Proceedings of the IISE Annual Conference & Expo 2024
A. Brown Greer, C. Contardo, J.-M. Frayret, eds. Bui, Eksioglu, Proano7318

#  Evaluating the Impact of Vaccine Hesitancy on the Allocation of Vital
Resources During COVID-19 Pandemic

Hieu Bui1 Sandra D. Eksioglu1 Ruben A. Proano2
1Department of Industrial Engineering
University of Arkansas  Fayetteville  AR 72701
2Department of Industrial and Systems Engineering
Rochester Institute of Technology  Rochester  NY 14623

###### Abstract

The COVID-19 pandemic highlighted significant challenges in the allocation of
vital healthcare resources. Existing epidemiological models, specifically
compartmental models, aimed to predict the spread of the COVID-19 virus and
its impact on the population, but they overlooked the influence of vaccine
hesitancy (VH) on disease dynamics, including the expected number of
hospitalizations and fatalities. We propose improvements to the susceptible-
exposed-infectious-recovered (SEIR) model for COVID-19 by incorporating the
influence of vaccination, VH, and resource availability on the disease
dynamics. We collect publicly available data and perform data analysis to
capture VH dynamic changes over time and develop scenario paths for VH. We
simulate the proposed compartmental model for each VH path to explain the
impacts of public attitudes toward vaccination, the impacts of healthcare
resources on patient outcomes, and the timing of vaccination rollout on the
progression and severity of the epidemic. Our analysis demonstrates that
reducing VH improves health outcomes, reinforcing the importance of addressing
VH to curb the spread of infectious diseases. Our results show that adequate
levels of critical healthcare resources are crucial for minimizing fatalities
and also highlight the life-saving impact of timely and effective vaccination
programs.

## Keywords

compartmental model, COVID-19, data analytics, healthcare resources, vaccine
hesitancy

##  1 Introduction

The emergence of infectious diseases with high transmission rates, such as
COVID-19, emphasizes the critical role of strategic public health
interventions to limit the spread of the disease, and reduce hospitalizations
and fatalities, such as vaccination campaigns, social distancing, and the
strategic allocation of vital resources [9]. The success of vaccination
campaigns is compromised by vaccine hesitancy (VH), which is the resistance or
refusal to be immunized despite the availability of vaccines. VH is a barrier
to achieving widespread immunity, challenging public compliance, and
complicating healthcare resource allocation[1].

Strategic public policies are essential to manage epidemics. The effectiveness
of these policies depends on the accurate prediction of the spread of the
disease. Compartmental epidemiological models, such as susceptible-infectious-
recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) models,
have been instrumental in this regard. The SIR model was introduced in the
early 20th century. It divides the population into Susceptible, Infectious,
and Recovered compartments to mimic the changes in the health state of a
homogenous population affected by an infectious disease [3]. The SEIR model
extends the SIR by including an Exposed state to account for individuals
exposed to a virus that show symptoms and become infectious after an
incubation period. Reviews by [4, 12] show that compartmental models have been
widely used to study COVID-19, providing a good representation of its
transmission dynamics. Traditional SEIR models are deterministic (i.e., assume
constant transition rates between compartments); however, several model
adaptations are stochastic and capture the inherent randomness in disease
transmission. For example, Kucgarski et al. [6] utilized stochastic
transmission dynamic models to estimate the transmission rate of COVID-19.
Kretzschmar et al. [5] applied stochastic mathematical models to estimate
effective reproduction numbers, accounting for delays in testing and
isolation. Despite the variety of available compartmental models, there is a
scarcity of models that explicitly analyze the impact of stochastic VH on the
spread of the disease and the demand for healthcare resources.

The compartmental epidemiological models are solved via systems of ordinary
differential equations to track population dynamics between different
compartments over time. For complex models such as those for COVID-19, closed-
form solutions of these ODEs are impractical, necessitating numerical methods
within simulation frameworks. These simulations numerically approximate ODE
solutions and can help clarify the nuances of disease propagation and the
potential impact of different interventions for short time intervals. However,
current simulation frameworks are often limited in scope, focusing on specific
scenarios that do not capture the entire spectrum of stochastic parameters.
This limitation can lead to a narrow understanding of potential outcomes.
Hence, there is a need for comprehensive models that explore a wider range of
variables and uncertainties.

Our study seeks to fill these gaps by presenting a compartmental model
designed specifically for COVID-19, which incorporates the stochasticity of VH
and constraints on healthcare resource access. We analyze the VH data provided
by surveys to create a comprehensive spectrum of VH trajectories (i.e.,
scenario paths). We use the model to answer these research questions relevant
to the COVID-19 pandemic: RQ1: _How do changes in public attitudes towards
vaccination affect epidemic outcomes?_ RQ2: _How do healthcare resources
impact patient outcomes during a pandemic?_ RQ3: _How do vaccine rollout times
affect the progression and severity of the epidemic?_

Finally, this research sets the foundation for more sophisticated mathematical
models, such as multi-stage stochastic program (MSP), aiming to cultivate a
healthcare strategy that is flexible and responsive to the spread of the
disease and people’s attitudes toward vaccination.

##  2 Methodology

The SEIR model divides the population into four compartments: Susceptible
($\boldsymbol{S}$), Exposed ($\boldsymbol{E}$), Infectious ($\boldsymbol{I}$),
and Recovered ($\boldsymbol{R}$). Individuals transition through these
compartments over time. The transition rate between compartments depends on
the specific characteristics of the infectious disease. Section 2.1 focuses on
our proposed compartmental model that extends the SEIR model for COVID-19 to
capture the impact of vaccine availability, VH stochasticity, and availability
of healthcare resources on the spread of the disease and health outcomes. Our
numerical analysis indicates that VH in the U.S. changed over time due to
different public health interventions. Section 2.2 summarizes our approach to
model uncertainty of VH.

###  2.1 COVID-19-Specific Compartmental Model

The proposed model called susceptible-vaccinated-exposed-infectious-
hospitalized-recovered (SVEIHR), summarizes the dynamic progression of
COVID-19 in the presence of vaccines (see Figure 1(a). Thus, unlike in the
SEIR model, individuals in $\boldsymbol{S}$ can transition to compartment
$\boldsymbol{E}$ or opt to vaccinate, moving forward to the vaccinated
compartment ($\boldsymbol{V}$). This transition is influenced by the expected
vaccination ($\rho$) and VH ($h$) rates. We consider $h$ to be a stochastic
parameter because VH differs by population group based on political, economic,
and social standing. VH changes over time due to many reasons, including
public health campaigns, interventions, and social media commentaries. Since
vaccine efficacy ($\epsilon$) is less than 100%, vaccinated individuals could
become exposed. These individuals transition to the compartment
($\boldsymbol{EV}$).

Figure 1: (a) SVEIHR: the proposed COVID-19 compartmental model. (b) and (c)
illustrate the impact of healthcare capacity on the number of deaths and
hospitalizations for a population of size 1.2M.

The model simulates different health trajectories post-exposure, based on
incubation rates $\alpha$ and severity of the disease. Individuals progress to
mild ($\boldsymbol{I^{\text{m}}}$), severe ($\boldsymbol{I^{\text{s}}}$), or
critical condition ($\boldsymbol{H^{\text{c}}}$) compartments with
probabilities $p^{\text{m}},p^{\text{s}},$ and $p^{\text{c}}$, respectively.
The vaccinated population is less likely to experience severe/critical health
outcomes and, if infected, is more likely to recover and move directly to the
compartment ($\boldsymbol{R}$).

The model factors the availability of healthcare resources, such as the number
of hospital beds (Bed) and ventilators (Vent), which are crucial in
determining patients’ health outcomes. To illustrate the significance of bed
and ventilator capacity on hospitalizations and fatalities, we conducted two
simulations using the SVEIHR model within a population of 1.2 million. The
first simulation operates under the assumption of unlimited bed and ventilator
availability, while the second imposes limitations on these critical
resources. The parameters used in these simulations, and other experiments
presented in this study, are summarized in Table 1. Several of the parameters
used were derived from the literature [11, 13, 8, 7]. We estimated other
parameters using a similar procedure as in [2]. The cumulative number of
hospitalizations and deaths for the first simulation are presented in Figure
1(b), and for the second simulation is presented in Figure 1(c). The results
indicate that the availability of beds and ventilators affects the number of
deaths. In the second simulation, we observe a visible spike in the cumulative
number of deaths after all the available beds are occupied. As a result, the
number of deaths in the first simulation totaled 5,239, and in the second
simulation, 28,507 deaths. These illustrative examples demonstrate the
potential consequences of resource shortages during a pandemic. They also
emphasize the importance of effective resource allocation strategies to reduce
mortality rates.

Table 1: Estimated weekly parameters for SVEIHR model. Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---|---|---|---|---
$\bar{T}$ | 20.00 | $\gamma^{v}$ | 1.00 | $\mu^{\text{ks}}$ | 0.70 | $p^{\text{c}}$ |  0.05 [11] | $\varsigma^{\text{c}}$ |  0.4 [11]
$\beta$ | 3.20 | $\gamma^{\text{m}}$ | 1.00 | $\mu^{\text{c}}$ |  0.35 [13] | $p^{\text{r},v}$ | 0.85 | $N$ | 1.2M
$\epsilon$ | 0.95 | $\gamma^{\text{s}}$ |  0.64 [13] | $\sigma$ |  0.64 [13] | $p^{\text{m},v}$ | 0.10 | $E_{0}$ | 100
$\rho$ | 0.10 | $\gamma^{\text{ks}}$ | 0.23 | $p^{\text{m}}$ |  0.80 [11] | $p^{\text{s},v}$ | 0.05 | $I^{\text{m}}_{0}$ | 50
$\alpha$ |  0.71 [8] | $\gamma^{\text{c}}$ |  0.33 [13] | $p^{\text{s}}$ |  0.15 [11] | $\varsigma^{\text{ks}}$ | 0.40 | $I^{\text{s}}_{0}$ | 20

Recovery ($\gamma$) and mortality ($\mu$) rates vary depending on the severity
of the disease and the availability of healthcare resources. The model assumes
that critically ill patients transition to the compartment ($\boldsymbol{D}$)
when ventilators are unavailable, emphasizing the dire consequences of
resource limitations during a pandemic.

