††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Creation of a tweezer array for cold atoms utilizing a generative neural
network

Zejian Ren  Microelectronics Thrust, The Hong Kong University of Science and
Technology (Guangzhou), Guangzhou, China  Xu Yan  Department of Physics, The
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong
Kong, China  Kai Wen  Microelectronics Thrust, The Hong Kong University of
Science and Technology (Guangzhou), Guangzhou, China  Huijin Chen
Microelectronics Thrust, The Hong Kong University of Science and Technology
(Guangzhou), Guangzhou, China  Elnur Hajiyev  Department of Physics, The Hong
Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong
Kong, China  Chengdong He  Department of Physics, The Hong Kong University of
Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China  Gyu-Boong
Jo<EMAIL_ADDRESS>Microelectronics Thrust, The Hong Kong University of Science
and Technology (Guangzhou), Guangzhou, China  Department of Physics, The Hong
Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong
Kong, China

###### Abstract

Optical tweezers have become essential tools for dynamically manipulating
objects, ranging from microspheres or biological molecules to neutral atoms.
In this study, we demonstrate the creation of tweezer arrays using a
generative neural network, which allows for the trapping of neutral atoms with
tunable atom arrays. We have successfully loaded cold strontium atoms into
various optical tweezer patterns generated by a spatial light modulator (SLM)
integrated with generative models. Our approach shortens the process time to
control the SLM wtih minimal time delay, eliminating the need for repeated re-
optimization of the hologram for the SLM.

## Introduction

An optical tweezer, a highly focused laser beam, have found applications in a
wide range of fields, including biology, chemistry, and material science
Minowa  _et al._ (2022). In recent years, optical tweezers have also been
widely exploited in quantum science Kaufman and Ni (2021), where they have
enabled the control and manipulation of individual atoms Endres  _et al._
(2016); Kim  _et al._ (2016) and molecules Cairncross  _et al._ (2021) in the
quantum regime with unprecedented precision. The ability to control quantum
systems at the individual level is crucial for the development of quantum
technologies, such as quantum computing Ebadi  _et al._ (2022), quantum
simulation Browaeys and Lahaye (2020) and quantum sensing Degen  _et al._
(2017).

To date, the creation of tweezer arrays using the spatial light modulator
(SLM) has been mostly achieved through classical algorithms like the
Gerchberg-Saxton (GS) algorithm Gerchberg (1972) and its variations Kim  _et
al._ (2019). However, these methods often face a trade-off between the quality
of reconstruction and the time required for calculations. Moreover, the time-
consuming nature of these algorithms limits their potential application in
quantum control with dynamic shot-to-shot re-configuration, as the quantum
system’s coherence time is finite Zhang  _et al._ (2016). Recently, the
emergence of artificial intelligence presents a new opportunity to optimize
control Wigley  _et al._ (2016); Milson  _et al._ (2023), analysis Zhang  _et
al._ (2019); Rem  _et al._ (2019); Bohrdt  _et al._ (2019); Ness  _et al._
(2020); Zhao  _et al._ (2021, 2022) and monitoring Chen  _et al._ (2023) of
quantum experiments, while also providing a new approach to holography He  _et
al._ (2016). In this work, we utilize the power of artificial intelligence and
propose a simple deep learning approach that utilizes generative neural
networks to generate atom arrays on demand based on the U-net model
Ronneberger  _et al._ (2015). Our work offers an effective way to advance
quantum control through the use of machine learning, which holds the potential
to drive dynamic control in optical-tweezer-based systems.

Figure 1:  Overview of the scheme for producing an atom array based on deep
learning To trap 88Sr atoms, a optical tweezer array is utilized. The Spatial
Light Modulator (SLM) is controlled by a generative neural network that is
based on the U-net model. This generative model is trained using a hologram
dataset consisting of tweezer images and corresponding holograms for the SLM.
Various tweezer configurations can be created on demand.

In the typical setup, a static optical pattern is generated by utilizing the
SLM, resulting in the formation of an optical tweezer array for neutral atoms.
To achieve this, a phase-only hologram (POH) needs to be loaded into the SLM.
This approach allows for the generation of arbitrary tweezer geometries,
providing flexibility and control. However, there is a trade-off between the
calculation time and the reconstruction time associated with this method.

One of the challenges with this technique is the need to re-optimize the
phase-only hologram whenever a new tweezer pattern is required. This re-
optimization step can be time-consuming and may hinder the real-time operation
of the atom array system. To address this limitation, our objective is to
leverage a trained generative neural network. By employing this neural
network, we aim to bypass the re-optimization process and realize the
generation of real-time atom arrays. By utilizing the trained generative
neural network, we anticipate significant improvements in the efficiency and
speed of generating atom arrays. This approach would eliminate the need for
repeated re-optimization of the phase-only hologram, saving valuable time and
resources. Ultimately, our goal is to enable the realization of real-time atom
arrays with enhanced flexibility and control, opening up new possibilities for
research and practical applications in the field of neutral atom manipulation.

## General idea

Fig. 1 illustrates the basic concept of using a neural network model to create
a tweezer atom array. Initially, we create two sets of data comprising the POH
and the corresponding light pattern produced by the SLM. Random tweezer
patterns with uniform light intensity are generated to form the datasets. We
obtain the light pattern from a monitoring CCD camera while each corresponding
POH is created by the GS algorithm Gerchberg (1972). Then, the designated
dataset is sequentially fed as input into the neural network based on the
U-net model Ronneberger  _et al._ (2015). We train the generative neural
network with the datasets consisting of paired matrix and phase data. Finally,
the trained network is subsequently utilized to predict the POH for generating
a tweezer pattern. The resultant POHs predicted by the generative model are
then transferred onto SLMs for trapping neutral atoms in the experiment. In
the experiment, we test the method by loading cold strontium atoms from a
magneto-optical trap to tweezer traps achieved by the SLM with generative
models Wen  _et al._ (2024). We successfully generated different
configurations of tweezers, with an emphasis on creating specific patterns
without experiencing any significant time delay.

Figure 2:  Diagram of the neural network based on the U-net model (a) The
architecture of generative neural network accepts a pair of images, followed
by the U-net model. (b) Down sampling and Up sampling block architecture. The
convolution (Conv) layer is used in down sampling block to increase the
channel number while the transpose convolution (TConv) layer is added to the
Up-sampling block in order to reconstruct the phase information

## Hologram dataset of the atomic array

As the first step to facilitate the training of the generative model, we
introduce a large-scale hologram dataset consisting of 2000 pairs of tweezer
images and corresponding holograms for the SLM (Holoeye; PLUTO-2.1-VIS-130).
Since the tweezer is usually formed in a far-field regime, we use the
traditional Fraunhofer zone method to create the dataset for training our
model, resulting in the optical reconstruction on the object plane. The
diffraction process in the GS algorithm is calculated according to the
Fraunhofer diffraction Hecht (1989).

$\displaystyle\hat{I}\left(x,y\right)$
$\displaystyle=|\hat{\mathbf{C}}(\bm{x},\bm{y})|^{2}=|\mathcal{F}\left\\{\exp[\mathrm{i}\bm{\Phi}\left(\bm{x}_{0},\bm{y}_{0}\right)]\right\\}|^{2}$
$\displaystyle=|{\mathcal{F}}\left\\{\exp[\mathrm{i}\bm{\varphi}\left(\bm{x}_{0},\bm{y}_{0}\right)]\cdot\exp\left[\mathrm{i}\frac{\pi}{\lambda
d}\left(\bm{x}_{0}^{2}+\bm{y}_{0}^{2}\right)\right]\right\\}|^{2},$

where $x,y,{x}_{0},{y}_{0}$ represent the coordinates on the object plane and
the hologram plane, $\mathbf{C}(\bm{x},\bm{y})$ represents the complex
amplitude distribution on the object plane, $\mathcal{F}$ denotes the Fourier
transform, $\bm{\varphi}$ is the phase generated by the network. In this
regime, the condition $d\lambda\gg\bm{x}_{0}^{2}+\bm{y}_{0}^{2}$ is well-
satisfied resulting in $exp\left[\mathrm{i}\frac{\pi}{\lambda
d}\left(\bm{x}_{0}^{2}+\bm{y}_{0}^{2}\right)\right]\\}$$\approx$1\. The
maximum intensity of the tweezer trap remains constant while the pattern of
the tweezer array is randomly selected to generate a dataset.

## Generative neural network based on the U-net model

The main idea of proposed scheme in this work is based on the U-Net structure
that is a commonly used segmentation model providing contextual information
Ronneberger  _et al._ (2015). The U-Net architecture consists of a contracting
down-sampling pathway that captures contextual information and a symmetric
expanding up-sampling pathway that facilitates accurate localization (see Fig.
2) Li  _et al._ (2022). Notably, the incorporation of skip connections from
the down-sampling to up-sampling pathway constitutes a pivotal feature of the
generative network, resulting in enhanced output image fidelity and detail
retention. Here the down-sampling path is a residual neural network,
consisting of down-sampling blocks and corresponding residual blocks. Each
block is composed of a set of batch normalization, nonlinearity(ReLU), and a
4$\times$4 convolutional layer stacked one above the other. And the residual
block effectively solves the degradation problem with skip connections He  _et
al._ (2016). The down-sampling is realized directly by convolutional layers
with a stride of two and a padding of one. The up-sampling path consists of
up-sampling blocks.

