11institutetext: JADBio - Gnosis DA S.A. 22institutetext: Computer Science
Department - University of Crete 22email:
<EMAIL_ADDRESS>
# A Meta-Level Learning Algorithm for Sequential Hyper-Parameter Space
Reduction in AutoML

Giorgos Borboudakis 11 Paulos Charonyktakis 11 Konstantinos Paraschakis 11
Ioannis Tsamardinos 1122

###### Abstract

AutoML platforms have numerous options for the algorithms to try for each step
of the analysis, i.e., different possible algorithms for imputation,
transformations, feature selection, and modelling. Finding the optimal
combination of algorithms and hyper-parameter values is computationally
expensive, as the number of combinations to explore leads to an exponential
explosion of the space. In this paper, we present the Sequential Hyper-
parameter Space Reduction (SHSR) algorithm that reduces the space for an
AutoML tool with negligible drop in its predictive performance. SHSR is a
meta-level learning algorithm that analyzes past runs of an AutoML tool on
several datasets and learns which hyper-parameter values to filter out from
consideration on a new dataset to analyze. SHSR is evaluated on 284
classification and 375 regression problems, showing an approximate $30\%$
reduction in execution time with a performance drop of less than $0.1\%$.

###### Keywords:

AutoML Algorithm Recommendation Algorithm Selection Hyper-parameter
Optimization

##  1 Introduction

AutoML platforms for predictive modelling try to solve the Combined Algorithm
Selection and Hyper-parameter Optimization (CASH) [24] problem. CASH optimizes
the algorithmic choices, as well as the hyper-parameter values of the machine
learning pipeline (hereafter called a configuration) that produces the final
model. Each configuration may contain several steps, such as algorithms for
missing value imputation, feature transformation, feature extraction, feature
selection, and of course, predictive modelling. All such choices can be
represented with hyper-parameters which form the decision variables of an
optimization problem with objective function the out-of-sample predictive
performance of the final model, called Hyper-Parameter Optimization (HPO). The
hyper-parameters form the configuration space over which to optimize.

There have been at least two complementary approaches to address this
optimization. The first approach is to employ black-box optimization
strategies to search the configuration space, such as grid search, random
search, and Sequential Bayesian Optimization [22]. These algorithms evaluate
the configurations on the dataset to analyze by training models. The second
approach is try to reduce or prioritize the exploration of the configuration
space based on prior information. Such algorithms, called meta-level learning
algorithms, analyze the past performance of configurations on other datasets
and try to predict their performance on a new dataset based on its
characteristics, called meta-features [21]. Obviously, the two approaches can
be combined with the meta-level algorithm selecting a subset of configurations
to explore, and the HPO algorithm performing an intelligent search within this
reduced space.

When optimizing over a continuous hyper-parameter, e.g., the cost $C$ of a
Support Vector Machine (SVM), several reasonable assumptions to facilitate to
explore the configuration space have been proposed, e.g., that neighboring $C$
values will lead to correlated model performances. In the Bayesian Sequential
Optimization these assumptions are captured by the kernel function used in the
Gaussian Process that fits the performance landscape. However, for discrete
choices and hyper-parameters, particularly the choice of which algorithm to
use at each analysis step, it is less clear if and which assumptions are
reasonable. Will an SVM perform well on a specific dataset, given the
performance of the Decision Tree? Meta-level learning algorithms can be proven
invaluable when optimizing categorical hyper-parameters.

In this paper, we propose such an algorithm called Sequential Hyper-Parameter
Space Reduction (SHSR). SHSR is a meta-level learning algorithm that analyzes
the past performances and execution times of configurations on a corpus of
datasets, stored in matrices $\mathbf{P}$ and $\mathbf{E}$, respectively. It
learns which configurations can be safely removed from consideration given the
meta-features of the dataset, without affecting the predictive performance of
the final model (within some user-defined tolerance threshold), while
simultaneously trying to minimize execution time. It then recursively applies
this step to the remaining configurations. By removing such values, SHSR
exponentially reduces the discrete part of the configuration space. To apply
SHSR, it is required that a set of configurations is run on a corpus of
datasets, and their performances and execution times are measured and stored
in matrices $\mathbf{P}$ and $\mathbf{E}$, respectively.

SHSR is evaluated on 284 classification and 375 regression problems on real
public datasets. The trade-offs between the tolerance threshold vs. the
computational savings are explored. In addition, it is shown that the
algorithm performs well when provided with incomplete matrices $\mathbf{P}$
and $\mathbf{E}$, i.e., when configurations are run only on a small fraction
of the datasets, practically providing the same results even if only 20% of
the values in $\mathbf{P}$ and $\mathbf{E}$ are present. For a very strict
tolerance threshold, SHSR achieves a relative drop in predictive performance
of less than 0.1%, with 30% computational savings. We note that time savings
are measured with respect to a simple grid search, where all choices are
discrete. For less strict thresholds, SHSR achieves computational savings of
50% and 40% with a relative performance drop of 1.5% and 0.1%) for
classification and regression, respectively.