###  2.2 Modeling of Uncertainties in Vaccine Hesitancy

Figure 2(a) presents the county-level VH estimates derived from the COVID-19
Trends and Impact Survey (CTIS) conducted by the Delphi Group at Carnegie
Mellon University, in partnership with Facebook [10]. Each line in the graph
represents the progression of VH over time in every county of Arkansas and New
York. The graph reveals an overall decline in VH rates over time. The graph
also reveals various vaccination attitudes across counties in these states.
These variations highlight the challenges of achieving widespread acceptance
of vaccines across the U.S. For instance, counties in Arkansas exhibit higher
VH than NY and, consequently, lower vaccination rates (see Figure 2(b)). This
variation is critical because it affects both the spread of the disease and
the need for healthcare resources.

Figure 2: Vaccine hesitancy and vaccination uptake trends of Arkansas and New
York counties.

We use historical VH estimates from CTIS data to determine the rate of change
of VH from one period to the next. For example, a rate change of -0.15 in
period $t$ indicates a 15% decrease in VH from period $t-1$ to $t$. We fit a
distribution to the rate of change. Numerical analysis shows that the monthly
rate of change of VH is normally distributed. The corresponding mean
($\mu_{t}$) and standard deviation ($\sigma_{t}$) also change from one month
to the next (see Figure 3(a)-(d)).

To represent the uncertainties associated with VH, we develop several scenario
paths, represented by $\mathit{VH}^{\omega}\in\Omega$. Each scenario path
consists of a series of VH values, one per time period,
$\mathit{VH}^{\omega}=\\{\mathit{VH}_{1}^{\omega},\mathit{VH}_{2}^{\omega}\ldots,\mathit{VH}_{T}^{\omega}\\}$,
where $T$ represents the length of the period of study. We create each
scenario path via the following procedure. Initially, we approximate the
normal distribution functions by a triangular, discrete distribution function
that takes values $\mu_{t},\mu_{t}-\sigma_{t},\mu_{t}+\sigma_{t}$ with
probability 0.158, 0.684 and 0.158 correspondingly. Next, we create the
scenario paths. Each scenario path begins with an initial $\mathit{VH}_{1}$,
which changes by $\mu_{t},\mu_{t}-\sigma_{t}$, or $\mu_{t}+\sigma_{t}$ each
period, with probability 0.158, 0.684 and 0.158, correspondingly.

Figure 3: (a)-(d) Distribution of the monthly rate of change of vaccine
hesitancy, (e) Example of VH scenario paths over four months.

Let us illustrate this via an example. Assume that $\mathit{VH}_{1}=0.415$ and
the length of our study period is $T=5$. Using our proposed approach, we
construct a total of $3^{4}=81$ unique scenario paths. Figure 3(e) presents
the different scenario paths for this example. Let $p^{\omega}$ represent the
probability associated with each scenario path. The graph highlights a
baseline, an optimistic, and a pessimistic scenario paths. The optimistic
scenario path assumes that the rate of change of VH in each time period is
$\mu_{t}-\sigma_{t}.$ The pessimistic scenario path assumes that the rate of
change of VH in each time period is $\mu_{t}+\sigma_{t}.$ As a result, the
probability associated with each of these two scenarios is
$0.158^{4}=6.23\times 10^{-4}$. In the baseline scenario path, the rate of
change in each period is $\mu_{t},$ and the probability of this scenario path
is $0.684^{4}=2.19\times 10^{-1}$.

##  3 Numerical Analysis and Results

Our numerical analysis uses historical VH estimates from CTIS data for five
months, from January 1 to May 7, 2021. We focus our study on this period
because COVID-19 vaccines were available in the U.S. Via our numerical
analysis, we address three important research questions. To address RQ1: _How
do changes in public attitudes towards vaccination affect epidemic outcomes?_
We simulate the SVEIHR model weekly for 20 weeks (5 months). We conduct a
total of $|\Omega|=81$ simulations, one for each
$\mathit{VH}^{\omega}\in\Omega$, and summarize the results in Figure 4.

In Figures 4(a) to 4(d), the blue line represents the optimistic VH scenario
path, which assumes a continuous decrease in VH, resulting in the highest
vaccination uptake. The orange line represents the pessimistic VH scenario
path, for which vaccination uptake is lowest. The results of Figure 4(b)
indicate that high vaccination rates lead to a noticeable decrease in the
number of infections. Results in Figure 4(c) indicate that high vaccination
rates reduce the number of patients whose needs for healthcare resources (such
as beds and ventilators) are not met. The death toll for the scenario paths
with higher vaccination rates is lower compared to those with lower
vaccination rates (see Figure 4(d)). The maximum difference in vaccination
among the different scenario paths, at the end of the study period, is 33,099
cases. The corresponding maximum difference in the number of infected is
8,079, the maximum difference in the number of patients whose needs for
healthcare resources were not met is 1,023, and the maximum difference in the
number of deaths is 3,323. These results highlight the significant influence
of VH on public health outcomes during a pandemic. Reducing VH is linked to
better health outcomes, which emphasizes the critical role of VH mitigation in
controlling the spread of infectious diseases.

Figure 4: Evaluating the impact of different scenario paths illustrated in
Figure 3(e) on the outcomes of SVEIHR model. The optimistic, pessimistic, and
baseline scenario paths are highlighted.

To address RQ2: _How do healthcare resources impact patient outcomes during a
pandemic?_ We design a total of 25 problems by varying the number of
ventilators and beds available. For each of these problems, we run a total of
81 simulations, each corresponding to a different scenario path
$\mathit{VH}^{\omega}\in\Omega$. Each simulation uses the SVEIHR model and
runs for 20 weeks (5 months). We summarize the results in Figure 5(a). The
numbers represent the expected number of fatalities for each problem. The
results reflect an increase in fatalities as the availability of resources
decreases, highlighting the direct correlation between healthcare
infrastructure and survival rates. The results show that maintaining a robust
supply of beds and ventilators is essential for keeping fatalities to a
minimum.

We develop a regression model using the data from the experiments. The
independent variables for this model are the number of beds ($a_{1}$) and
ventilators ($a_{2}$), and the dependent variable is the expected number of
fatalities ($\bar{D}$). Based on model (1), for every available bed ($a_{1}$),
the expected number of deaths reduces by 0.827, and every additional available
ventilator ($a_{2}$) reduces the expected number of deaths by 1.511.

Figure 5: (a) Expected number of fatalities relative to the number of beds and
ventilators available. (b) Changes in weekly VH over time for the baseline
scenario path. (c) Expected number of fatalities over time.

$\bar{D}=38657.33-0.827a_{1}-1.511a_{2}$ (1)

To address RQ3: _How do vaccine rollout times affect the progression and
severity of the epidemic?_ We create a total of 6 problems by changing the
timing of vaccine rollouts. One of the problems assumes that vaccines were
available at the beginning of the study period. Each of the remaining problems
assumes a delay of 1 or 2, …, or 5 months in vaccine rollout. For each
problem, we run a total of $|\Omega|$ simulations, one for each scenario path
$\mathit{VH}^{\omega}$. Each simulation runs the SVEIHR model for 20 weeks (5
months). We summarize the results in Figures 5(b) and (c).

In our model, we control vaccination rollout via VH rate, $h$. Figure 5(b)
presents the changes in $h$ over time for each problem, for the baseline
scenario path only. The changes in $h$ for the baseline scenario path are
presented by the black line. If no one is vaccinated during the study period,
VH is 100%. This problem is represented by the straight line at $h=1.$ If
vaccine rollout is delayed by 1 month, $h=1$ for the 1st month, and next, $h$
decreases continuously as people are vaccinated. The specific decrease in $h$
over time depends on the specific scenario path. In Figure 5(b), the changes
in $h$ follow the changes of the baseline scenario path.

Figure 5(c) presents the expected number of fatalities for each problem. The
results show that the earliest vaccinations begin, the highest the decrease in
fatalities. The expected number of fatalities at the end of the study period
is 4.88 times higher if vaccines were not available versus vaccines being
available at the start of the study period. We notice that the difference in
the expected number of fatalities when vaccines are rolled out in the last
month versus vaccines never being available is statistically insignificant.
These results underscore the life-saving potential of prompt and efficient
immunization programs.

##  4 Conclusions and Future Research Directions

This study presents an extension of compartmental epidemiological models
developed to estimate the spread of the COVID-19 virus and its impacts on the
population. The proposed model captures the influence of vaccine availability,
VH, and the availability of healthcare resources on disease dynamics, the
expected number of hospitalizations, and the expected number of fatalities. We
simulate the proposed model for different VH scenario paths and different
problem settings. Our analysis highlights the significant influence of ($i$)
VH on public health outcomes during the pandemic; ($ii$) critical healthcare
resources on the expected number of fatalities; and ($iii$) timing of vaccine
rollouts on the expected number of fatalities.

Our data analysis shows that VH changes over time and regions of the U.S. The
rate of change in VH from one period of time to the next is normally
distributed. The proposed simulations of SVEIHR model highlight the impact of
stochastic VH on model outcomes. Our next steps will focus on extending this
work by developing a multi-stage stochastic program (MSP) that supports
resource allocation decisions under uncertainty. The proposed model will
consider budget limitations and will ensure fairness within the healthcare
system. We plan to use an MSP because it facilitates sequential decision-
making over time and effectively addresses resource availability and evolving
uncertainties of VH, infection rate, etc. Such models mimic reality since
decision-makers continually refine their decisions as new data emerge to
ensure a proactive and adaptive response to a pandemic.