Figure 3: Experiment scheme generating atomic arrays (a) A 515 nm tweezer beam
generated by the SLM is projected to the glass cell through the bottom
objective lens. A 461-nm probe beam is directed from top to bottom and
captured by a camera. (b) Energy-level diagram for bosonic strontium atom.
Broad-line 461 nm and narrow-line 689 nm transitions are utilized to generate
MOTs, respectively. (c) Sketch of the experimental sequence used to produce
and load the tweezer arrays by projecting light from a SLM. Figure 4: Various
atom arrays produced by the generative neural network. (a) Experimental
absorption image of $\sim$20 tweezer sites, where each bright spot corresponds
to an ensemble of $\sim$30 88Sr atoms. Various lattices including square,
triangular and honeycomb geometry are formed by the trained generative neural
network. The lattice constant is 15 $\mu$m. Absorption images are recorded
with a CMOS camera. (b) With the atomic signal at the center of the tweezer
trap, we define the SNR as the ratio between $I_{signal}$ and $I_{noise}$. (c)
A comparison of algorithm runtime and tweezer array quality reveals that the
generative neural network (green) surpasses the GS algorithm (purple) when it
comes to the number of iterations needed. The SNR of the tweezer array is
achieved without requiring iteration.

The trained generative network outputs the phase value which is to be similar
to the desired phase pattern (i.e. POH) in the SLM plane. For training, we
compare the output and the part of phase value in training dataset with Loss
function $\mathcal{L}=\lvert\varphi_{out}-\varphi_{train}\rvert$. The
generative network are implemented and trained using paddlepaddle 2.42 Ma  _et
al._ (2019) on an NVIDIA RTX 3080 GPU with Adam optimizer. The learning rate
is 0.00001 and training runs for 6000 epochs. The training of U-Net neural
network takes 20 hours.

## Loading strontium atoms into the tweezer array

A tweezer array generated by the computer-generated holography (CGH) based on
the trained generative neural network is readily usable in a neutral atom
experiment Wen  _et al._ (2024). Here we produce a large sample of cold 88Sr
atoms in a narrow-line magneto-optical trap (MOT) in a glass cell (see Fig. 3)
Wen  _et al._ (2024). To obtain such a narrow-line MOT, a 88Sr atomic beam
generated by an oven in a custom-designed vacuum chamber is slowed down and
captured in a broad-line 461 nm two-dimensional MOT. Subsequently, a 461 nm
laser beam with low power, referred to as the "pushing beam" exerts a force on
the atoms in the 2D-MOT, guiding them into a 461 nm 3D MOT in a separate glass
cell as shown in Fig. 3. Finally, we create a 689 nm narrow-line MOT in which
a cold gas of 2$\times 10^{6}$ 88Sr atoms at the temperature of 2-5 $\mu$K are
prepared at the center of the glass cell. The sample preparation process
typically takes approximately 1 second Wen  _et al._ (2024). To create a
tweezer array, programmable CGHs are generated by the use of phase SLMs
(PLUTO-2.1-VIS-130) at the lens’ back focal plane of the objective lens as
described in Fig. 3. A 515 nm laser beam of 0.5W is sent to the SLM, resulting
in about twenty tweezer traps with the trap depth of $k_{B}\times$1.2 mK where
$k_{B}$ is the Boltzmann constant.

## Experimental analysis

With a tweezer array being switched on, a few tens of Sr atoms are trapped in
each micro trap, which allows us to examine the quality of the tweezer light
pattern generated. To record the number of trapped atoms in each trap, a
collimated 461nm laser beam is sent through the Sr atom array, and the optical
density (OD) is recorded by the CMOS camera (Dhyana 400BSI V3) Wen  _et al._
(2024). To avoid the unwanted reflections, such as stripes and Newtons rings,
we employ a double-exposure scheme. The first exposure is performed while the
atoms are trapped in the optical tweezers within 10us while the second
reference exposure is performed shortly after 0.5 s as a reference image. The
OD is calculated according to $OD=ln\frac{I_{atom}-I_{ref}}{I_{atom}-I_{ref}}$
where $I_{atom}$ is the first image with atoms and $I_{ref}$ is the image
without atom.

Fig. 4(a) showcases a variety of arrays generated by the generative neural
network. These arrays consist of square, triangular, and hexagonal lattices,
with each tweezer trap containing tens of Sr atoms. In order to analyze these
arrays, we extract the signal-to-noise ratio (SNR) from the absorption images
shown in Fig.ure 4(a). This is done by examining the maximum optical density
at the center of each tweezer trap ($I_{signal}$) and comparing it to the
background fluctuation ($I_{noise}$), as explained in Fig. 4(b).

The main benefit of using the generative neural network for obtaining CGHs is
that this approach overcomes the trade-off between computation time and the
quality of the final atom arrays. The generative neural network outperforms
the GS algorithm and the SNR of the tweezer array is achieved without
requiring iteration. For example, the neural network can generate the POH in
0.16s with the SNR 14dB as shown in Fig. 4(c). In contrast, atom arrays
produced by the traditional GS algorithm face the problem of speckle noise due
to the initial random phase and amplitude. As a result, several iterations are
inevitably required to achieve a comparable SNR, which takes about 6 seconds.
In comparing the runtime of the algorithm and the quality of the tweezer
array, therefore, it is evident that the generative neural network outperforms
the GS algorithm in terms of the required number of iterations.

In the current architecture, there is a slight variation of approximately
10$\%$ in the trap depth of individual tweezers from site to site. However,
this variation does not significantly affect the loading efficiency from the
MOT. Nonetheless, we expect that accurate control of the trap depth will be
required when implementing laser cooling techniques Tuchendler  _et al._
(2008), such as Raman sideband or Sisyphus cooling, which would also enable
fluorescence imaging. To produce a more uniform atom array and further reduce
the computational complexity, many extensions to the current method are worth
investigating. One is training the generative network with weighted GS
algorithm Di Leonardo  _et al._ (2007). For every atom array point, weighted
coefficient feedback from every iteration will enhance uniformity. The current
network architecture is based on the universal U-Net. It is conceivable that
other advanced network architectures, such as generative adversarial network
and graph neural network can also be applicable.

## Conclusion

In summary, we have introduced a method for generating tweezer arrays to trap
cold atoms utilizing state-of-the-art generative neural networks. Our findings
demonstrate the remarkable capability of the U-net based deep learning
approach in successfully producing arrays of Sr atoms. Importantly, this
method eliminates the need for time-consuming iterative procedures such as the
GS algorithm. Furthermore, the Sr atom array discussed in this work is not
limited to its current form but can be easily applied to fermionic isotopes.
This would open up exciting possibilities for exploring the SU($N$) physics in
a tweezer trap He  _et al._ (2019).

## Acknowledgement

GBJ acknowledges support from the RGC through 16306119, 16302420, 16302821,
16306321, 16306922, 16302123, C6009-20G, N-HKUST636-22, and RFS2122-6S04. RZJ
acknowledges support from the CPSF through 2022M720890. CH acknowledges
support from the RGC for RGC Postdoctoral fellowship.

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# Feedback equivalence and uniform ensemble reachability

Michael Schönlein  Institute for Mathematics, University of Würzburg, Germany
<EMAIL_ADDRESS>
###### Abstract

This paper considers feedback methods for ensemble reachability of parameter-
dependent linear systems $(A(\theta),B(\theta))$, where the parameter $\theta$
is varying over a compact Jordan arc in the complex plane. Recently, pointwise
testable sufficient conditions for uniform ensemble reachability have been
developed. Beside the necessity of pointwise reachablility these conditions
put restrictions on the spectra of the matrices $A(\theta)$ and the Hermite
indices of the pair $(A(\theta),B(\theta))$. In this paper we show that these
conditions can be ensured by applying a suitable feedback transformation if
the pair $(A(\theta),B(\theta))$ is pointwise reachable and it Kronecker
indices are independent from the parameter.

###### keywords:

Parameter-dependent linear systems , ensemble reachability , canonical from ,
Kronecker indices , Hermite indices , Feedback equivalence

###### MSC:

[2010] 93B05 , 93B52 , 93B55

††journal: Journal of LaTeX Templatesmytitlenotemytitlenotefootnotetext: This
work was supported by the German Research Foundation (DFG) under grant SCHO
1780/1-1.

##  1 Introduction

An emerging field in mathematical systems and control theory refers to the
task of controlling a large, potentially infinite, number of states, or
systems, using a single input function or a single feedback controller. Main
goals of a control theory for such problems is to establish fundamental system
theoretic methods and results in the context, i.e., to prove the existence of
parameter independent open-loop and closed-loop controllers, develop methods
for constructing them and tackle relevant system theoretic tasks. This is a
huge area and the term ensemble control has been established to refer to this
area of research, cf. [9, Section 2.4].

Ensembles arise in statistical approaches to linear systems, where the states
are replaced by probability density functions. The design of controllers that
morph one probability density function into another one then become control
problems of the Fokker-Planck equation or the Liouville transport equations,
cf. [4, 9, 11, 17, 42, 44, 43].

Ensemble control also embraces the situation of uncertainties in the model
parameters. In this cases the task is to control a parameter-dependent system
with a single or a few open-loop inputs which are independent from the unknown
model parameters [29]. Recently, there has been much interest in motion
control problems for spatio-temporal systems and infinite platoons of vehicles
[5, 12, 32, 33]. Using Fourier-transform techniques, such control systems can
be identified with parameter-dependent families of linear systems, cf. [5,
13].

Besides, this topic is not entirely new and other terms then ensemble control
are also present in the literature such as simultaneous control or control of
families of systems, cf. [8, 23, 25, 31, 37]. Other closely related topics are
robust control [3], and the blending problem, as considered in [38]. Sontag
(together with Hautus and Wang) considered stabilization and pole-shifting for
parameterized families of systems, cf. [25, 35, 37]. Besides, the series of
papers by Ghosh [19, 20, 21, 22] investigates the possibility of
simultaneously solving classical control problems for finitely many systems.

For recent contributions to the controllability problem for parameter-
dependent systems we refer to [1, 2], [10], [28, 29, 30] and [41]. Agrachev
and Sarychev consider ensemble controllability for nonlinear drift-free
parameter-dependent systems and provide a characterization in terms of Lie-
brackets. In the same direction, the work of Chen [10] also treats nonlinear
systems and considers Lie extensions. We note that these approaches do not
apply to the setting in this thesis. In [28] a characterization for ensemble
controllability for time-varying parameter-dependent linear systems is
presented, which is based on the singular value decomposition of the
reachability operator.