This paper is structured as follows: Section 2 overviews the literature of
meta-level learning and dataset characterization. Section 3 presents our
proposed algorithm, SHSR, while Section 4 describes SHSR’s experimental
evaluation. Specifically, in Section 4.1, we describe the experimental setup,
while section 4.2 presents the evaluation results. Finally, Section 5
summarizes our conclusions and future work, while Section 6 discusses the
limitations of this work.

##  2 Related Work

Meta-level learning, often called meta-learning, is a heavily overloaded term,
and has been used to refer to different problems in the literature, such as
algorithm recommendation, ensembling and transfer learning [14]. We will
hereafter focus on the Algorithm Recommendation Problem [7], which deals with
learning to recommend configurations to try for a new problem, based on past
runs. This is typically done by first characterizing datasets using measures
(also called meta-features) that describe their properties, and then learning
to predict the predictive performance and/or running time of configurations.
Next, we will provide a brief overview of different meta-features and meta-
level learning methods. Interested readers may refer to [28] for an overview
of meta-level learning, and to [10] on how it is used in the context of
AutoML.

###  2.1 Dataset Characterization

[20, 19] introduced the idea of dataset characterization for meta-level
learning, and studied the effect of simple measures on the performance of
learning algorithms. [16] describe several dataset characterization measures,
and group them into simple (e.g., number of samples), statistical (e.g.,
average absolute correlation between feature pairs) and information-theoretic
measures (e.g., average entropy of features). Additional categories and types
of measures have been introduced over time: model-based measures, which are
extracted from models induced from the meta-features (e.g., the tree depth of
a decision tree trained on the meta-features), landmarking measures, which use
the performance of simple and fast learning algorithms (e.g., accuracy of
Naive Bayes), and other measures (e.g., silhouette index of k-means
clustering). We refer the reader to [21] for a comprehensive review of meta-
features.

###  2.2 Meta-level Learning

To the best of our knowledge, the algorithm recommendation problem was first
addressed in [20]. The authors used simple meta-features, and studied whether
learning to recommend algorithms based on them leads to improvements. [1]
extended that work, by (a) considering additional simple meta-features and (b)
introducing the idea of learning interpretable rules for algorithm
recommendation, and evaluated the method on synthetic data. [3] were the first
to apply meta-level learning on real-world data. They used simple, statistical
and information-theoretic meta-features, and applied decision trees to learn
whether an algorithm should be applied on a given dataset or not. The work was
subsequently extended to use regression and instance-based models to directly
predict the predictive performance of algorithms on new datasets [8]. [4] use
a KNN based algorithm on dataset meta-features to find similarities and then
they apply a performance/computational time multi-criterion to rank algorithms
(more on this in our Comparison Section below). More recently, collaborative
filtering methods [17, 23, 30, 31] were proposed to solve the algorithm
recommendation problem, inspired by the Netflix challenge. These methods allow
for incomplete inputs (i.e., not all configurations are run on all datasets).
However, they suffer from the cold-start problem, but there exist approaches
that try to avoid that by using meta-features [17]. For a detailed review of
meta-level learning methods, we point the reader to [15, 11].

##  3 The Sequential Hyper-parameter Space Reduction Algorithm for Meta-level
Learning

In this section we introduce the Sequential Hyper-parameter Space Reduction
algorithm (SHSR) for algorithm recommendation. SHSR uses the predictive
performances and execution times of groups of configurations on past analyses,
and returns models which are used to filter out unpromising groups, while
simultaneously trying to minimize execution time. Configuration groups can
contain one or multiple configurations (e.g., all configurations with the same
modelling algorithm), are not necessarily mutually exclusive (i.e., the same
configuration might be present in multiple groups), and the input can be
partial (i.e., results for some configurations might not be present for all
past analyses). SHSR is shown in Algorithm 1. We proceed with a detailed
description of SHSR.

Let $G$, $D$, and $F$ denote the number of configuration groups, datasets, and
meta-features, respectively. SHSR takes as input: (a) $G\times D$ matrices
$\mathbf{P}$ and $\mathbf{E}$ containing the performance ratios and execution
times of all configuration groups, (b) an $F\times D$ matrix $\mathbf{X}$ of
meta-features, (c) a threshold $T$, and (d) a list of $G$ sets Active, with
$\textsc{Active}[g]$ initialized to all datasets for which results for $g$ are
present. For a given group $g$ and dataset $d$, the performance ratio
$\mathbf{P}_{g,d}$ is defined as the maximum performance over configurations
in group $g$ on dataset $d$, divided by the maximum performance over all
configurations. The execution time $\mathbf{P}_{g,d}$ is defined as the sum of
execution times of all configurations in $g$. The output is a sequence of
models $(\textsc{Model}[g_{1}],\dots,\textsc{Model}[g_{k}])$, where
$\textsc{Model}[g]$ is a model predicting the performance ratio achieved
without group $g$ on a dataset $d$ based on its meta-features
$\mathbf{X}_{*,d}$.