## References

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OR

    operations research
SIR

    susceptible-infectious-recovered
SEIR

    susceptible-exposed-infectious-recovered
SVEIHR

    susceptible-vaccinated-exposed-infectious-hospitalized-recovered
MSP

    multi-stage stochastic program
VH

    vaccine hesitancy
CTIS

    COVID-19 Trends and Impact Survey
ODE

    ordinary differential equation
  *[VH]: vaccine hesitancy
  *[SEIR]: susceptible-exposed-infectious-recovered
  *[SIR]: susceptible-infectious-recovered
  *[ODEs]: ordinary differential equation
  *[ODE]: ordinary differential equation
  *[MSP]: multi-stage stochastic program
  *[SVEIHR]: susceptible-vaccinated-exposed-infectious-hospitalized-recovered
  *[CTIS]: COVID-19 Trends and Impact Survey


# Realization of the transient dynamics of dimension-varying control systems

Naqi Fan  Lijun Zhang<EMAIL_ADDRESS>College of Intelligent Systems
Science and Engineering, Harbin Engineering University, Harbin, 150001, China.
School of Marine Technology, Northwestern Polytechnical University, Xi’an,
710129, China.

###### Abstract

Dynamic evolution behaviors of dimension-varying control systems often appear
in the genetic regulatory network and the vehicle clutch system etc. An
interesting and significant study on dimension-varying control systems is how
to realize the dimension-varying (the transient dynamics) process smoothly
between the different dimensional subsystems. The quotient space approach is
considered as an effective tool to model the transient dynamics of dimension-
varying control systems. This paper investigates the realization problem of
the transient dynamics of dimension-varying control systems. By revealing the
structure of the controllable subspace for the linear system on quotient
space, we propose the condition for the realization of the transient dynamics
of dimension-varying control systems, based on which a new scheme for modeling
the transient dynamics is given. Moreover, the proposed scheme justifies the
existing result for the strategy for modelling the transient dynamics. A
numeric example is given to illustrate our theoretical results.

###### keywords:

Dimension-varying control system, transient dynamics, controllability,
quotient space, realization.

††journal: Journal of LaTeX Templatescor1cor1footnotetext: This work was
supported by the Natural Science Basic Research Program of Shaanxi Province
under Grant no. 2021JZ-12.

##  1 Introduction

Dimension-varying control systems, also known as cross-dimensional control
systems, are used for modeling complex systems with state spaces of different
dimensions. In practice, many dynamic behaviours can be modeled by a
dimension-varying control system. For example, in the internet or some other
service-based networks, some users may join or withdraw every now and then
[1]. In a genetic regulatory network, cells may die or birth at any time [2,
3]. Docking, undocking, departure and joining of spacecrafts have “short
periods” of dimension-varying process [4]. During the dimension-varying
process, the dimension-varying control system evolves from one model to
another model of different dimensions.

In the past, researchers did some work on dimension-varying control systems,
where they usually treated dimension-varying control systems as switching
systems and investigated the stability and control design problem based on the
switching system theory[5, 6, 7, 4]. However, this scheme ignores the
transient dynamics of the system during the dimension-varying process. In
fact, this transient period may be long enough so that the dynamics during
this process can not be ignored [4, 8]. For example, in automotive
engineering, the vehicle clutch system shown as Figure 1 can be described as
2-dimensional system or 1-dimensional system depending on whether the clutch
is “disengaged” or “engaged” [9, 10]. It takes about 1 second to complete the
transient process of “disengaged” or “engaged”. The processes of docking,
undocking, departure and joining in spacecrafts formation take even longer [6,
7, 4]. Therefore, it is significant to study the transient dynamics of
dimension-varying control systems. Modeling and control design for the
transient dynamics of dimension-varying control systems can improve the
performance of mechanical or other systems.

However, for decades, mathematically few researches on the transient dynamics
of dimension-varying control systems appear due to the lack of proper
theoretical tool. Fortunately, Prof. Cheng [11] in 2019 developed the theory
of dimension-varying control systems, where he investigated the underlying
mathematics of dimension-varying control systems and deliver a masterful
exposition of dimension-varying control systems to model and analyse the
dynamics of dimension-varying process of linear systems. Typically, based on
quotient space theory, Cheng in [10] proposed a projecting system of
dimension-varying control systems on a developed quotient space which consists
of the equivalent class of the state of different dimensional systems. In
consequence, the study on the transient dynamics of dimension-varying control
systems is treated as the investigation on the corresponding projecting
systems on quotient space.

Figure 1: The vehicle clutch system.

Specifically, for the given dimension-varying control system consisting of two
linear subsystems with various dimensions:

$\displaystyle\Sigma_{1}:\dot{x}(t)=A_{1}x(t)+B_{1}u_{1}(t),\quad
x\in\mathbb{R}^{p},$ (1)
$\displaystyle\Sigma_{2}:\dot{y}(t)=A_{2}y(t)+B_{2}u_{2}(t),\quad
y\in\mathbb{R}^{q},$

where $x(t)$ and $y(t)$ are states of subsystems $\Sigma_{1}$ and $\Sigma_{2}$
respectively, $A_{1}\in\mathbb{R}^{p\times p}$, $A_{2}\in\mathbb{R}^{q\times
q}$, $B_{1}\in\mathbb{R}^{p\times r}$ and $B_{2}\in\mathbb{R}^{q\times s}$ are
constant matrices that characterise the subsystems, $u_{i}(t)$ $(i=1,2)$
stands for the controlled input of subsystem $\Sigma_{i}$. During the
dimension-varying process, system (1) will evolve from $\Sigma_{1}$ to
$\Sigma_{2}$ (or $\Sigma_{2}$ to $\Sigma_{1}$). For system (1), a well-known
challenge is that different dimensions of $\Sigma_{1}$ and $\Sigma_{2}$ lead
to an intractable modeling of the transient dynamics. In order to overcome
this challenge, Cheng [12, 11] first proposed that the equivalence of linear
systems on spaces of different dimensions and constructed a quotient space
based on the equivalence, and then modeled the transient dynamics of
dimension-varying control systems by projecting the dimension-varying control
system onto quotient space. As a result, the transient dynamics of system (1)
can be realized if one can design $u_{1}(t)$ and $u_{2}(t)$ such that
$\Sigma_{1}$ ($\Sigma_{2}$) can evolve to $\Sigma_{2}$ ($\Sigma_{1}$) during
the transient period. Thereby, in theory, the transient process of dimension-
varying control systems can be realized as smoothly as possible via the
control design.

Recently the theory of dimension-varying control systems proposed by Cheng
[11] has attracted considerable attention. Zhang et al. [13] given a set of
basis for the cross-dimensional state space of cross-dimensional systems to
show it is of countably infinite dimension. Feng et al. [14] considered the
variation of a class of cross-dimensional linear systems and established an
algorithm for calculating the state dimensions after a transition time. Other
new published research on dimension-varying control systems, please refer to
[15, 16, 17, 18]. However, to the best of our knowledge, other than Cheng’s
works [12, 11, 10], the above mentioned works have never referred to the
realization problem of the transient dynamics of dimension-varying control
systems. What is worth saying, Cheng [10] applied his theory to modeling the
transient dynamics of vehicle clutch systems (Figure 1), based on which he had
designed the control for dimension-varying process of clutch systems. The
control implementation for the transient dynamics of vehicle clutch systems in
[10] urges us a query: What the condition for “the control implementation” is.

_Cheng’s works[12, 11, 10] provide us a promising direction to investigate the
transient dynamics of the dimension-varying control system, in which a
fundamental theoretical framework has been built. However, as for the
realization of the transient dynamics of dimension-varying control systems,
several intriguing and challenging issues arise:_

  1. 1.

_Whether the realization problem of system (_ 1) is equivalent to that of its
projecting system?

  2. 2.

_What conditions need to be satisfied for system (_ 1) to realize transient
process?

  3. 3.

_What conditions should be satisfied to model the transient dynamics of
dimension-varying control systems?_

Motivated by the above issues, we proceed to study the realization problem of
transient dynamics of dimension-varying control systems in our paper.
Specifically, from the viewpoint of the controllability theory of the system,
this paper aims at providing a theoretical basis for modeling the transient
dynamics of dimension-varying control systems on quotient space. By analyzing
the controllable subspace of the linear system on quotient space, we first
show that the realization of dimension-varying control systems can be
equivalent to that of its projecting system on quotient space. Specifically,
we present the coordinate transformation of linear system on quotient space
for controllability decomposition, in which we show that the relation between
the controllability of dimension-varying control systems and that of its
projecting systems. Furthermore, we present a condition for the realization of
the transient dynamics of dimension-varying control systems based on the
controllability of the projecting system. In particular, based on the proposed
condition, a new scheme for modeling the transient dynamics of dimension-
varying control systems is developed, based on which the realization of the
transient dynamics of vehicle clutch systems in [10] can be justified.

The structure of this paper is organized as follows: In section 2, we give
some preliminaries on quotient space and formulate problems. Section 3
analyzes the controllability of linear systems on quotient space and gives the
coordinate transformation for controllability decomposition. The condition for
the realization of the transient dynamics of dimension-varying control systems
is presented in Section 4. Section 5 presents the condition for modeling the
transient dynamics of dimension-varying control systems, where we also recall
an example in [11] to illustrate the effectiveness of our proposed the
theoretical results. The conclusion is drawn in Section 6.

##  2 Preliminaries and Problem Description

In this section, we sketch fundamental definitions and concepts of quotient
space, and introduce linear systems on quotient space. For more details, the
reader is suggested to refer to [19]. At the end of this section, the
considered problems about the realization of the transient dynamics of
dimension-varying control systems are put forth.