A recent observation from the control of probability densities is that the
range of achievable tasks becomes much wider by using a mixture of open-loop
and feedback controllers. In [9] and [14] it is shown that with pure open-loop
controls only the mean value can be controlled, whereas by using additionally
a feedback controller it is possible to control the mean value as well as the
variance of the probability distribution. In the context of parameter-
dependent systems and ensemble reachability, however, feedback methods have
not been addressed so far. This paper devoted to this problem. In our analysis
embraces the Hermite and the Kronecker indices. We note that in a series of
papers Baragãna, Zaballa and co-workers tackled the relation between the
Hermite indices and the controllability indices as well as the their behavior
under perturbation and feedback, cf. [6, 7, 40].

The organization of the paper is as follows. In Section 2 we introduce the
class of systems under consideration. It also provides the definition of
uniform ensemble reachability and recalls a known set of sufficient conditions
that prepare the ground for the analysis in this paper. In this section we
recall the definition of the Kronecker and Hermite indices and adapt the
notion of feedback equivalence to parameter-dependent systems. In Section 3 we
show that a pair of parameter-dependent matrices is restricted feedback
equivalent to the Brunovsky from if its Kronecker indices do not depend on the
parameter. Section 4 contains the main results of the paper. That is, we show
that for single-input pairs $(A,b)$ there is a continuous parameter-dependent
feedback $f$ such that $(A+bf,b)$ is uniformly ensemble reachable if the pair
is pointwise reachable. In this section we also show that in the for multi-
input pairs $(A,B)$ there is feedback transformation so that the feedback
transformed pair becomes uniformly ensemble reachable if it is pointwise
reachable and the Kronecker indices are constant. In Section 5 we investigate
the controlled harmonic oscillator in light of the present context. For this
example, we tackle the harder problem of deriving a parameter-independent
feedback. Moreover, under the additional assumption that the desired terminal
states satisfy a Lipschitz condition, we also provide an estimate how the
feedback gain influences the approximation of the terminal states.

##  2 Problem statement, notation and known results

In this paper we consider the reachability properties of parameter-dependent
linear systems with the additional constraint that the open-loop control input
is independent of the parameter. To investigate continuous-time systems

$\displaystyle\tfrac{\partial}{\partial t}x(t,\theta)$
$\displaystyle=A(\theta)x(t,\theta)+B(\theta)u(t)$ (1)

and discrete-time systems

$\displaystyle x_{t+1}(\theta)$
$\displaystyle=A(\theta)x_{t}(\theta)+B(\theta)u_{t}$ (2)

along the same lines the initial condition is in both cases
$x(0,\theta)=x_{0}(\theta)=0$ for each parameter $\theta\in\mathbf{P}$. The
parameter space $\mathbf{P}$ is assumed to be a Jordan arc in the complex
plane, i.e. $\mathbf{P}$ is the image of a continuous and bijective function
defined on a compact interval. The matrix-valued functions
$A:\mathbf{P}\mapsto\mathbb{C}^{n\times n}$ and
$B:\mathbf{P}\mapsto\mathbb{C}^{n\times m}$ are assumed to be continuous and
we will use the short notation $(A,B)\in C_{n,n}(\mathbf{P})\times
C_{n,m}(\mathbf{P})$ to express this. Moreover, we denote by
$C_{n}(\mathbf{P})$ the space of continuous functions from $\mathbf{P}$ to
$\mathbb{C}^{n}$. The time domain is $\mathbb{N}_{0}$ in the discrete-time
case or $[0,\infty)$ in the continuous-time case, and the inputs are $u\in
L^{1}_{\text{loc}}\left([0,\infty),\mathbb{C}^{m}\right)$ or
$u=(u_{0},u_{1},\dots),u_{i}\in\mathbb{C}^{m}$, respectively. Let

$\displaystyle\varphi(T,0,u)(\theta)=\int_{0}^{T}\mathrm{e}^{A(\theta)(T-s)}B(\theta)u(s)\mathrm{d}s$

and

$\displaystyle\varphi(T,0,u)(\theta)=\sum_{k=0}^{T-1}A(\theta)^{k}B(\theta)u_{T-1-k}$

denote the solutions of (1) and (2), respectively.

The central notion of reachability that will be considered in this paper is as
follows. A pair $(A,B)$ is called _uniformly ensemble reachable (from zero)_ ,
if for any $f\in C_{n}(\mathbf{P})$ and any $\varepsilon>0$ there exist $T>0$
and $u\in L^{1}_{m}([0,T])$ or $u=(u_{0},u_{1},\dots,u_{T-1}),\
u_{i}\in\mathbb{C}^{m}$ such that

$\displaystyle\|\varphi(T,0,u)-f\|_{\infty}=\sup_{\theta\in\mathbf{P}}\|\varphi(T,0,u)(\theta)-f(\theta)\|<\varepsilon.$

We note that the notion ensemble reachability coincides with approximate
reachability for the infinite-dimensional system, cf. [15]. Also, we note that
in continuous-time ensemble reachability is independent from the final time,
i.e. if it holds for some $T>0$ is also holds for every $T>0$, cf. [16, 39].
Thus, for continuous-time systems ensemble reachability is equivalent to
complete ensemble reachability, i.e. for every $x_{0},f\in C_{n}(\mathbf{P})$,
for every $\varepsilon>0$ and for every $T>0$ there is an input $u\in
L^{1}_{m}([0,T])$ such that $\|\varphi(T,x_{0},u)-f\|_{\infty}<\varepsilon$.
Also we recall that exact ensemble reachability (i.e. $\varepsilon=0$) is
never possible if the parameter space is infinite. In contrast, if
$\mathbf{P}=\\{\theta_{1},...,\theta_{N}\\}$ is finite, ensemble reachability
boils down to the classical finite-dimensional reachability of the
corresponding parallel connection defined by the pair

$\displaystyle\begin{pmatrix}A(\theta_{1})&&\\\ &\ddots&\\\
&&A(\theta_{N})\end{pmatrix},\quad\begin{pmatrix}B(\theta_{1})\\\ \vdots\\\
B(\theta_{N})\end{pmatrix}$

Problem statement: In this paper we explore how the application of feedback
methods enlarges the class of the parameter-dependent linear systems that are
uniformly ensemble reachable. We consider inputs of the form

$\displaystyle u(t,x)=F(\theta)x(t,\theta)+u(t),\quad F\in
C_{n,m}(\mathbf{P}).$

and aim at conditions on $(A,B)\in C_{n,n}(\mathbf{P})\times
C_{n,m}(\mathbf{P})$ guaranteeing the existence of a feedback $F\in
C_{n,m}(\mathbf{P})$ and an open-loop input $u$ such that the mixed open-loop
and feedback controlled pair $(A+BF,B)$ is uniformly ensemble reachable.

In recent years, some effort has been spent to derive necessary and sufficient
conditions for uniform ensemble reachability that are verifiable just in terms
of the matrices $A(\theta)$ and $B(\theta)$, cf. [15, 30]. Exemplary, we
recall the following set of sufficient conditions, cf. [15, Corollary 4].

In the case that $\mathbf{P}$ is a Jordan arc a pair $(A,B)$ is uniformly
ensemble reachable if the following conditions are satisfied:

  1. (N1)

$(A(\theta),B(\theta))$ is reachable for all $\theta\in\mathbf{P}$.

  2. (N2)

For all distinct parameters $\theta,\theta^{\prime}\in\mathbf{P}$, the spectra
$\sigma\big{(}A(\theta)\big{)}$ and $\sigma\big{(}A(\theta^{\prime})\big{)}$
are disjoint.

  3. (S)

For each $\theta\in\mathbf{P}$, the eigenvalues of $A(\theta)$ are simple.

  4. (H)

The Hermite indices $h_{1}(\theta),\dots,h_{m}(\theta)$ of
$(A(\theta),B(\theta))$ are independent of $\theta\in\mathbf{P}$.

Before we recall the definition of the Hermite indices, some comments are in
order. Condition (N1) is also necessary for uniform ensemble reachability.
Moreover, for single-input systems condition (N2) is also necessary for
uniform ensemble reachability. The label for condition (S2) is chosen such
that the notation is consistent with the labeling in [34].

Next we recall relevant lists of indices from finite-dimensional linear
control theory, cf. [27]. Let $(A,B)\in\mathbb{C}^{n\times
n}\times\mathbb{C}^{n\times m}$ be a reachable pair, i.e.

