of the finite-dimensional linear964
equation ˙z=Cz satisfying Z(0) = I. As each column of Φ(t) is a linear combination965
of the columns of Z(t), this set of columns is linearly independent if and only if M966
has maximal rank. As c≤n, it follows that Mhas rank c, so that the same is true967
of Φ(0). The same argument applies to Ψ(0).968
8.4. Proof of Lemma 4.6.Let x∈ M−1
1(X) and y∈ M−1
1(Y). Then, we have969
M1(tx + (1 −t)y) = M0(tx + (1 −t)y) + Z970
=t(M0(x) + Z) + (1 −t)(M0(y) + Z)971
=tM1(x) + (1 −t)M1(y)972
=tX + (1 −t)Y.973
974
Consequently, tx + (1 −t)y∈ M−1
1(tX + (1 −t)Y) for all t∈[0,1], x∈ M−1
1(X) and975
y∈ M−1
1(Y), from which the inclusion (4.2) follows.976
8.4.1. Proof of Theorem 4.11.Before we begin, we remark that by virtue of977
convexity, any local optimum of the program (YP) is automatially a global optimum.978
As such, we will refer to these as global optimums.979
Let ˜
Cbe a GPCC, let Pbe a global optimum for (Y) and Bbe a global optimum980
for the problem (YP). Suppose by way of contradiction that Bis not a global optimum981
for (Y). By Proposition 3.1, there must exist a global optimum B∗, and as ˜
Cis a982
GPCC we have ˜
C◦Mk(B∗)<˜
C◦Mk(B). But P=Mk(B), and the latter inequality983
implies Pis not optimal for the program (P˜
C), a contradiction. Therefore, Bis a984
global optimum for (Y).985
Now, let ˜
Cbe a LPCC, let Pbe a local optimum for (Y) and Bbe a global986
optimum for the problem (YP). Suppose by way of contradiction that Bis not a local987
optimum for (Y). Then, there exists a sequence Bn→Bsuch that C(Bn)<C(B).988
By continuity of Mk:Uk→(Rc×c)k, we have Pn:= Mk(Bn)→ Mk(B) = P, but989
local optimality of Pimplies that for nsufficiently large, ˜
C(P)≤˜
C(Pn). Using the990
LPCC property (type 1 or type 2) of ˜
C, it follows that C(B)≤ C(Bn) for nsufficiently991
large, which is a contradiction. The result follows.992
8.5. Proof of Theorem 4.12.Radial unboundedness follows from radial un-993
boundedness of C:U → R+. For continuity, we remark that ˜
Cis the optimal value994
function associated to the family of convex programs (YP) indexed by the param-995
eter P∈ Pk. We have M−1
k:Pk⇒Ukis closed-valued, convex-valued, and can996
be shown to be continuous (see the characterization provided by Lemma 4.5). The997
cost Cis convex and continuous, and the solution set is bounded for each Pdue to998
the radial unboundedness of C. Using [Theorem 1, [20]], we obtain continuity of ˜
C.999
Using Lemma 4.6, Lemma 4.8 and the convexity of C, if we use the subscript notation1000
This manuscript is for review purposes only.

STABILIZATION AND BIFURCATION SUPPRESSION 29
P= (P0, . . . , Pk−1) for arbitrary k-probe elements, then1001
˜
C(tX + (1 −t)Y) = min{C(V) : V∈ M−1
k(tX + (1 −t)Y)}1002
= min{C(V0, . . . , Vk−1) : Vi∈ M−1
1(tXi+ (1 −t)Yi)}1003
≤min{C(V0, . . . , Vk−1) : Vi∈tM−1
1(Xi)⊕(1 −t)M−1
1(Yi)}1004
=
k−1
X
i=0
min{C(txi+ (1 −t)yi) : xi∈ M−1
1(Xi), yi∈ M−1
1(Yi)}1005
≤
k−1
X
i=0
min{tC(xi) + (1 −t)C(yi) : xi∈ M−1
1(Xi), yi∈ M−1
1(Yi)}1006
≤t˜
C(X) + (1 −t)˜
C(Y),1007
1008
and we conclude that ˜
Cis convex.1009
Next we prove that ˜
Cis a GPCC. If B∗is a global optimum and Bis feasible,1010
then1011
˜
C◦ Mk(B∗) = min
x∈M−1
k(Mk(B∗))
C(X) = C(B∗)1012
≤min
x∈M−1
k(Mk(B))
C(X) = ˜
C◦ Mk(B).1013
1014
Bis also a global minimum if and only if C(B∗) = C(B), and in this case the calculation
above is simplified. We have
˜
C◦ Mk(B∗) = C(B∗) = C(B) = ˜
C◦ Mk(B).
Our final task is to prove that ˜
Cis a LPCC. We have1015
˜
C◦ Mk(B) = min
X∈M−1
k(Mk(B))
C(X)≤ C(B)1016
1017
because B∈ M−1
k(Mk(B)). Equality holds precisely when Bsatisfies (4.3).1018
8.6. Proof of Corollary 5.2.Under the assumption of the corollary, the set1019
{x≥0 : ∀|| ≤ x, ρ(M,h(B)) ≤e−γ /h}is bounded above, and therefore has a1020
supremum, η. By continuity of the spectral radius, it follows that Bis a feasible1021
solution for the program (Yη). To see that it is optimal, we recall that due to part1022
2 of Proposition 3.7, we have the inclusion e
Yη(h;γ)⊆e
Y(0, h;γ). Since Bminimizes1023
(locally or globally) C:e
Y(0, h;γ)→Rand is feasible for (Yη), it must also minimize1024
(locally or globally) C:e
Yη(h;γ)→R.1025
This manuscript is for review purposes only.