Some notations are given first. Symbol $\otimes$ is Kronecker product of
matrices. One-entry vector is
$\mathbf{1}_{n}=[\underbrace{1,\cdots,1}_{n}]^{T}$. Denote the $k\times k$
matrix with its entries being $\frac{1}{k}$ by $\mathbf{J}_{k}$. The set of
mix-dimensional vectors is defined as
$\mathcal{V}:=\bigcup_{n=1}^{\infty}\mathcal{V}_{n}$, where $\mathcal{V}_{n}$
is an $n$-dimensional vector space. For brevity, we assume
$\mathcal{V}_{n}=\mathbb{R}^{n}$. The set of all matrices is defined as
$\mathcal{M}:=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}\mathcal{M}_{m\times
n}$, where $\mathcal{M}_{m\times n}$ represents the set of $m\times n$
dimensional real matrices.

###  2.1 Preliminaries

######  Definition 1.

[19] 1) Let $x,y\in\mathcal{V}$. $x$ and $y$ are said to be equivalent,
denoted by $x\leftrightarrow y$, if there exist $\mathbf{1}_{\alpha}$ and
$\mathbf{1}_{\beta}$, such that
$x\otimes\mathbf{1}_{\alpha}=y\otimes\mathbf{1}_{\beta}.$

2) The equivalent class is denoted by

$\bar{x}=\\{y\in\mathcal{V}\mid y\leftrightarrow x\\}.$

3) A vector $x_{1}\in\bar{x}$ is irreducible, if there are no $y$ and
$\mathbf{1}_{s}$, $s>1$, such that $x_{1}=y\otimes\mathbf{1}_{s}$. Then, all
the elements in $\bar{x}$ can be expressed as
$x_{i}=x_{1}\otimes\mathbf{1}_{i},i=1,2,3,\ldots$, and $x_{1}$ is an unique
irreducible element.

4) The quotient vector space of $\mathcal{V}$ under equivalence relation
$\leftrightarrow$, denoted by $\Omega$, is

$\Omega:=\\{\bar{x}\mid x\in\mathcal{V}\\}.$

######  Definition 2.

[19] Let $x\in\mathcal{V}_{m}$, $y\in\mathcal{V}_{n}$ and $t=m\vee n$ be the
least common multiple of $m$ and $n$. An addition of $x$ and $y$ is defined as
$x\vec{|+}y:=\left(x\otimes\mathbf{1}_{t/m}\right)+\left(y\otimes\mathbf{1}_{t/n}\right)\in\mathcal{V}_{t}$.
Correspondingly, the subtraction is defined as
$x\vec{\vdash}y:=x\vec{|+}(-y).$ Then, for $\bar{x},\bar{y}\in\Omega$ and
$a\in\mathbb{R}$,

$\displaystyle\bar{x}\vec{|+}\bar{y}$ $\displaystyle:=\overline{x\vec{|+}y},$
$\displaystyle\bar{x}\vec{\vdash}$ $\displaystyle:=\overline{x\vec{\vdash}y},$
$\displaystyle a\bar{x}$ $\displaystyle:=\overline{ax}.$

Definition 2 shows the quotient space $\Omega$ is a vector space.

######  Definition 3.

[19] Let $A,B\in\mathcal{M}$. $A$ and $B$ are said to be equivalent, denoted
by $A\thickapprox B$, if there exist $\mathbf{J}_{\alpha}$ and
$\mathbf{J}_{\beta}$, such that
$A\otimes\mathbf{J}_{\alpha}=B\otimes\mathbf{J}_{\beta}.$ The equivalent class
is denoted by $\hat{A}=\\{B\mid B\approx A\\}.$ Then the quotient space is
denoted by $\Xi:=\mathcal{M}/\approx.$

######  Definition 4.

[19] 1)Let $A\in\mathcal{M}_{m\times n},B\in\mathcal{M}_{p\times q}$, and
$t=n\vee p$. Then the second semi-tensor product of $A$ and $B$, denoted by
$A\circ B$, is defined as

$A\circ B:=\left(A\otimes J_{t/n}\right)\left(B\otimes J_{t/p}\right).$

2)Define a product on $\Xi$ as $\hat{A}\circ\hat{B}:=\widehat{A\circ B}.$

The following definition presents a vector equivalence for two matrices, which
is used to describe linear systems on quotient space.

######  Definition 5.

[19] Let $B,D\in\mathcal{M}$. $B$ and $D$ are said to be vector equivalent,
denoted by $B\leftrightarrow D$, if there exist $\mathbf{1}_{\alpha}$ and
$\mathbf{1}_{\beta}$, such that
$B\otimes\mathbf{1}_{\alpha}=D\otimes\mathbf{1}_{\beta}.$ The equivalent class
of $B$ is denoted by $\bar{B}=\\{D\mid D\leftrightarrow B\\}.$

######  Definition 6.

[19] 1) Let $A\in\mathcal{M}_{m\times n}\subset\mathcal{M}$,
$x\in\mathcal{V}_{r}\subset\mathcal{V}$ and $t=n\vee r$. Then the product of
$A$ and $x$, denoted by $\vec{\circ}$, is defined as

$A\vec{\circ}x:=\left(A\otimes\mathbf{J}_{t/n}\right)\left(x\otimes\mathbf{1}_{t/r}\right).$
(2)

2) The action of $\Xi$ on $\Omega$ is denoted as
$\hat{A}\vec{\circ}\bar{x}:=\overline{A\vec{\circ}x}$.

Viewing a matrix $B$ as a set of column vectors, the above definition can be
easily extended to the case of two matrices $A$ and $B$:

######  Definition 7.

1) Let $A\in\mathcal{M}_{m\times n}$, $B\in\mathcal{M}_{r\times s}$ and
$t=n\vee r$. Then

$A\vec{\circ}B:=\left(A\otimes\mathbf{J}_{t/n}\right)\left(B\otimes\mathbf{1}_{t/r}\right).$
(3)

2) The action on $\Xi$ is denoted as
$\hat{A}\vec{\circ}\bar{B}:=\overline{A\vec{\circ}B}$.

By Definition 2, Cheng [10] has defined the projecting system of the linear
system on quotient space as follows.

######  Definition 8.

[10] 1) Consider a linear system

$\displaystyle\dot{x}=Ax(t)+Bu(t),\quad x(t)\in\mathbb{R}^{r}.$ (4)

The following system on quotient space $\Omega$ is called the projecting
system of (4)

$\displaystyle\dot{\bar{x}}(t)=\hat{A}\vec{\circ}\bar{x}(t)+\bar{B}u(t),\quad\bar{x}(t)\in\Omega.$
(5)

In turn, system (4) is called the lifting system of (5) on $\mathbb{R}^{r}$,
if $A\in\hat{A}$ and $B\in\bar{B}$.

2) Let $\Theta_{0}$ be a linear control system on quotient space and
$\Theta_{n}$ be its lifting system on $\mathbb{R}^{n}$. Then all the lifting
systems are said to be equivalent.

###  2.2 Problem Description

Note that a system on quotient space is a set of equivalent systems of
different dimensions, hence the systems of different dimensions can be
“lifted” up to a space of the identical dimension. This makes it possible to
model the transient dynamics of dimension-varying control systems on quotient
space. Therefore, by Definition 8, the projection of system (1) on quotient
space can be described as follows.

$\displaystyle\bar{\Sigma}_{1}:\dot{\bar{x}}(t)=\hat{A}_{1}\vec{\circ}\bar{x}(t)+\bar{B}_{1}u_{1}(t),\quad\bar{x}\in\bar{\mathbb{R}}^{p},$
(6)
$\displaystyle\bar{\Sigma}_{2}:\dot{\bar{y}}(t)=\hat{A}_{2}\vec{\circ}\bar{y}(t)+\bar{B}_{2}u_{2}(t),\quad\bar{y}\in\bar{\mathbb{R}}^{q},$

where $\bar{\mathbb{R}}^{p}=\\{\bar{x}|x\in{\mathbb{R}}^{p}\\}$ and
$\bar{\mathbb{R}}^{q}=\\{\bar{y}|y\in{\mathbb{R}}^{q}\\}$. For simplify,
hereafter we use the notation $\Omega$ to represent $\bar{\mathbb{R}}^{p}$ or
$\bar{\mathbb{R}}^{q}$. Based on system (6), we can convert the transient
dynamics from $\Sigma_{1}$ to $\Sigma_{2}$ of system (1) into that of
$\bar{\Sigma}_{1}$ to $\bar{\Sigma}_{2}$ on quotient space during the
transient period $[t_{0},t_{e}]$.

######  Definition 9.

[10] For system (1), the transient dynamics is said to be properly realized
from a given starting state ${x}(t_{0})\in\mathbb{R}^{p}$ to
${y}(t_{e})\in\mathbb{R}^{q}$ during the transient period $[t_{0},t_{e}]$, if,
for system (6), there are controls $u_{1}(t)$ and $u_{2}(t)$ such that a given
starting state $\bar{x}(t_{0})$ can be controlled to the state
$\bar{y}(t_{e})$ on the quotient space $\Omega$.

Here several intriguing and challenging problems arise:

  1. 1.

Whether the realization problem of system (1) is equivalent to that of system
(6)?

  2. 2.

What conditions need to be satisfied for system (1) to realize transient
process?

  3. 3.

What conditions should be satisfied to model the transient dynamics of
dimension-varying control systems?

Now, we will discuss each of these problems in details.

##  3 Some Analyses for Linear Systems on Quotient Space

In this section, our purpose is to illustrate the realization problem of
system (1) is equivalent to that of system (6) by studying the controllable
subspace of the projecting system (5) on quotient space. Furthermore, we
present the coordinate transformation of the projecting system which will
support our sequent research.