$\displaystyle\operatorname{rank}\begin{pmatrix}B&AB&\cdots&A^{n-1}B\end{pmatrix}=n.$

Let $b_{i}$ denote the $i$th column of $B$. Selecting from left to right the
first linear independent vectors

$\displaystyle
b_{1},b_{2},...,b_{m},Ab_{1},...,Ab_{m},A^{2}b_{1},...,A^{2}b_{m},\ldots,A^{n-1}b_{1},\ldots,A^{n-1}b_{m}$
(3)

one obtains a list of basis vectors of the reachability subspace as

$\displaystyle
b_{1},\ldots,A^{\kappa_{1}-1}b_{1},\ldots,b_{m},\ldots,A^{\kappa_{m}-1}b_{m}.$

The integers $\kappa(A,B)=(\kappa_{1},\dots,\kappa_{m})$ are called the
_Kronecker indices_ of $(A,B)$, where $\kappa_{i}:=0$ if the vector $b_{i}$
has not been selected. Associated to the Kronecker indices
$\kappa=(\kappa_{1},...,\kappa_{m})$ we define the pair
$(A_{\kappa},B_{\kappa})$ given by

$\displaystyle A_{\kappa}=\begin{pmatrix}{A_{\kappa_{1}}}&{0}&{\cdots}&{0}\\\
{0}&{A_{{\kappa_{2}}}}&{\cdots}&{0}\\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{0}&{0}&{\cdots}&{A_{\kappa_{m}}}\end{pmatrix}\quad\text{ and }\quad
B_{\kappa}=\begin{pmatrix}{b_{\kappa_{1}}}&{0}&{\cdots}&{0}\\\
{0}&{b_{\kappa_{2}}}&{\cdots}&{0}\\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{0}&{0}&{\cdots}&{b_{\kappa_{m}}}\end{pmatrix},$ (4)

where for $\kappa_{i}\geq 1$ the blocks $A_{\kappa_{i}}$ have size
$k_{i}\times k_{i}$ and the $b_{\kappa_{i}}$ are columns of size $k_{i}$ and
have the following form

$\displaystyle A_{\kappa_{i}}=\left(\begin{array}[]{ c c c c c
}{0}&{0}&{0}&{\dots}&{0}\\\ {1}&{0}&{0}&{\dots}&{0}\\\
{\vdots}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\\ {0}&{\ddots}&{1}&{0}&{0}\\\
{0}&{0}&{\ldots}&1&{0}\end{array}\right),\quad
b_{k_{i}}=\left(\begin{array}[]{ c }{1}\\\ {0}\\\ {\vdots}\\\ {0}\\\
{0}\end{array}\right)$

and if $\kappa_{i}=0$ the block $A_{\kappa_{i}}$ is absent and
$b_{\kappa_{i}}=0$.

Besides, selecting from left to right the first independent vectors

$\displaystyle
b_{1},Ab_{1},...,A^{n-1}b_{1},b_{2},Ab_{2},...,A^{n-1}b_{2},...,b_{m},Ab_{m},\ldots,A^{n-1}b_{m}$

one obtains another list of basis vectors of the reachability subspace

$\displaystyle
b_{1},\ldots,A^{h_{1}-1}b_{1},\ldots,b_{m},\ldots,A^{h_{m}-1}b_{m}.$

The integers ${h}(A,B)=(h_{1},\dots,h_{m})$ are called the _Hermite indices_ ,
where $h_{i}:=0$ if the vector $b_{i}$ has not been selected. Recall that a
pair $(A,B)$ is reachable if and only if

$\displaystyle\sum_{j=1}^{m}\kappa_{j}=\sum_{j=1}^{m}h_{j}=n.$

Furthermore, we briefly adapt the notion of feedback equivalence from finite-
dimensional linear systems to the present context of parameter-dependent
linear systems. For a more comprehensive exposition, we refer to [18, Sections
6.1 and 6.3], [36, Section 5.2].

A triple $(T,F,S)\in C_{n,n}(\mathbf{P})\times C_{m,n}(\mathbf{P})\times
C_{m,m}(\mathbf{P})$ is called a _restricted feedback transformation_ if it
satisfies $T(\theta)\in\operatorname{GL}_{n}(\mathbb{C})$ and
$S(\theta)\in\mathbf{U}_{m}$ for every $\theta\in\mathbf{P}$, i.e. $T(\theta)$
is invertible and $U(\theta)$ is upper triangular so that all diagonal entries
are $1$. The term restricted refers to the requirement that
$S(\theta)\in\mathbf{U}_{m}$. In contrast, $(T,F,S)\in
C_{n,n}(\mathbf{P})\times C_{m,n}(\mathbf{P})\times C_{m,m}(\mathbf{P})$ would
be called a _feedback transformation_ if
$T(\theta)\in\operatorname{GL}_{n}(\mathbb{C}^{n})$ and
$S(\theta)\in\operatorname{GL}_{m}(\mathbb{C})$ for every
$\theta\in\mathbf{P}$.

In the following, for $M\in C_{n,m}(\mathbf{P})$ and $N\in
C_{m,p}(\mathbf{P})$ we will write $M\,N$ as a short notation for
$M(\theta)N(\theta)$ for every $\theta\in\mathbf{P}$. Moreover, the set of a
restricted feedback transformations defines the _restricted feedback group_
$\mathscr{F}_{n,m}(\mathbf{P})$, where the composition of two elements
$(T_{1},F_{1},S_{1})$ and $(T_{2},F_{2},S_{2})$ is given by

$\displaystyle\left(T_{1},F_{1},S_{1}\right)\circ\left(T_{2},F_{2},S_{2}\right)=\left(T_{1}T_{2},F_{1}T_{2}+S_{1}F_{2},S_{1}S_{2}\right).$

Further, the neutral element of the restricted feedback group
$\mathscr{F}_{n,m}(\mathbf{P})$ is $\left(I_{n},0_{n,m},I_{m}\right)$ and the
inverse of an element $(T,F,S)$ is given by
$\left(T^{-1},-S^{-1}FT^{-1},S^{-1}\right)$. The restricted feedback group
$\mathscr{F}_{n,m}(\mathbf{P})$ acts on a pair $(A,B)\in
C_{n,n}(\mathbf{P})\times C_{n,m}(\mathbf{P})$ as follows

$\displaystyle(A,B)\quad\mapsto_{(T,F,S)}\quad\left(T(A-BS^{-1}F)T^{-1},TBS^{-1}\right)$

For notational convenience we will write

$\displaystyle(T,F,S)\cdot(A,B):=\left(T(A-BS^{-1}F)T^{-1},TBS^{-1}\right).$

Two pairs $(A_{1},B_{1}),(A_{2},B_{2})$ in $C_{n,n}(\mathbf{P})\times
C_{n,m}(\mathbf{P})$ are called _restricted feedback equivalent on_
$\mathbf{P}$, denoted by $\sim_{\mathbf{P}}$, if there exists
$(T,F,S)\in\mathscr{F}_{n,m}(\mathbf{P})$ such that

$\displaystyle(A_{2},B_{2})=(T,F,S)\cdot(A_{1},B_{1}).$

Note that $(A_{1},B_{1})\sim_{\mathbf{P}}(A_{2},B_{2})$ if and only if there
is a restricted feedback transformation $(T,F,S)$ such that

$\displaystyle\begin{split}TA_{1}-A_{2}T&=B_{2}F\\\
TB_{1}&=B_{2}S.\end{split}$ (5)

Also, we recall that the Kronecker indices $\kappa(A,B)$ are invariant under
restricted feedback transformations, cf. [18, Lemma 6.16], i.e. for all
$(T,F,S)\in\mathscr{F}_{n,m}(\mathbf{P})$ it holds

$\displaystyle\kappa(A,B)(\theta)=\kappa\left(T(A+BF)T^{-1},TBS^{-1}\right)(\theta).$

For future use we recap the following well-known result.

###### Lemma 1

Let $T\in C_{n,n}(\mathbf{P})$ and suppose that
$T(\theta)=\big{(}t_{1}(\theta),\cdots,t_{n}(\theta)\big{)}\in\operatorname{GL}_{n}(\mathbb{C})$
for every $\theta\in\mathbf{P}$.

  1. (a)

Then the mapping $\theta\mapsto T(\theta)^{-1}$ is continuous.

  2. (b)

Let $v\colon\mathbf{P}\to\mathbb{C}^{n}$ be continuous. Then, the coordinates
$\alpha_{1}(\theta),\dots,\alpha_{n}(\theta)$ of $v(\theta)$ with respect to
the basis $t_{1}(\theta),\dots,t_{n}(\theta)$ depend continuously on $\theta$.

##  3 Feedback Equivalence for parameter-dependent linear systems and
canonical forms

In this section we show that every pointwise reachable pair $(A,B)\in
C_{n,n}(\mathbf{P})\times C_{n,m}(\mathbf{P})$ with constant Kronecker indices
is restricted feedback equivalent to $(A_{\kappa},B_{\kappa})$. This result
will be used in the proof of Theorem 3 and might be of independent interest.
The proof follows the exposition in [18], where the finite-dimensional case is
treated. The main step in the subsequent proof will be to conclude that the
constructed restricted feedback transformation in [18, Proof of Theorem 6.18]
is continuous in the parameter.

###### Theorem 1

Suppose that $(A,B)\in C_{n,n}(\mathbf{P})\times C_{n,m}(\mathbf{P})$ has
constant Kronecker indices $\kappa(A,B)(\theta)=(\kappa_{1},...,\kappa_{m})$
satisfying $\sum_{i=1}^{m}\kappa_{i}=n$, then
$(A,B)\sim_{\mathbf{P}}(A_{\kappa},B_{\kappa})$.

###### Proof.

Let $\kappa(A,B)=(\kappa_{1},\cdots,\kappa_{m})$ denote the constant Kronecker
indices. We will construct a suitable feedback transformation
$(T,F,S)\in\mathscr{F}_{n,m}(\mathbf{P})$ in four steps.