###  3.1 The controllable subspace of linear system on $\Omega$

######  Lemma 1.

Let $\mathcal{C}=\operatorname{span}\left\\{B,AB,\cdots,A^{r-1}B\right\\}$ be
the controllable subspace of system (4), the controllable subspace of the
projecting system (5), denoted by $\bar{\mathcal{C}}$, is

$\overline{\mathcal{C}}=\operatorname{span}\\{\bar{B},\overline{AB},\overline{A^{2}B},\cdots,\overline{A^{r-1}B}\\}$

###### Proof.

It is easy to calculate that the trajectory of the projecting system (5) with
the initial $\bar{x}(t_{0})$ is

$\bar{x}(t)=\mathrm{e}^{\hat{A}t}\vec{\circ}\bar{x}(t_{0})+\int_{t_{0}}^{t}\mathrm{e}^{\hat{A}(t-\tau)}\vec{\circ}\bar{B}u(\tau)\mathrm{d}\tau.$
(7)

Let $\mathbf{0}:=\\{[\underbrace{0,0,\cdots,0}_{n}]^{T}\mid n=1,2,\cdots\\}$.
Suppose $\bar{x}(t_{0})$ is driven to the origin $\bar{x}(t)=\mathbf{0}$, then
(7) is further written as

$\bar{x}(t_{0})=-\int_{t_{0}}^{t}\mathrm{e}^{\hat{A}(t_{0}-\tau)}\vec{\circ}\bar{B}u(\tau)\mathrm{d}\tau.$
(8)

It can be obtained from Definition 3 that the rank of $\hat{A}$ is equal to
that of $A$. Using Cayley¨CHamilton theorem [20], we have the following Taylor
expansion,

$\mathrm{e}^{\hat{A}(t_{0}-\tau)}=\sum_{j=0}^{r-1}\lambda_{j}(t_{0}-\tau)\hat{A}^{j},$

where $\hat{A}^{j}=\hat{A}^{0}\circ\hat{A}^{1}\circ\cdots\hat{A}^{j}$.

Let $\gamma_{j}=\int_{t_{0}}^{t}\lambda_{j}(t_{0}-\tau)u(\tau)\mathrm{d}\tau$,
(8) is rewritten as

$\displaystyle\bar{x}(t_{0})$
$\displaystyle=-\sum_{j=0}^{r-1}\hat{A}^{j}\vec{\circ}\bar{B}\gamma_{j}$ (9)
$\displaystyle=-[\begin{array}[]{llll}\bar{B}&\hat{A}\vec{\circ}\bar{B}&\cdots&\hat{A}^{r-1}\vec{\circ}\bar{B}\end{array}][\begin{array}[]{cccc}\gamma_{0}&\gamma_{1}&\cdots&\gamma_{r-1}\end{array}]^{T}$

It follows from Definition 4 that $\hat{A}^{j}=\widehat{A^{j}}$. Together with
Definition 7, we have
$\hat{A}^{j}\vec{\circ}\bar{B}=\widehat{A^{j}}\vec{\circ}\bar{B}=\overline{A^{j}\vec{\circ}B}$.
Furthermore, note that the column number of $A$ is equal to the row number of
$B$, we have $\overline{A^{j}\vec{\circ}B}=\overline{A^{j}B}$. Thus, (9) can
be expressed as

$\displaystyle\bar{x}(t_{0})$
$\displaystyle=-[\begin{array}[]{llll}\bar{B}&\overline{A\vec{\circ}B}&\cdots&\overline{{A}^{r-1}\vec{\circ}B}\end{array}][\begin{array}[]{cccc}\gamma_{0}&\gamma_{1}&\cdots&\gamma_{r-1}\end{array}]^{T}$
$\displaystyle=-[\begin{array}[]{llll}\bar{B}&\overline{AB}&\cdots&\overline{{A}^{r-1}B}\end{array}][\begin{array}[]{cccc}\gamma_{0}&\gamma_{1}&\cdots&\gamma_{r-1}\end{array}]^{T}$

Note that $\Omega$ is a vector space (recall Definition 2), the controllable
subspace of system (5) can be expressed as

$\overline{\mathcal{C}}=\operatorname{span}\\{\bar{B},\overline{AB},\overline{A^{2}B},\cdots,\overline{A^{r-1}B}\\}$

∎

In contrast with $\mathcal{C}$, we can see that elements of
$\bar{\mathcal{C}}$ are equivalence classes of elements of $\mathcal{C}$,
which implies that controllable states of the projecting system (5) are
equivalent classes of controllable states of system (4).

Lemma 1 shows us that, say, if the initial state $x(t_{0})$ is controlled to
the state $x(t_{e})$ for system (4), correspondingly $\bar{x}(t_{0})$ can be
controlled to $\bar{x}(t_{e})$ for system (5) on $\Omega$. This means that the
study on the transient dynamics of system (1) may be performed by its
corresponding projecting (6) on quotient space. That is, for dimension-varying
control systems, instead of studying the realization of the transient dynamics
of dimension-varying control systems, we may study the corresponding problem
on quotient space. In view of this observation, we restrict attention to the
projecting system (6).

###  3.2 The coordinate transformation of linear system on $\Omega$

To proceed with our discussion, we only need to focus on the transient dynamic
of the projecting system (6). This subsection concerns the coordinate
transformation of system (6) aiming at investigating the controllability
decomposition on quotient space, which paves the way for the study of the
realization problem of the transient dynamics on quotient space.

######  Definition 10.

[11] Let $A\in\mathcal{M}_{m\times
n}\subset\mathcal{M},x\in\mathcal{V}_{r}\subset\mathcal{V}.$ Assume $t=n\vee
r$. Then the product of $A$ and $x$ is defined as

$A\overrightarrow{\ltimes}x:=\left(A\otimes
I_{t/n}\right)\left(x\otimes\mathbf{1}_{t/r}\right).$ (10)

Now, suppose $y\in\bar{y}$ and $y_{\pi}\in\bar{y}_{\pi}$ are two irreducible
vectors of same dimension, where $\bar{y},\bar{y}_{\pi}\in\Omega$ are two
vectors on quotient space $\Omega$, as seen in Definition 1. we have the
following Lemma.

######  Lemma 2.

Let $T_{\Lambda}$ be a coordinate transformation matrix such that
$y_{\pi}=T_{\Lambda}y$. Then $\langle T_{\Lambda}\rangle$ is said to be a
“pseudo-coordinate transformation matrix” such that $\bar{y}_{\pi}=\langle
T_{\Lambda}\rangle\overrightarrow{\ltimes}\bar{y}$, where $\langle
T_{\Lambda}\rangle=\\{T_{\Lambda},T_{\Lambda}\otimes
I_{2},\ldots,T_{\Lambda}\otimes I_{n}\\}$.

###### Proof.

It is sufficient to show that, for any $\xi\in\bar{y}$,
$\xi_{\pi}=T_{\Lambda}\overrightarrow{\ltimes}x\in\bar{y}_{\pi}$ holds.

Since $\xi\in\bar{y}$, then $\xi=y\otimes 1_{\alpha}$, where $y\in\bar{y}$ is
a irreducible vector. By Lemma 1, we have

$\displaystyle\xi_{\pi}=$ $\displaystyle
T_{\Lambda}\overrightarrow{\ltimes}\xi=T_{\Lambda}\overrightarrow{\ltimes}(y\otimes\mathbf{1}_{\alpha})$
(11) $\displaystyle=$ $\displaystyle(T_{\Lambda}\otimes
I_{\alpha})(y\otimes\mathbf{1}_{\alpha})$ $\displaystyle=$ $\displaystyle
T_{\Lambda}y\otimes\mathbf{1}_{\alpha}$ $\displaystyle=$ $\displaystyle
y_{\pi}\otimes\mathbf{1}_{\alpha}.$

Hence we have $\xi_{\pi}\leftrightarrow y_{\pi}$ which implies
$\xi_{\pi}\in\bar{y}_{\pi}$. I.e., since the arbitrariness of $\xi$, from the
derivation of (11), we have $\bar{y}_{\pi}=\langle
T_{\Lambda}\rangle\overrightarrow{\ltimes}\bar{y}$, where $\langle
T_{\Lambda}\rangle$ is determined according to the second line of (11).

In addition, the coordinate transformation matrix $T_{\Lambda}$ implies any
element of $\langle T_{\Lambda}\rangle$ is non-singular so that $\langle
T_{\Lambda}\rangle$ can be viewed as performing the coordinate transformation
ability for the vector $\bar{y}$. ∎

We note that it is clear that $\langle T_{\Lambda}\rangle$ is not a real
coordinate transformation matrix, so we call it “pseudo-coordinate
transformation matrix”. For statement ease, hereafter the terminology
“coordinate transformation matrix” is applied to $\langle T_{\Lambda}\rangle$
without cause confusion.

Then, using Lemma 2, the coordinate transformation of projecting systems on
$\Omega$ can be easily obtained.

######  Lemma 3.

Suppose that the lifting system (4) is transformed into the following system
under the coordinate transformation $x_{\pi}=Tx$,

$\displaystyle\dot{x}_{\pi}=A_{\pi}(t)x_{\pi}(t)+B_{\pi}(t)u(t),\quad
x_{\pi}(t)\in\mathbb{R}^{r},$ (12)

Then, correspondingly, its projecting system (5) on quotient space under the
“coordinate transformation” $\bar{x}_{\pi}=\langle
T\rangle\overrightarrow{\ltimes}\bar{x}$ can be converted into

$\displaystyle\dot{\bar{x}}_{\pi}(t)=\hat{A}_{\pi}\vec{\circ}\bar{x}_{\pi}(t)+\bar{B}_{\pi}u(t),\quad\bar{x}(t)\in\Omega.$
(13)

Lemma 2 and Lemma 3 provide us a proper “coordinate transformation” to bring
systems into the controllability decomposition on quotient space, which is
significant for us to proceed our study on the the realization of the
transient dynamics of dimension-varying control systems on quotient space.

##  4 Realization of the transient dynamics of dimension-varying control
systems on $\Omega$

In this section, a condition is presented for the realization of the transient
dynamics of dimension-varying control systems. To formulate our main result
precisely, we shall need some more notations and terminology.