Step 1: First we apply a transformation of the form $(I,0,U)$. By definition
and since $(A,B)$ is pointwise reachable, it holds that

$\displaystyle
b_{1}(\theta),...,A(\theta)^{\kappa_{1}-1}b(\theta)_{1},b(\theta)_{2},...,A(\theta)^{\kappa_{2}-1}b(\theta)_{2},...,b(\theta)_{m},...,A(\theta)^{\kappa_{m}-1}b(\theta)_{m}$

is a basis of $\mathbb{C}^{n}$ for each $\theta\in\mathbf{P}$. By
construction, for every $i=1,...,m$ there are functions
$\alpha_{ij}\colon\mathbf{P}\to\mathbb{C}$ and
$\beta_{ijl}\colon\mathbf{P}\to\mathbb{C}$ such that

$\displaystyle{A(\theta)^{\kappa_{i}}b(\theta)_{i}}={\sum_{j<i}}\alpha_{ji}(\theta)A(\theta)^{\kappa_{i}}b(\theta)_{j}+\sum_{l=1}^{\kappa_{i}}A(\theta)^{l-1}\sum_{j=1}^{m}\beta_{ijl}(\theta)b(\theta)_{j}$
(6)

Since the functions $\theta\mapsto A(\theta)^{l}b(\theta)_{i}$ are continuous
for every $l=0,1,2,...$ and $i=1,...,m$, by Lemma 1 (b) we have that
$\alpha_{ji}\in C(\mathbf{P})$ and $\beta_{ijl}\in C(\mathbf{P})$. Then, we
define the continuous upper-triangular matrix

$\displaystyle
U(\theta)=\begin{pmatrix}1&-\alpha_{12}(\theta)&-\alpha_{13}(\theta)&\cdots&-\alpha_{1m}(\theta)\\\
0&1&-\alpha_{23}(\theta)&\cdots&-\alpha_{2m}(\theta)\\\
\vdots&0&1&\ddots&\vdots\\\ &&\ddots&\ddots&-\alpha_{(m-1)\,m}(\theta)\\\
0&&&0&1\end{pmatrix}.$

Then, for $\tilde{B}(\theta):=B(\theta)U(\theta)$ the columns of $\tilde{B}$
and $B$ are related as follows

$\displaystyle{\tilde{b}(\theta)_{i}}=b(\theta)_{i}-\sum_{j<i}\alpha_{ji}(\theta)b(\theta)_{j},\quad
i=1,\ldots,m,$

or equivalently

$\displaystyle{b(\theta)_{i}}=\tilde{b}(\theta)_{i}-\sum_{j<i}\tilde{\alpha}_{ji}(\theta)\tilde{b}(\theta)_{j},$

where $\tilde{\alpha}_{ji}(\theta)$ denote the entries of $U^{-1}$. Note that
$\tilde{\alpha}_{ji}\colon\mathbf{P}\to\mathbb{C}$ are continuous functions.
For each $i=1,...,m$ it follows from (6) that

$\displaystyle\begin{split}A(\theta)^{\kappa_{i}}{\tilde{b}(\theta)_{i}}&=\sum_{l=1}^{\kappa_{i}}A(\theta)^{l-1}\sum_{j=1}^{m}{\beta}_{ijl}(\theta){b(\theta)_{j}}\\\
&=\sum_{l=1}^{\kappa_{i}}A(\theta)^{l-1}\sum_{j=1}^{m}{\beta}_{ijl}(\theta)\left(\tilde{b}(\theta)_{j}-\sum_{\mu<j}\tilde{\alpha}_{\mu
j}\tilde{b}(\theta)_{\mu}\right)\\\
&=\sum_{l=1}^{\kappa_{i}}A(\theta)^{l-1}\sum_{j=1}^{m}{\tilde{\beta}}_{ijl}(\theta){\tilde{b}(\theta)_{j}},\end{split}$
(7)

with $\tilde{\beta}_{ijl}\in C(\mathbf{P})$. To see that these functions are
continuous, observe that the $\tilde{\beta}_{ijl}$ are compositions of the
continuous functions $\tilde{\alpha}_{ji}$ and ${\beta}_{ijl}$.

Step 2: In this step we will construct a continuous transformation
$(T^{-1},0,I)$ so that the $0$\- and $1$-entries are at the right places. To
this end, for $i=1,\dots,m$ and $l:=2,\dots\kappa_{i}$ we define continuous
vectors

$\displaystyle\mathrm{v}_{1i}(\theta)$
$\displaystyle:={\tilde{b}(\theta)_{i}}$
$\displaystyle\mathrm{v}_{li}(\theta)$
$\displaystyle:=A(\theta)^{l-1}{\tilde{b}}(\theta)_{i}-\sum_{\mu=1}^{l-1}A(\theta)^{l-1-\mu}\sum_{j=1}^{m}\tilde{\beta}_{ij(\kappa_{i}+1-\mu)}(\theta)\tilde{b}(\theta)_{j}$

and the transformation

$\displaystyle
T(\theta):=\begin{pmatrix}\mathrm{v}_{11}(\theta)&\cdots&\mathrm{v}_{\kappa_{1}1}(\theta)&\mathrm{v}_{12}(\theta)&\cdots&\mathrm{v}_{\kappa_{2}2}(\theta)&\cdots&\mathrm{v}_{1m}(\theta)&\cdots&\mathrm{v}_{\kappa_{m}m}(\theta)\end{pmatrix}.$

To see that $T(\theta)$ is continuously invertible, by Lemma 1 (b) it suffices
to show that for every $\theta\in\mathbf{P}$ the columns of $T(\theta)$ define
a basis of $\mathbb{C}^{n}$. Indeed, fix $\theta^{*}\in\mathbf{P}$ and let

$\displaystyle\mathscr{X}:=\operatorname{span}\\{\mathrm{v}_{li}(\theta^{*})\,|\,i=1,...,m,\,l=1,...,\kappa_{i}\\}.$

Since $(A(\theta^{*}),B(\theta^{*}))$ is reachable the claim follows by
verifying that $\mathscr{X}$ is $A(\theta^{*})$-invariant. To ease notation,
we drop the dependence on $\theta^{*}$ and shortly write

$\gamma_{il}:=\sum_{j=1}^{m}\tilde{\beta}_{ijl}\tilde{b}_{j}=\sum_{j=1}^{m}\tilde{\beta}_{ijl}\mathrm{v}_{1j}\in\mathscr{X}$

for a moment. Then, for $i=1,...,m$ and $l<\kappa_{i}$ we have

$\displaystyle\begin{split}A\mathrm{v}_{li}&=A^{(l+1)-1}\tilde{b}_{i}-A\left(A^{l-2}\gamma_{i\kappa_{i}}+A^{l-3}\gamma_{i(\kappa_{i}-1)}+\cdots\gamma_{i(\kappa_{i}-l)}\right)-\gamma_{i(\kappa_{i}-(l+1))}+\gamma_{i(\kappa_{i}-(l+1))}\\\
&=A^{(l+1)-1}\tilde{b}_{i}-\sum_{\mu=1}^{(l+1)-1}A^{(l+1)-1-\mu}\gamma_{i(\kappa_{i}+1-\mu)}+\gamma_{i(\kappa_{1}-(l+1))}=\mathrm{v}_{(l+1)i}+\gamma_{i(\kappa_{1}-(l+1))}\in\mathscr{X}.\end{split}$
(8)

For $l=\kappa_{i}$ we use the same reasoning as above together with
$\eqref{eq:dependency2}$ and obtain

$\begin{split}A\mathrm{v}_{\kappa_{i}i}&=A^{\kappa_{i}}\tilde{b}_{i}-A\left(A^{\kappa_{i}-2}\gamma_{ik_{i}}+A^{\kappa_{i}-3}\gamma_{i(k_{i}-1)}+\cdots+\gamma_{i2}\right)-\gamma_{i1}+\gamma_{i1}=\gamma_{i1}\in\mathscr{X}.\end{split}$
(9)

Step 3: Structure of $T(\theta)^{-1}A(\theta)T(\theta)$. From (8) and (9) we
get the following block structure

$\displaystyle
T(\theta)^{-1}A(\theta)T(\theta)=\begin{pmatrix}{\tilde{A}(\theta)_{11}}&{\tilde{A}(\theta)_{12}}&{\cdots}&{\tilde{A}(\theta)_{1m}}\\\
{\tilde{A}(\theta)_{21}}&{\tilde{A}(\theta)_{22}}&{\cdots}&{\tilde{A}(\theta)_{2m}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{\tilde{A}(\theta)_{m1}}&{\tilde{A}(\theta)_{m2}}&{\cdots}&{\tilde{A}(\theta)_{mm}}\end{pmatrix},$

where the diagonal blocks have the form

$\displaystyle\tilde{A}(\theta)_{ii}=\begin{pmatrix}{\tilde{\beta}_{ii1}(\theta)}&{\tilde{\beta}_{ii2}(\theta)}&{\cdots}&{\cdots}&{\tilde{\beta}_{ii\kappa_{i}}(\theta)}\\\
{1}&{0}&{\cdots}&{\cdots}&{0}\\\ {0}&{1}&{\ddots}&{}&{\vdots}\\\
{\vdots}&{\ddots}&{\ddots}&{0}&{\vdots}\\\ {0}&{\cdots}&{0}&{1}&{0}\\\
\end{pmatrix}\in\mathbb{C}^{\kappa_{i}\times\kappa_{i}}$

and the off-diagonal blocks have the form

$\displaystyle\tilde{A}(\theta)_{ij}=\begin{pmatrix}{\tilde{\beta}_{ji1}(\theta)}&{\tilde{\beta}_{ji2}(\theta)}&{\cdots}&{\tilde{\beta}_{ji\kappa_{i}}(\theta)}\\\
{0}&{0}&{\cdots}&{0}\\\ {\vdots}&{\vdots}&{}&{\vdots}\\\
{0}&{0}&{\dots}&{0}\\\
\end{pmatrix}\in\mathbb{C}^{\kappa_{j}\times\kappa_{i}},\quad i\neq j.$

Let ${\bf\operatorname{e}}_{k}$ denote the $k$-th standard basis vector in
$\mathbb{C}^{n}$. It follows

$\displaystyle
T(\theta){\bf\operatorname{e}}_{k}=\begin{cases}\mathrm{v}_{k\,1}(\theta)&\text{
if }k\leq\kappa_{1}\\\ \mathrm{v}_{(k-\kappa_{1})\,2}(\theta)&\text{ if
}\kappa_{1}<k\leq\kappa_{1}+\kappa_{2}\\\ \vdots&\\\
\mathrm{v}_{(k-\kappa_{1}-\cdots-\kappa_{m-1})\,m}(\theta)&\text{ if
}\kappa_{1}+\cdots+\kappa_{m-1}<k\end{cases}$

and equivalently, we have

$\displaystyle
T^{-1}\mathrm{v}_{li}(\theta)=\begin{cases}{\bf\operatorname{e}}_{l}&\text{ if
}i=1,\,l=1,...,\kappa_{1}\\\ {\bf\operatorname{e}}_{\kappa_{1}+l}&\text{ if
}i=2,\,l=1,...,\kappa_{2}\\\ \vdots\\\
{\bf\operatorname{e}}_{\kappa_{1}+\cdots+\kappa_{m-1}+l}&\text{ if
}i=m\,,l=1,...,\kappa_{m}.\end{cases}$ (10)