Let $\mathcal{W}$ be a vector space. Denote
$\bar{\mathcal{W}}:=\\{\bar{\omega}\mid\omega\in\mathcal{W}\\}$. It is easy to
see $\text{dim}(\bar{\mathcal{W}})=\text{dim}(\mathcal{W})$.

######  Definition 11.

Let $\mathcal{V}_{m}$ and $\mathcal{V}_{n}$ be two vector space. The sum of
$\mathcal{V}_{m}$ and $\mathcal{V}_{n}$ is defined as
$\mathcal{V}_{m}|\vec{+}\mathcal{V}_{n}:=\left\\{x|\vec{+}y|x\in\mathcal{V}_{m},y\in\mathcal{V}_{n}\right\\}$.
Then the sum of $\bar{\mathcal{V}}_{m}$ and $\bar{\mathcal{V}}_{n}$ is defined
as
$\bar{\mathcal{V}}_{m}|\vec{+}\bar{\mathcal{V}}_{n}:=\overline{\mathcal{V}_{m}|\vec{+}\mathcal{V}_{n}}.$

Given a vector
$\bm{\varepsilon}=(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{m})^{T}\in\mathcal{V}_{m}$
and a vector space $\mathcal{V}_{n}$, where $m\leq n$. The _embedded mapping_
of $\bm{\varepsilon}$ on $\mathcal{V}_{n}$ is defined as
$\varphi_{n}(\bm{\varepsilon}):=(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{m},\underbrace{0,0,\ldots,0}_{n-m})^{T}$,
then we denote the embedded space by
$\mathcal{V}_{n}^{m}=\text{span}\\{\varphi_{n}(\bm{\varepsilon})\\}$. That is,
the missing elements of $\bm{\varepsilon}$ is filled up with $0$.

######  Definition 12.

Let $\mathcal{V}_{m}$ and $\mathcal{V}_{n}$ be two vector space and $m\leq n$.
the direct sum of $\mathcal{V}_{n}^{m}$ and $\mathcal{V}_{n}$ is defined as
$\mathcal{V}_{n}^{m}\vec{\oplus}\mathcal{V}_{n}:=\left\\{\varphi_{n}(x)+y|\varphi_{n}(x)\cap
y=\emptyset,x\in\mathcal{V}_{m},y\in\mathcal{V}_{n}\right\\}$. Then the direct
sum of $\bar{\mathcal{V}}_{n}^{m}$ and $\bar{\mathcal{V}}_{n}$ is defined as
$\bar{\mathcal{V}}_{n}^{m}\vec{\oplus}\bar{\mathcal{V}}_{n}:=\overline{\mathcal{V}_{n}^{m}\vec{\oplus}\mathcal{V}_{n}}$.

Without loss of generality, we assume $p<q$ for system (1). Then a condition
for realization of transient dynamics of dimension-varying control system (6)
is given as follows.

######  Theorem 1.

Consider system (6). Let $\bar{\mathcal{C}}_{i},i=1,2$ be controllable
subspace of $\bar{\Sigma}_{i}$. Then $\bar{x}(t_{0})$ can be controlled to a
state $\bar{y}(t_{e})$ during the transient period $[t_{0},t_{e}]$ on quotient
space $\Omega$ if there exists
$\bar{\mathcal{C}}_{2^{*}}\subset\bar{\mathcal{C}}_{2}$ such that

$\bar{\mathcal{C}}_{1}\vec{\oplus}\bar{\mathcal{C}}_{2^{*}}=\bar{\mathbb{R}}^{q},$

where $\bar{\mathbb{R}}_{q}=\\{\bar{y}~{}|~{}y\in\mathbb{R}^{q}\\}$.

###### Proof.

With a mild modification, the system $\Sigma_{1}$ might be described as
follows:

$\left\\{\begin{array}[]{l}\dot{x}=Ax+Bu_{1},\quad x\in\mathbb{R}^{p},\\\
\dot{x^{c}}=\mathbf{0}_{q-p},\quad
x^{c}\in\mathbb{R}^{q-p},\end{array}\right.$ (14)

where $\mathbf{0}_{q-p}=[\underbrace{0,\ldots,0}_{q-p}]$. Let $w=(x,x^{c})$,
then the projecting system of (14) on quotient space $\Omega$ is denoted by

$\bar{\Sigma}_{3}:\dot{\bar{w}}(t)=\hat{A}_{3}\vec{\circ}\bar{w}(t)+\bar{B}_{3}u_{1}(t),\quad\bar{w}\in\Omega.$

Note that $\bar{\Sigma}_{3}$ has the same controllable space as
$\bar{\Sigma}_{1}$. Then the transient dynamics of system (6) is identical to
the transient dynamics of the the following system.

$\displaystyle\bar{\Sigma}_{3}:\dot{\bar{w}}(t)=\hat{A}_{3}\vec{\circ}\bar{w}(t)+\bar{B}_{3}u_{1}(t),\quad\bar{w}\in\Omega,$
(15)
$\displaystyle\bar{\Sigma}_{2}:\dot{\bar{y}}(t)=\hat{A}_{2}\vec{\circ}\bar{y}(t)+\bar{B}_{2}u_{2}(t),\quad\bar{y}\in\Omega.$

Therefore, we next study the transient dynamics of system (15).

For system (15), by Definition 9, we need to show that there exist controls
$u_{1}$ and $u_{2}$ such that the transient dynamics of system (15) can be
realized from a given state $\bar{w}(t_{0})$ of system $\bar{\Sigma}_{3}$ to
the state $\bar{y}(t_{e})$ of system $\bar{\Sigma}_{2}$ during the transient
period $[t_{0},t_{e}]$.

Let $\mathcal{C}_{3}$ and $\bar{\mathcal{C}}_{3}$ be the controllable subspace
of system (14) and system $\bar{\Sigma}_{3}$, respectively. With the
assumption $p<q$, it can be seen that $\operatorname{dim}(\mathcal{C}_{3})<q.$

Next we consider two cases.

Case 1: $\operatorname{dim}(\bar{\mathcal{C}}_{2})=q.$

Since $\operatorname{dim}(\mathcal{C}_{3})<q$, there exist a coordinate
transformation $\mathbf{w}=T_{1}w$ such that system (14) can be converted into

$\left\\{\begin{array}[]{l}\dot{\mathbf{w}}_{1}=A_{3}^{11}\mathbf{w}_{1}+A_{3}^{12}\mathbf{w}_{2}+B_{3}u_{1},\\\
\dot{\mathbf{w}}_{2}=A_{3}^{22}\mathbf{w}_{2}.\end{array}\right.$ (16)

Then, by Lemma 3, accordingly there exist the coordinate transformation
$\bar{\mathbf{w}}=\langle T_{1}\rangle\vec{\ltimes}\bar{w}$ such that
$\bar{\Sigma}_{3}$ is converted into

$\left\\{\begin{array}[]{l}\dot{\bar{\mathbf{w}}}_{1}=\hat{A}_{3}^{11}\vec{\circ}\bar{\mathbf{w}}_{1}+\hat{A}_{3}^{12}\vec{\circ}\bar{\mathbf{w}}_{2}+\bar{B}_{3}u_{1},\\\
\dot{\bar{\mathbf{w}}}_{2}=\hat{A}_{3}^{22}\vec{\circ}\bar{\mathbf{w}}_{2}.\end{array}\right.$
(17)

Note that $\operatorname{dim}(\bar{\mathcal{C}}_{2})=q$, similarly, under the
“coordinate transformation” $\bar{\mathbf{y}}=\langle
T_{1}\rangle\vec{\ltimes}\bar{y}$, $\bar{\Sigma}_{2}$ is converted onto

$\left\\{\begin{array}[]{l}\dot{\bar{\mathbf{y}}}_{1}=\hat{A}_{2}^{11}\vec{\circ}\bar{\mathbf{y}}_{1}+\hat{A}_{2}^{12}\vec{\circ}\bar{\mathbf{y}}_{2}+\bar{B}_{2}^{1}u_{2}\\\
\dot{\bar{\mathbf{y}}}_{2}=\hat{A}_{2}^{21}\vec{\circ}\bar{\mathbf{y}}_{1}+\hat{A}_{2}^{22}\vec{\circ}\bar{\mathbf{y}}_{2}+\bar{B}_{2}^{2}u_{2}\end{array}\right.$
(18)

Let $\bar{w}(t_{0})=(\bar{\mathbf{w}}_{1}(t_{0}),\bar{\mathbf{w}}_{2}(t_{0}))$
and
$\bar{y}(t_{e})=(\bar{\mathbf{y}}_{1}(t_{e}),\bar{\mathbf{y}}_{2}(t_{e}))$.
Then it is enough to show that the transient dynamics can be realized from
system (17) to system (18). Clearly $\bar{\mathbf{w}}_{1}(t_{0})$ is
controllable. Next, we only need to consider how to control
$\bar{\mathbf{w}}_{2}(t_{0})$ to $\bar{\mathbf{y}}_{2}(t_{e})$. Note that
$(\hat{A}_{2}^{22},\bar{B}_{2}^{2})$ is controllable in (18), by Definition 9,
there is $u_{2}$ such that $\bar{\mathbf{w}}_{2}(t_{0})$ is controlled to
$\bar{\mathbf{y}}_{2}(t_{e})$. Thus, $\bar{w}(t_{0})$ can be controlled to
$\bar{y}(t_{e})$.

Case 2: $\operatorname{dim}(\bar{\mathcal{C}}_{2})<q.$

Let $r=\operatorname{dim}(\bar{\mathcal{C}}_{1})$, clearly
$\bar{\mathbf{w}}_{2}(t)\in\bar{\mathbb{R}}^{q-p}\vec{\oplus}\bar{\mathbb{R}}^{p-r}$.
By the assumption
$\bar{\mathcal{C}}_{1}\vec{\oplus}\bar{\mathcal{C}}_{2^{*}}=\bar{\mathbb{R}}_{q}$
with $\bar{\mathcal{C}}_{2^{*}}\subset\bar{\mathcal{C}}_{2}$, it follows
$\bar{\mathcal{C}}_{2^{*}}=\bar{\mathbb{R}}^{q-p}\vec{\oplus}\bar{\mathbb{R}}^{p-r}$.
i.e., $\bar{\mathbf{w}}_{2}(t)\subset\bar{\mathcal{C}}_{2}$. So we have

$\bar{\mathbb{R}}^{q}=\bar{\mathcal{C}}_{1}\vec{\oplus}\bar{\mathbb{R}}^{q-p}\vec{\oplus}\bar{\mathbb{R}}^{p-r}.$
(19)

Based on (19), by Lemma 3, there is a proper “coordinate transformation”
$\bar{\mathbf{w}}=\langle T_{2}\rangle\vec{\ltimes}w$ to further split
$\bar{\Sigma}_{3}$ into

$\left\\{\begin{array}[]{l}\dot{\bar{\mathbf{w}}}_{1}=\hat{A}_{3}^{11}\vec{\circ}\bar{{\mathbf{w}}}_{1}+\hat{A}_{3}^{12}\vec{\circ}\bar{\mathbf{w}}_{2}^{1}+\hat{A}_{3}^{13}\vec{\circ}\bar{\mathbf{w}}_{2}^{2}+\bar{B}_{3}^{1}u_{1},\quad{\bar{\mathbf{w}}}_{1}\in\bar{\mathbb{R}}^{r},\\\
\dot{\bar{\mathbf{w}}}_{2}^{1}=\hat{A}_{3}^{22}\vec{\circ}\bar{\mathbf{w}}_{2}^{1}+\hat{A}_{3}^{23}\vec{\circ}\bar{\mathbf{w}}_{2}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad{\bar{\mathbf{w}}}_{2}^{1}\in\bar{\mathbb{R}}^{p-r},\\\
\dot{\bar{\mathbf{w}}}_{2}^{2}=\hat{A}_{3}^{33}\vec{\circ}\bar{\mathbf{w}}^{2}_{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad{\bar{\mathbf{w}}}_{2}^{2}\in\bar{\mathbb{R}}^{q-p}.\end{array}\right.$
(20)

Using the same “coordinate transformation matrix” $\langle T_{2}\rangle$ ,
$\bar{\Sigma}_{2}$ can be split into

$\displaystyle\left\\{\begin{array}[]{l}\dot{\bar{\mathbf{y}}}_{1}^{1}=\hat{A}_{2}^{11}\vec{\circ}\bar{\mathbf{y}}_{1}^{1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad{\bar{\mathbf{y}}}_{1}^{1}\in\bar{\mathbb{R}}^{r},\\\
\dot{\bar{\mathbf{y}}}_{1}^{2}=\hat{A}_{2}^{21}\vec{\circ}\bar{\mathbf{y}}_{1}^{1}+\hat{A}_{2}^{22}\vec{\circ}\bar{\mathbf{y}}_{1}^{2}+\hat{A}_{2}^{23}\vec{\circ}\bar{\mathbf{y}}_{2}+\bar{B}_{2}^{1}u_{2},\quad{\bar{\mathbf{y}}}_{1}^{2}\in\bar{\mathbb{R}}^{p-r},\\\
\dot{\bar{\mathbf{y}}}_{2}=\hat{A}_{2}^{31}\vec{\circ}\bar{\mathbf{y}}_{1}^{1}+\hat{A}_{2}^{32}\vec{\circ}\bar{\mathbf{y}}_{1}^{2}+\hat{A}_{2}^{33}\vec{\circ}\bar{\mathbf{y}}_{2}+\bar{B}_{2}^{2}u_{2},\quad{\bar{\mathbf{y}}}_{2}\in\bar{\mathbb{R}}^{q-p}.\end{array}\right.$
(21)

Now we analyze the transient dynamics of system (20) and system (21).

For system (20), since $(\hat{A}_{3}^{11},\bar{B}_{3}^{1})$ is controllable,
there is the control $u_{1}$ such that the state $\bar{\mathbf{w}}_{1}(t)$ is
controlled to $\bar{\mathbf{y}}_{1}^{1}(t_{e})$. Again since
$(\hat{A}_{2}^{22},\hat{B}_{2}^{1})$ and $(\hat{A}_{2}^{33},\hat{B}_{2}^{2})$
are controllable, there is the control $u_{2}$ such that the states
$\bar{\mathbf{w}}^{1}_{2}(t)$ and $\bar{\mathbf{w}}_{2}^{2}(t)$ are
controllable. That is, for system (20) and system (21), we can find control
$u_{1}$ and $u_{2}$ such that the transient dynamic from the starting state
$\bar{w}(t_{0})$ to the destination state $\bar{y}(t_{e})$ during the
transient period $[t_{0},t_{e}]$. ∎

As a direct consequence, the version of Theorem 1 on Euclidean space can be
stated as follows.

######  Corollary 1.

Consider system (1). Let $\mathcal{C}_{i},i=1,2$ be the controllable subspace
of $\Sigma_{i}$. Then the dimension transience of system (1) is properly
realized if there exists $\mathcal{C}_{2^{*}}\subset\mathcal{C}_{2}$ such that

$\mathcal{C}_{1}\vec{\oplus}\mathcal{C}_{2^{*}}=\mathbb{R}^{q}.$

##  5 Condition for Modeling the Transient Dynamics of Dimension-Varying
Systems

A necessary condition for modeling the transient dynamics is proposed in this
section, based on which a new scheme for modeling the transient dynamics of
dimension-varying control systems is developed. Accordingly, a numerical
example is given to illustrate the effectiveness of our proposed the
theoretical results.

Based on Definition 9, the transient dynamics of dimension-varying control
system (6) on quotient space $\Omega$ is modeled as the following unified
form:

$\dot{\bar{z}}=\bar{F}(\bar{z},u_{1},u_{2}),\bar{z}\in\Omega.$ (22)

(22) is called _transient dynamics model_. Our purpose is to realize the
transient process of (6) with the help of the transient dynamics model (22) on
$\Omega$. Based on Theorem 1, a necessary condition for modeling the transient
dynamics can be obtained naturally.

######  Corollary 2.

With the same notations as Theorem 1. Let $\bar{\mathcal{C}_{z}}$ be the the
controllable subspace of (22). Then a necessary condition for modeling the
transient dynamics of system (6) is as follows:

$\bar{\mathcal{C}}_{1}+\bar{\mathcal{C}}_{2}\subset\bar{\mathcal{C}}_{z}.$
(23)

Next, we will demonstrate that the modelling strategy of the transient
dynamics of dimension-varying control systems proposed by Prof. Cheng [11]
satisfies the condition of Corollary 2.

Let $n=p\vee q$ be the least common multiple of $p$ and $q$. Then (6) can be
“lifted” to $\mathbb{R}^{n}$ as

$\begin{gathered}\Sigma_{z}^{1}:\dot{z}(t)=\mathbf{A_{1}}z(t)+\mathbf{B_{1}}u_{1}(t),\quad
z(t)\in\mathbb{R}^{n},\\\
\Sigma_{z}^{2}:\dot{z}(t)=\mathbf{A_{2}}z(t)+\mathbf{B_{2}}u_{2}(t),\quad
z(t)\in\mathbb{R}^{n},\end{gathered}$ (24)

where

$\begin{gathered}\mathbf{A_{1}}=A_{1}\otimes\mathbf{J}_{n/p},\mathbf{B_{1}}=B_{1}\otimes\mathbf{1}_{n/p},\\\
\mathbf{A_{2}}=A_{2}\otimes\mathbf{J}_{n/q},\mathbf{B_{2}}=B_{2}\otimes\mathbf{1}_{n/q}.\end{gathered}$

Cheng [11] modelled the transient dynamics of system (24) as a linear
combination form as follows:

$\dot{z}(t)=\left[\mu\mathbf{A}_{1}+(1-\mu)\mathbf{A}_{2}\right]z(t)+\mu\mathbf{B}_{1}u_{1}+(1-\mu)\mathbf{B}_{2}u_{2}.$
(25)

where $\mu=\frac{m_{1}}{m_{1}+m_{2}}$, $m_{1}$ and $m_{2}$ are “formal masses”
of the two systems. For notational convenience, let
$\mathbf{A}^{*}=\mu\mathbf{A}_{1}+(1-\mu)\mathbf{A}_{2}$,
$\mathbf{B}_{1}^{*}=\mu\mathbf{B}_{1}$ and
$\mathbf{B}_{2}^{*}=1-\mu\mathbf{B}_{2}$. Then the transient dynamics model
(22) of system (24) can be specified as

$\dot{\bar{z}}(t)=\hat{\mathbf{A}}^{*}\vec{\circ}\bar{z}(t)+\bar{\mathbf{B}}_{1}^{*}u_{1}+\bar{\mathbf{B}}_{2}^{*}u_{2}.$
(26)

######  Proposition 1.

With the same notations as Theorem 1. Let $\bar{\mathcal{C}}^{*}_{z}$ be the
controllable subspace of the transient dynamics (26). Then for system (24)

$\bar{\mathcal{C}}_{1}+\bar{\mathcal{C}}_{2}\subset\bar{\mathcal{C}}^{*}_{z}.$

###### Proof.

The solution of (26) with $\bar{z}(0)=\bar{z}_{0}$ is given by

$\bar{z}(t)=\mathrm{e}^{\hat{\mathbf{A}}^{*}(t-t_{0})}\vec{\circ}\bar{z}_{0}+\int_{t_{0}}^{t}\mathrm{e}^{\hat{\mathbf{A}}^{*}(t-\tau)}\vec{\circ}\bar{\mathbf{B}}_{1}^{*}u_{1}(\tau)\mathrm{d}\tau+\int_{t_{0}}^{t}\mathrm{e}^{\hat{\mathbf{A}}^{*}(t-\tau)}\vec{\circ}\bar{\mathbf{B}}_{2}^{*}u_{2}(\tau)\mathrm{d}\tau$