From (10) it follows that the transformation $(T^{-1},0,I)$ acts on the matrix
$\tilde{B}$ as follows

$\displaystyle
T(\theta)^{-1}\tilde{B}(\theta)=\begin{pmatrix}{\bf\operatorname{e}}_{1}&{\bf\operatorname{e}}_{\kappa_{1}+1}&\cdots&{\bf\operatorname{e}}_{\kappa_{1}+\dots+\kappa_{m-1}+1}\end{pmatrix}.$

Step 4: The final step is to transform $T(\theta)^{-1}A(\theta)T(\theta)$ into
the desired form $A_{\kappa}$. In doing so, the numbers
$\tilde{\beta}_{ijl}(\theta)$ in the blocks of ${A}(\theta)_{ij}$ have to be
eliminated. This will be achieved by applying the continuous feedback
transformation $(I,F(\theta),I)$, where $F(\theta)$ is defined as the block
matrix

$\displaystyle
F(\theta)=\begin{pmatrix}F_{1}(\theta)&\cdots&F_{m}(\theta)\end{pmatrix}$

where

$\displaystyle
F_{i}(\theta)=\begin{pmatrix}-\tilde{\beta}_{i11}(\theta)&\cdots&-\tilde{\beta}_{i1\kappa_{i}}(\theta)\\\
\ \vdots&&\vdots\\\
-\tilde{\beta}_{im1}(\theta)&\cdots&-\tilde{\beta}_{im\kappa_{i}}(\theta)\
\end{pmatrix}\in\mathbb{C}^{m\times\kappa_{i}}.$

Hence, the application of the restricted feedback transformation $(I,F,I)$ to
the pair $(T^{-1}AT,B_{\kappa})$ leads to

$\displaystyle
T(\theta)^{-1}A(\theta)T(\theta)-B_{\kappa}F(\theta)=A_{\kappa}.$

In summary, for the restricted feedback transformation $(T^{-1},F,U^{-1})$ we
have

$\displaystyle T(\theta)A(\theta)-A_{\kappa}T(\theta)^{-1}$
$\displaystyle=B_{\kappa}F(\theta)$ $\displaystyle T(\theta)^{-1}B(\theta)$
$\displaystyle=B_{\kappa}U(\theta)^{-1}$

and, by (5), the claim follows.

∎

##  4 Main Results

The main results of the paper explore the possibility to derive uniformly
ensemble reachable systems by using a mixture of open-loop inputs and feedback
controllers. We begin with the single-input case. In this case there is only
one Hermite index and it is equal to $n$ if condition (N1) holds. Also, the
conditions (N2) and (S2) put restrictions on the spectra of the matrices
$A(\theta)$. Under the assumption that the pairs are reachable for every
parameter, the spectra of the matrices can be assigned arbitrarily by the the
Pole-Shifting Theorem, cf. [18, Theorem 6.23]. We obtain the following result.

###### Theorem 2

Let $\mathbf{P}$ be a Jordan arc and assume that $(A,b)\in
C_{n,n}(\mathbf{P})\times C_{n}(\mathbf{P})$ is pointwise reachable. Then,
there is a continuous feedback $f\in C_{1,n}(\mathbf{P})$ such that the
feedback pair $(A+bf,b)$ is uniformly ensemble reachable.

###### Proof.

Since $\mathbf{P}$ is Jordan arc, there is a continuous and injective function
$\gamma\colon[0,1]\to\mathbf{P}$. Then, for $1\leq l\leq k<n$ we define the
injective mappings

$\displaystyle\lambda_{l}(\theta):=e^{2\pi
i\,\left(\gamma^{-1}(\theta)\tfrac{l-1}{k}+(1-\gamma^{-1}(\theta))\left(\tfrac{l}{k}-\tfrac{1}{k+1}\right)\right)}\in\partial\mathbb{D}.$

and $k<l\leq n$

$\displaystyle\lambda_{l}(\theta):=(l+1)-\gamma^{-1}(\theta)\in\mathbb{R}.$

Consequently, for all $k\neq l$ we have

$\displaystyle\lambda_{l}(\mathbf{P})\cap\lambda_{k}(\mathbf{P})=\emptyset.$

Next we define the family of monic polynomials
$(p_{\theta})_{\theta\in\mathbf{P}}$ by

$\displaystyle p_{\theta}(z):=\prod_{i=1}^{n}(z-\lambda_{i}(\theta)).$

Then, by Ackermann’s Formula [18, Theorem 6.20], the family

$\displaystyle
f(\theta):=(0,\dots,0,1)R(A(\theta),b(\theta))^{-1}p_{\theta}(A(\theta))$

of state-feedback, where
$R(A(\theta),b(\theta))=\begin{pmatrix}b(\theta)&A(\theta)b(\theta)&\cdots&A(\theta)^{n-1}b(\theta)\end{pmatrix}$
is the reachability matrix, satisfies

$\displaystyle\det(zI-A(\theta)+b(\theta)f(\theta))=p_{\theta}(z).$

Thus, the spectral conditions (N2) and (S2) are fulfilled and it remains to
prove that $\theta\mapsto F(\theta)$ is continuous on $\mathbf{P}$. Since
$A(\theta)$ is continuous and $p_{\theta}$ is a polynomial, $\theta\mapsto
p_{\theta}(A(\theta))$ is also continuous. Furthermore the reachability matrix
$R(A(\theta),b(\theta))$ is continuous and invertible for every
$\theta\in\mathbf{P}$. So, by Lemma 1 (a) its inverse is also continuous.
Hence, $F(\cdot)$ is continuous on $\mathbf{P}$. Finally we note that for
single-input systems condition (N1) implies condition (H). This shows the
assertion. ∎

To treat the multi-input case, we note that, it is well-known that the Hermite
indices are not invariant under feedback, cf. [7] and we have the following
statement.

###### Theorem 3

Let $\mathbf{P}$ be a Jordan arc and assume that the pair $(A,B)\in
C_{n,n}(\mathbf{P})\times C_{n,m}(\mathbf{P})$ is pointwise reachable and the
Kronecker indices are constant. Then, there exists a restricted feedback
transformation $(T,F,S)\in\mathscr{F}_{n,m}(\mathbf{P})$ such that the pair

$\displaystyle(\tilde{A},\tilde{B})=(T,F,S)\cdot(A,B)$

is uniformly ensemble reachable.

###### Proof.

Let $\kappa(A,B)=(\kappa_{1},...,\kappa_{m})$ denote the constant Kronecker
indices of the pair $(A,B)$. The proof is carried out in three steps.

Step 1: Constructing a pair $(\tilde{A},\tilde{B})$ satisfying (N1), (N2),
(S2) and (H). We define the pair $(\tilde{A},\tilde{B})$ by

$\displaystyle\tilde{A}(\theta):=\left(\begin{array}[]{ c c c c c
}{0}&{0}&{\ldots}&{0}&{a_{0}(\theta)}\\\
{1}&{0}&{\ldots}&{0}&{a_{1}(\theta)}\\\ {0}&{1}&{\dots}&{0}&{a_{2}(\theta)}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\\
{0}&{0}&{\ldots}&{1}&{a_{n-1}(\theta)}\end{array}\right)$

and

$\displaystyle\tilde{B}:=\begin{pmatrix}{\bf\operatorname{e}}_{1}&e_{1+\kappa_{1}}&{\bf\operatorname{e}}_{1+\kappa_{1}+\kappa_{2}}&\cdots&{\bf\operatorname{e}}_{1+\kappa_{1}+\dots+\kappa_{m}}&0&\cdots&0\end{pmatrix}.$

If $\kappa_{1}=n$, we define
$\tilde{B}:=\begin{pmatrix}{\bf\operatorname{e}}_{1}&0&\cdots&0\end{pmatrix}$.

To show that the pair $(\tilde{A},\tilde{B})$ satisfies the conditions (N1),
(N2), (S) and (H), observe that for $i=1,\dots,n-1$ one has

$\tilde{A}(\theta)^{i}{\bf\operatorname{e}}_{1}={\bf\operatorname{e}}_{i+1}\qquad\forall\,\,\theta\in\mathbf{P}.$

Thus, the pair $(\tilde{A},\tilde{B})$ has constant Hermite indices
$h(\tilde{A},\tilde{B})=(n,0,\dots,0)$. This shows that conditions (N1) and
(H) are satisfied. As the functions $a_{0},...,a_{n-1}$ are the coefficients
of the characteristic polynomial of $\tilde{A}$, choosing them as in the proof
of Theorem 2 the spectral conditions (N2) and (S2) are satisfied and the pair
$(\tilde{A},\tilde{B})$ is uniformly ensemble reachable.

Step 2: $(\tilde{A},\tilde{B})$ has the same Kronecker indices as $(A,B)$.

We exemplary treat the first Kronecker index. The others follow from the same
reasoning. Note that for all $i=1,2,...,n-1$ it holds

$\displaystyle\tilde{A}(\theta)^{i}\tilde{b}_{1}=\tilde{A}(\theta)^{i}{\bf\operatorname{e}}_{1}={\bf\operatorname{e}}_{i+1}$

and

$\displaystyle\tilde{A}(\theta)^{i}\tilde{b}_{l}=\tilde{A}(\theta)^{i}{\bf\operatorname{e}}_{1+\kappa_{1}+\cdots+\kappa_{l-1}}={\bf\operatorname{e}}_{1+\kappa_{1}+\cdots+\kappa_{l-1}+i}$

for all $l=2,...,m-1$ and $i=1,...,n-(1+\kappa_{1}+\cdots+\kappa_{l-1})$.
Thus, for all $i<\kappa_{1}$ the vectors $\tilde{A}(\theta)^{i}\tilde{b}_{1}$
are linear independent from the vectors $\tilde{b}_{1},....,\tilde{b}_{m}$ and
$\tilde{A}(\theta)^{i}\tilde{b}_{1},...,A(\theta)^{i}\tilde{b}_{m}$. For
$i=\kappa_{1}$ one has
$\tilde{A}(\theta)^{\kappa_{1}}{\bf\operatorname{e}}_{1}={\bf\operatorname{e}}_{\kappa_{1}+1}=\tilde{b}_{2}$,
which has already been selected. Thus, the first Kronecker index
$k_{1}(\tilde{A},\tilde{B})=\kappa_{1}$.

Step 3: Application of Theorem 1. By Theorem 1 there are feedback
transformations $(T,F,S)$ and $(\tilde{T},\tilde{F},\tilde{S})$ such that

$\displaystyle(T,F,S)\cdot(A,B)=(A_{k},B_{k})=(\tilde{T},\tilde{F},\tilde{S})\cdot(\tilde{A},\tilde{B}).$

Thus, it holds

$\displaystyle(\tilde{A},\tilde{B})=\left((\tilde{T},\tilde{F},\tilde{S})^{-1}\circ(T,F,S)\right)\cdot(A,B).$

This shows the assertion. ∎

We note that another well-known list is given by the controllability indices,
cf. [36] and [26]. In [18, p. 301]111In [18] the controllability are called
reachability indices. it is pointed out that if all Kronecker indices are non-
zero, the controllability indices are obtained from the Kronecker indices by
reordering them in decreasing form. Therefore, since the Kronecker indices in
Example 1 are non-zero, the example also shows that constant Hermite indices
is independent from constant controllability indices and vice versa.

###### Remark 1

Following the proof of Theorem 14 in [36], Theorem 1 can easily be modified to
show that for every pair $(A,B)\in C_{n,n}(\mathbf{P})\times
C_{n,m}(\mathbf{P})$ with constant controllability indices that sum up to $n$
there is a feedback transformation $(T,F,S)$ such that $(T,F,S)\cdot(A,B)$ is
uniformly ensemble reachable.

The following Example 1 (a) is taken from [26] and can be used to show that
the constancy of the Kronecker indices is independent from the constancy of
the Hermite indices and controllability indices.

###### Example 1

Let $\mathbf{P}=[-1,1]$.

  1. (a)

Consider the matrix pair $(A_{1},B_{1})$ defined by

$\displaystyle A_{1}(\theta)=\begin{pmatrix}0&1&0&0\\\
2\theta^{2}&0&0&2\theta\\\ 0&0&0&1\\\ 0&-2\theta&0&0\\\ \end{pmatrix}\qquad
B_{1}(\theta)=\begin{pmatrix}0&0\\\ 1&0\\\ 0&0\\\ 0&1\\\ \end{pmatrix}.$

The columns of the Kalman matrix are

$\displaystyle\begin{pmatrix}0\\\ 1\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 0\\\ 1\\\
\end{pmatrix},\begin{pmatrix}1\\\ 0\\\ 0\\\ -2\theta\\\
\end{pmatrix},\begin{pmatrix}0\\\ 2\theta\\\ 1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ -2\theta\\\ 0\\\
\end{pmatrix},\begin{pmatrix}2\theta\\\ 0\\\ 0\\\ -4\theta^{2}\\\
\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ -4\theta^{3}\\\ -4\theta^{2}\\\ 0\\\
\end{pmatrix}.$

Hence, the pair $(A_{1},B_{1})$ has constant Kronecker indices
$\kappa(A_{1},B_{1})\equiv(2,2)$ and the Hermite indices are given by

$\displaystyle h(A_{1},B_{1})(\theta)=\begin{cases}(3,1)&\text{ if }\theta\neq
0\\\ (2,2)&\text{ if }\theta=0.\\\ \end{cases}$

  2. (b)

Consider the pair $(A_{2},B_{2})$ defined by

$\displaystyle A_{2}(\theta)=\begin{pmatrix}0&0&2&\theta^{2}-\tfrac{1}{2}\\\
1&0&0&1\\\ 0&1&0&0\\\ 0&0&0&0\\\ \end{pmatrix}\qquad
B_{2}(\theta)=\begin{pmatrix}0&0\\\ 1&0\\\ 0&0\\\ 0&1\\\ \end{pmatrix}.$

The columns of the Kalman matrix are

$\displaystyle\begin{pmatrix}0\\\ 1\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 0\\\ 1\\\
\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}\theta^{2}-\tfrac{1}{2}\\\ 1\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}2\\\ 0\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ \theta^{2}-\tfrac{1}{2}\\\ 1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ 2\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}2\\\ 0\\\ \theta^{2}-\tfrac{1}{2}\\\ 0\\\
\end{pmatrix}.$

Hence, the pair $(A_{2},B_{2})$ has constant Hermite indices
$h(A_{2},B_{2})\equiv(3,1)$ and the Kronecker indices are given by

$\displaystyle\kappa(A_{2},B_{2})(\theta)=\begin{cases}(3,1)&\text{ if
}\theta^{2}\neq\tfrac{1}{2}\\\ (2,2)&\text{ if }\theta^{2}=\tfrac{1}{2}.\\\
\end{cases}$

##  5 Open-Loop and Feedback Controlled Harmonic Oscillators

In this section, we consider an ensemble of controlled harmonic oscillators
and investigate the possibility to use a mixture of an open-loop and constant
feedback controller of the form

$u(t,y)=ky(t,\theta)+u(t),\quad k\in\mathbb{R}.$

For notational convenience we denote the feedback gain by $k\in\mathbb{R}$
instead of $F$ as in the previous section. Then, let $g(\theta)$ denote the
input function associated with the parameter
$\theta\in\mathbf{P}:=[-\theta^{*},\theta^{*}]\subset\mathbb{R}$. The dynamic
equation reads as follows

$\displaystyle\tfrac{\partial^{2}}{\partial
t^{2}}y(t,\theta)+\theta^{2}y(t,\theta)=g(\theta)\,\left(ky(t,\theta)+u(t)\right).$
(11)

In order to establish conditions guaranteeing the existence of a
$k\in\mathbb{R}$ such that (11) is uniformly ensemble reachable we consider
the corresponding first order system

$\displaystyle\tfrac{\partial}{\partial
t}x(t,\theta)=A_{k}(\theta)x(t,\theta)+b_{g}(\theta)u(t)$ (12)

with

$\displaystyle A_{k}(\theta):=\begin{pmatrix}0&1\\\
kg(\theta)-\theta^{2}&0\end{pmatrix},\quad b_{g}(\theta):=\begin{pmatrix}0\\\
g(\theta)\end{pmatrix}.$ (13)

Note that for $\mathbf{P}=[0,\theta^{*}]$ it follows from [15, Theorem 4] that
the family of controlled harmonic oscillators (12) is uniformly ensemble
reachable by means of pure open-loop controller, i.e. $k=0$. The following
result states conditions such that the application of
$u(t,y)=ky(t,\theta)+u(t)$ yields uniform ensemble reachability over the
parameter space $[-\theta^{*},\theta^{*}]$. Our first result is as follows.

###### Proposition 1

Let $\theta^{*}>0$ and $\mathbf{P}=[-\theta^{*},\theta^{*}]$ and suppose that
$g\in C^{1}(\mathbf{P})$ is zero-free and strictly monotone. Then, for
$k^{*}:=\max_{\theta\in\mathbf{P}}\tfrac{2\theta}{g^{\prime}(\theta)}$ the
open-loop and feedback controlled harmonic oscillators (11) are uniformly
ensemble reachable for all $k>k^{*}$ .

###### Proof.

First, note that the Kalman matrix for (12) is given by

$g(\theta)\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

Since $g(\theta)\neq 0$ for all $\theta\in[-\theta^{*},\theta^{*}]$, the
Kalman matrix has rank $2$, i.e. (12) is reachable for every $\theta$. The
characteristic polynomial of $A_{k}(\theta)$ is given by

$\displaystyle z^{2}-(kg(\theta)-\theta^{2}).$

By [15, Theorem 4] it suffices to show that there is a $k^{*}\in\mathbb{R}$
such that the functions $h_{k}\in C^{1}(\mathbf{P})$,
$h_{k}(\theta):=kg(\theta)-\theta^{2}$ are injective for all $k>k^{*}$. To
this end, let
$k^{*}:=\max_{\theta\in\mathbf{P}}\tfrac{2\theta}{g^{\prime}(\theta)}$. Then,
for all $k>k^{*}$ one has
$h^{\prime}_{k}(\theta):=kg^{\prime}(\theta)-2\theta>0$ and, thus, the
functions $h_{k}$ are injective on $\mathbf{P}$ for all $k>k^{*}$. This shows
the assertion. ∎

We note that in the proof above it is also sufficient to pick $k^{*}$ such
that $h^{\prime}_{k}(\theta)<0$ for all $\theta\in\mathbf{P}$ and for all
$k<k^{*}$. An appropriate choice in this case would be
$k^{*}:=\min_{\theta\in\mathbf{P}}\tfrac{2\theta}{g^{\prime}(\theta)}$.
Depending on the particular situation at hand it might be suitable to use the
latter. Subsequently we investigate how to get an error bound and the
influence of the feedback gain $k$ on it. Before we do so, we recap useful
properties of Lipschitz continuous functions.