Similar to the proof of Lemma 1, the controllable subspace of (26) is

$\displaystyle\bar{\mathcal{C}}^{*}_{z}$
$\displaystyle=\operatorname{span}\\{[\begin{array}[]{llll}\bar{\mathbf{B}}^{*}_{1}&\overline{\mathbf{A}^{*}\mathbf{B}^{*}_{1}}&\cdots&\overline{(\mathbf{A}^{*})^{n-1}\mathbf{B}^{*}_{1}}\end{array}]\\}+\operatorname{span}\\{[\begin{array}[]{llll}\bar{\mathbf{B}}^{*}_{2}&\overline{\mathbf{A}^{*}\mathbf{B}^{*}_{2}}&\cdots&\overline{(\mathbf{A}^{*})^{n-1}\mathbf{B}^{*}_{2}}\end{array}]\\},$

Briefly,

$\bar{\mathcal{C}}^{*}_{z}=\operatorname{span}\\{\bar{\mathbf{C}}_{1}^{*}\\}+\operatorname{span}\\{\bar{\mathbf{C}}_{2}^{*}\\}.$

We note that
$\bar{\mathcal{C}}_{1}\subset\operatorname{span}\\{\bar{\mathbf{C}_{1}^{*}}\\}$
and
$\bar{\mathcal{C}}_{2}\subset\operatorname{span}\\{\bar{\mathbf{C}}_{2}\\}$.
Thus,

$\bar{\mathcal{C}}_{1}+\bar{\mathcal{C}}_{2}\subset\bar{\mathcal{C}}^{*}_{z}.$

∎

######  Remark 1.

We note that the transient dynamics model (25) proposed in [11] is represented
as a convex combination form. However, from the proof of Proposition 1, we can
relax the restriction of the parameter $\mu$ to a generalized case. In fact,
it is enough that to design the transient dynamics model of system (24)
satisfies condition (23). For example, we can choose the two parameters
$\alpha,\beta\in\mathbb{R^{+}}$ such that the transient dynamics

$\dot{z}(t)=\left[\alpha\mathbf{A}_{1}+\beta\mathbf{A}_{2}\right]z(t)+\alpha\mathbf{B}_{1}u_{1}+\beta\mathbf{B}_{2}u_{2},$
(27)

satisfies (23).

The following numerical example from Cheng [11] is illustrated to verify the
condition of Corollary 2. In particular, the parameters of the designed
transient dynamics model need not satisfy the convex combination constraint,
as required in Cheng [11].

######  Example 1.

[11] Consider a dimension-varying control system, which has two models
$(A_{1},B_{1})$ and $(A_{2},B_{2})$ as

$\displaystyle\dot{x}(t)=A_{1}x(t)+B_{1}u_{1}(t),\quad x\in\mathbb{R}^{2},$
(28) $\displaystyle\dot{y}(t)=A_{2}y(t)+B_{2}u_{2}(t),\quad
y\in\mathbb{R}^{3},$

where

$\begin{array}[]{ll}A_{1}=\left[\begin{array}[]{cccccc}0&1\\\
0&0\end{array}\right],&B_{1}=\left[\begin{array}[]{cccccc}0\\\
1\end{array}\right],\\\ A_{2}=\left[\begin{array}[]{cccccc}0&0&1\\\ 0&0&0\\\
0&1&0\end{array}\right],&B_{2}=\left[\begin{array}[]{cccccc}0\\\ 1\\\
0\end{array}\right].\end{array}$

Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be controllable subspaces of
$(A_{1},B_{1})$ and $(A_{2},B_{2})$, respectively. Then, we have

$\mathcal{C}_{1}=\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\ 1\\\
\end{array}\right],\left[\begin{array}[]{l}1\\\
0\end{array}\right]\right\\},\mathcal{C}_{2}=\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\
1\\\ 0\end{array}\right],\left[\begin{array}[]{l}1\\\ 0\\\
0\end{array}\right],\left[\begin{array}[]{l}0\\\ 0\\\
1\end{array}\right]\right\\}.$

According to (27), let $\alpha=\frac{3}{2},\beta=\frac{1}{2}$. The transient
dynamics of (28) is calculated as follows

$\dot{z}(t)=\left[\begin{array}[]{cccccc}0&0&0&1/2&3/4&3/4\\\
0&0&0&1/2&3/4&3/4\\\ 0&0&0&1/2&1/2&1/2\\\ 0&0&0&0&0&0\\\ 0&0&1/4&1/4&0&0\\\
0&0&1/4&1/4&0&0\end{array}\right]z(t)+\left[\begin{array}[]{l}0\\\ 0\\\ 0\\\
3/2\\\ 3/2\\\ 3/2\end{array}\right]u_{1}(t)+\left[\begin{array}[]{l}0\\\ 0\\\
1/2\\\ 1/2\\\ 0\\\ 0\end{array}\right]u_{2}(t),\quad z\in\mathbb{R}^{6}.$

Its controllability matrix is

$C_{z_{1}}=\left[\begin{array}[]{cccccccccccc}0&3&9/16&27/32&9/64&27/128&0&1/4&3/8&3/32&3/32&3/128\\\
0&3&9/16&27/32&9/64&27/128&0&1/4&3/8&3/32&3/32&3/128\\\
0&4/9&3/8&9/16&3/32&9/64&1/2&1/4&1/4&1/16&1/16&1/64\\\
3/2&0&0&0&0&0&1/2&0&0&0&0&0\\\
3/2&3/8&9/16&3/32&9/64&27/128&0&1/4&3/8&3/32&3/32&3/128\\\
3/2&3/8&9/16&3/32&9/64&27/128&0&1/4&3/8&3/32&3/32&3/128\\\
\end{array}\right].$

Then, we obtain the controllable subspace of (28) is

$\mathcal{C}_{z_{1}}=\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\
0\\\ 0\\\ 3/2\\\ 3/2\\\ 3/2\end{array}\right],\left[\begin{array}[]{l}3\\\
3\\\ 9/4\\\ 0\\\ 3/8\\\ 3/8\end{array}\right],\left[\begin{array}[]{l}0\\\
0\\\ 1/2\\\ 1/2\\\ 0\\\ 0\end{array}\right],\left[\begin{array}[]{l}1/4\\\
1/4\\\ 1/4\\\ 0\\\ 1/4\\\ 1/4\end{array}\right]\right\\}.$

Thus, we can verify that

$\displaystyle\bar{\mathcal{C}}_{1}+\bar{\mathcal{C}}_{2}=$
$\displaystyle\operatorname{span}\left\\{\overline{\left[\begin{array}[]{l}0\\\
1\\\ \end{array}\right]},\overline{\left[\begin{array}[]{l}1\\\
0\end{array}\right]}\right\\}+\operatorname{span}\left\\{\overline{\left[\begin{array}[]{l}0\\\
1\\\ 0\\\ \end{array}\right]},\overline{\left[\begin{array}[]{l}1\\\ 0\\\
0\end{array}\right]},\overline{\left[\begin{array}[]{l}0\\\ 0\\\
1\end{array}\right]}\right\\}$ $\displaystyle=$
$\displaystyle\operatorname{span}\left\\{\overline{\left[\begin{array}[]{l}0\\\
0\\\ 0\\\ 1\\\ 1\\\ 1\\\
\end{array}\right]},\overline{\left[\begin{array}[]{l}1\\\ 1\\\ 1\\\ 0\\\ 0\\\
0\end{array}\right]}\right\\}+\operatorname{span}\left\\{\overline{\left[\begin{array}[]{l}0\\\
0\\\ 1\\\ 1\\\ 0\\\ 0\\\
\end{array}\right]},\overline{\left[\begin{array}[]{l}1\\\ 1\\\ 0\\\ 0\\\ 0\\\
0\end{array}\right]},\overline{\left[\begin{array}[]{l}0\\\ 0\\\ 0\\\ 0\\\
1\\\ 1\end{array}\right]}\right\\}\subseteq\bar{\mathcal{C}}_{z_{1}}.$

In addition, it is easy to see that there exists
$\mathcal{C}_{2^{*}}=\operatorname{span}\left\\{\left[\begin{array}[]{lll}0&0&1\end{array}\right]^{T}\right\\}\subseteq\mathcal{C}_{2}$
such that

$\displaystyle\mathcal{C}_{1}\vec{\oplus}\mathcal{C}_{2^{*}}=$
$\displaystyle\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\ 1\\\
\end{array}\right],\left[\begin{array}[]{l}1\\\
0\end{array}\right]\right\\}\vec{\oplus}\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\
0\\\ 1\\\ \end{array}\right]\right\\}$ $\displaystyle=$
$\displaystyle\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\ 1\\\
0\\\ \end{array}\right],\left[\begin{array}[]{l}1\\\ 0\\\
0\end{array}\right]\right\\}+\operatorname{span}\left\\{\left[\begin{array}[]{l}0\\\
0\\\ 1\\\ \end{array}\right]\right\\}=\mathbb{R}_{3}.$

By Corollary 1, the dimension-varying control system (1) can realize the
transient dynamics.

##  6 Conclusion

The realization problem of the transient dynamics of dimension-varying control
systems has been investigated in this work. By analyzing the controllable
subspaces of linear systems on quotient space, a condition for the realization
of the transient process for dimension-varying control systems has been given,
based on which a new strategy for modelling the transient dynamics of
dimension-varying control systems on quotient space is presented. As a result,
theoretically we prove that the dynamic evolution between systems of different
dimensions can be implemented by designing the control of the transient
dynamics model on quotient space. The correctness of theoretical results is
verified by a numerical example.

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