###### Lemma 2

Let $I$ and $J$ be compact intervals and suppose that $f\colon I\to\mathbb{R}$
and $g\colon J\to\mathbb{R}$ satisfy a Lipschitz condition with $L_{f}>0$ and
$L_{g}>0$, respectively.

  1. (i)

If $g(J)\subset I$, then the composition $f\circ g$ satisfies a Lipschitz
condition with $L_{f}L_{g}>0$.

  2. (ii)

If $I=J$, then the product $fg\colon I\to\mathbb{R}$ satisfies a Lipschitz
condition with $L_{f}M_{g}+L_{g}M_{f}>0$, where $M_{f}:=\max_{x\in I}|f(x)|$
and $M_{g}:=\max_{x\in I}|g(x)|$.

  3. (iii)

If $f$ is zero-free, then $\tfrac{1}{f}$ satisfies a Lipschitz condition with
$\tfrac{L_{f}}{m_{f}^{2}}$, where $m_{f}=\min_{x\in I}|f(x)|$.

  4. (iv)

If $f$ is continuously differentiable and strictly monotone, then the inverse
$f^{-1}$ satisfies a Lipschitz condition with $L_{f^{-1}}=(\min_{x\in
I}|f^{\prime}(x)|)^{-1}$.

In order to formulate the following result for continuous-time and discrete-
time controlled harmonic oscillators, we state the error in terms of
$\|p(A)b-f\|_{\infty}$ and note that in the continuous-time case, the
subsequent estimate for $\|p(A)b-f\|_{\infty}$ has to be combined with an
estimate for $\|\varphi(T,u,0)-p(A)b\|_{\infty}$, cf. [34, Section 4].

###### Proposition 2

Suppose that the assumptions of Proposition 1 hold and suppose that $f\in
C_{2}(\mathbf{P})$ satisfies a Lipschitz condition. Let

$\displaystyle
k^{*}:=\max\left\\{\tfrac{\theta^{*}}{\min_{\theta\in\mathbf{P}}|g(\theta)|},\,\max_{\theta\in\mathbf{P}}\tfrac{2\theta}{g^{\prime}(\theta)}\right\\}.$

Then, for every $k>k^{*}$ there is a sequence of polynomials $(p_{n})_{n}$ of
the degree of $2n+1$ with $n\geq 3$ such that for
$c_{g,\mathbf{P}}:=\left|g(\theta^{*})-g(-\theta^{*})\,\right|$ one has

$\displaystyle\|p_{n}(A_{k})b_{g}-f\|_{\infty}\leq\tfrac{1}{g(-\theta^{*})}\,\left(\tfrac{4M_{f}}{\min_{\theta\in\mathbf{P}}|g(\theta)|}+\,\tfrac{k\,c_{g,\mathbf{P}}}{2\min_{\theta\in\mathbf{P}}|kg^{\prime}(\theta)-2\theta|}\left(M_{f}\,\tfrac{L_{g}}{m_{g}^{2}}+\tfrac{L_{f}}{m_{g}}\right)\right)\sqrt{\frac{\log
n}{n}}.$

###### Proof.

Since $g\in C^{1}(\mathbf{P})$ is non zero and strictly monotone we assume
w.l.o.g. that $g(\theta)>0$ for every $\theta\in\mathbf{P}$. Then, for every
$k>k^{*}$ it holds

$\displaystyle h_{k}(\theta):=kg(\theta)-\theta^{2}>0\qquad\text{ and }\qquad
h_{k}^{\prime}(\theta)=kg^{\prime}(\theta)-2\theta>0$

for all $\theta\in\mathbf{P}$. Then, the continuous transformation

$\displaystyle T(\theta)^{-1}:=\tfrac{1}{g(\theta)}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ (14)

yields that

$\displaystyle
T(\theta)^{-1}A_{k}(\theta)T(\theta)=\begin{pmatrix}0&h_{k}(\theta)\\\
1&0\end{pmatrix},\quad T(\theta)^{-1}\begin{pmatrix}0\\\
g(\theta)\end{pmatrix}=\begin{pmatrix}1\\\ 0\end{pmatrix}.$

For every polynomial $p$ we get

$\displaystyle\|p(A_{k})b_{g}-f\|_{\infty}\leq\frac{1}{\min_{\theta\in\mathbf{P}}g(\theta)}\,\left\|p\left(\begin{pmatrix}0&h_{k}\\\
1&0\end{pmatrix}\right)\begin{pmatrix}1\\\
0\end{pmatrix}-\tfrac{1}{g}\begin{pmatrix}f_{2}\\\
f_{1}\end{pmatrix}\right\|_{\infty}.$

As in [15, Proof of Theorem 4], we consider a sequence of polynomials
$(p_{n})_{n\in\mathbb{N}}$ of the form

$\displaystyle p_{n}(z):=q_{n}\big{(}z^{2}\big{)}+r_{n}\big{(}z^{2}\big{)}z,$

for some polynomials $q_{n}$ and $r_{n}$ and obtain

$\displaystyle p_{n}\left(\begin{pmatrix}0&h_{k}(\theta)\\\
1&0\end{pmatrix}\right)\begin{pmatrix}1\\\
0\end{pmatrix}=\begin{pmatrix}q_{n}(h_{k}(\theta))\\\
r_{n}(h_{k}(\theta))\end{pmatrix}.$

Hence, for $z\in h_{k}(\mathbf{P})$ we have to investigate the terms

$\displaystyle|q_{n}(z)-\tfrac{f_{2}}{g}\circ
h_{k}^{-1}(z)|\quad\text{and}\quad|r_{n}(z)-\tfrac{f_{1}}{g}\circ
h_{k}^{-1}(z)|.$

By Lemma 2, the functions $\tilde{f}_{1}\colon h_{k}(\mathbf{P})\to\mathbb{R}$
and $\tilde{f}_{2}\colon h_{k}(\mathbf{P})\to\mathbb{R}$ defined by

$\displaystyle\tilde{f}_{1}(z):=\frac{f_{2}(h_{k}^{-1}(z))}{g(h_{k}^{-1}(z))}\quad\text{
and }\quad\tilde{f}_{2}(z):=\frac{f_{1}(h_{k}^{-1}(z))}{g(h_{k}^{-1}(z))},$

satisfy a Lipschitz condition with

$\displaystyle
L_{\tilde{f}_{1}}=L_{h_{k}^{-1}}\left(M_{f_{2}}\,\frac{L_{g}}{m_{g}^{2}}+\frac{L_{f}}{m_{g}}\right)\quad\text{
and }\quad
L_{\tilde{f}_{2}}=L_{h_{k}^{-1}}\left(M_{f_{1}}\,\frac{L_{g}}{m_{g}^{2}}+\frac{L_{f}}{m_{g}}\right),$

respectively. By Lemma 2 (iv) one has
$L_{h_{k}^{-1}}=\frac{1}{\min_{\theta\in\mathbf{P}}|kg^{\prime}(\theta)-2\theta|}$.
Then, we take $q_{n}$ and $r_{n}$ as the $n$th Bernstein polynomials to the
functions $\tilde{f}_{1}$ and $\tilde{f}_{2}$, respectively, i.e.

$\displaystyle q_{n}(z):=B_{n,\tilde{f}_{2}}(z)\quad\text{ and }\quad
r_{n}(z):=B_{n,\tilde{f}_{1}}(z).$

Then, by [24, Theorem 1] we have

$\|p_{n}(A_{k})b_{g}-f\|_{\infty}\leq\tfrac{1}{g(-\theta^{*})}\,\left\|p_{n}\left(\begin{pmatrix}0&h_{k}(\theta)\\\
1&0\end{pmatrix}\right)\binom{1}{0}-\tfrac{1}{g}\binom{f_{2}}{f_{1}}\right\|_{\infty}\\\
\leq\tfrac{1}{g(-\theta^{*})}\,\max_{i=1,2}|B_{n,\tilde{f}_{i}}(z)-\tilde{f}_{i}(z)|_{\infty}\leq\tfrac{1}{g(-\theta^{*})}\,\max_{i=1,2}\,\left\\{4M_{\tilde{f}_{i}}+\frac{k\,c_{g,\mathbf{P}}L_{\tilde{f}_{i}}}{2}\right\\}\sqrt{\frac{\log
n}{n}}\\\
\leq\tfrac{1}{g(-\theta^{*})}\,\left(\frac{4M_{f}}{\min_{\theta\in\mathbf{P}}|g(\theta)|}+\,\frac{k\,c_{g,\mathbf{P}}}{2\min_{\theta\in\mathbf{P}}|kg^{\prime}(\theta)-2\theta|}\left(M_{f}\,\frac{L_{g}}{m_{g}^{2}}+\frac{L_{f}}{m_{g}}\right)\right)\sqrt{\frac{\log
n}{n}}$

for $n\geq 3$. This shows the assertion. ∎

The latter error bound shows that the approximation is getting better the
larger the feedback gain is. In terms of the eigenvalues of the matrices
$A_{k}(\theta)$ it can be observed that the eigenvalue arcs
$\theta\mapsto\sqrt{kg(\theta)-\theta^{2}}$ and
$\theta\mapsto-\sqrt{kg(\theta)-\theta^{2}}$ are located on the positive and
the negative real line, respectively, and the gap between them is increasing
with the feedback gain $k$. Also, we emphasize that for pure open-loop inputs,
i.e. $k=0$, the pair $(A_{0},b_{g})$ is not uniformly ensemble reachable over
the parameter set $\mathbf{P}=[-\theta^{*},\theta^{*}]$. This corresponds to
the fact that the traces of the eigenvalue arcs $\theta\mapsto i\theta$ and
$\theta\mapsto-i\theta$ coincide. By using a feedback gain $k\in\mathbb{R}$
the traces can be separated and the assumptions in [15, Theorem 3.1.1 (c)] are
satisfied such that the pair $(A_{k},b_{g})$ is uniformly ensemble
reachability.

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