SHSR starts by creating one model per configuration group $g$, and computing
the time saved if that group were to be removed from consideration. For this,
an outcome $\mathbf{y}$ is created, which contains the maximum performance for
each dataset in $\textsc{Active}[g]$ over all groups except $g$, and a model
$\textsc{Model}[g]$ for $\mathbf{y}$ is fitted using meta-features
$\mathbf{X}$. Next, $\textsc{Covered}[g]$ is computed, by applying
$\textsc{Model}[g]$ on all $\textsc{Active}[g]$, and selecting only the ones
for which the model predicts a performance ratio at least as large as $T$. In
other words, $\textsc{Covered}[g]$ contains all datasets for which group $g$
can be excluded, as the remaining groups are sufficient to achieve a high
performance ratio. Finally, the time saved $\textsc{TimeSavings}[g]$ is
computed as the sum of execution times of all $\textsc{Covered}[g]$. Once this
is done, the group $g^{*}$ with the highest time savings is selected, and SHSR
is called recursively with $\textsc{Covered}[g^{*}]$ removed from
$\textsc{Active}[g]$ (in the algorithm we slightly abuse notation for the sake
of brevity). SHSR stops once no more time savings can be achieved, which
happens if no more datasets can be covered by the removal of any group.

Finally, to decide which groups to run on a new dataset $d^{\prime}$, the
models $(\textsc{Model}[g_{1}],\dots,\textsc{Model}[g_{k}])$ are applied in
sequence, and a group $g_{i}$ is removed from consideration if
$\textsc{Predict}(\textsc{Model}[g_{i}],\mathbf{X_{*,d^{\prime}}})\geq T$.

Algorithm 1 Sequential Hyper-parameter Space Reduction

Constants: Performance ratios $\mathbf{P}$, Execution times $\mathbf{E}$,
Meta-features $\mathbf{X}$, Threshold $T$
Input: Active datasets per configuration group Active
Output: Sequence of regression models

1:for each configuration group $g$ do

2: $\mathbf{y}_{d}\leftarrow\max_{i}\mathbf{P}_{i,d},i\neq g,\forall
d\in\textsc{Active}[g]$

3: $\textsc{Model}[g]\leftarrow\textsc{FitModel}(\mathbf{y},\mathbf{X})$

4: $\textsc{Covered}[g]\leftarrow\\{d\mid
d\in\textsc{Active}[g]\wedge\textsc{Predict}(\textsc{Model}[g],\mathbf{X_{*,d}})\geq
T\\}$

5:
$\textsc{TimeSavings}[g]\leftarrow\sum_{d}\mathbf{E}_{g,d},d\in\textsc{Covered}[g]$

6:end for

7:$g^{*}\leftarrow\operatorname*{arg\,max}_{g}\textsc{TimeSavings}[g]$

8:if $\textsc{TimeSavings}[g^{*}]=0$ then

9: return $\emptyset$

10:end if

11:return
$(\textsc{Model}[g^{*}])\cup\textsc{SHSR}(\textsc{Active}\setminus\textsc{Covered}[g^{*}])$

##  4 Experimental Evaluation

In this section we evaluate the ability of SHSR to filter out groups of
configurations, and measure its impact on running time and predictive
performance. To this end, we collected classification and regression datasets
and trained a set of predefined configurations on them, in order to obtain
performance estimates and execution times for SHSR. The full analysis required
training $\sim$85M models, taking a total of $\sim$58K core hours. We proceed
with a detailed description of the experimental setup. Results are presented
in Section 4.2.

###  4.1 Experimental Setup

####  4.1.1 Datasets

Figure 1: Sample size vs feature size for regression and classification
datasets. The x-axis shows the sample size, while the y-axis shows the feature
size. For the classification datasets, the color intensity varies depending on
the class distribution. Both axes are in $\log_{10}$ scale.

We collected a total of 659 datasets, out of which 284 are binary
classification problems and 375 are regression problems. Datasets were
selected to cover a wide range of problems, with varying sample sizes, number
of variables and, in case of classification problems, class distributions; see
Fig. 1 for a summary. All datasets were downloaded from OpenML [29] (licensed
under CC-BY 4.0) and from BioDataome [13] (publicly available). A list of all
datasets and their sources can be found on the project’s GitHub
repository111https://github.com/JADBio/SHSR.

####  4.1.2 Implementation

For model training we used JADBio [26]. JADBio is an automated machine
learning platform specifically designed for life scientists and molecular
data. It uses a fully automated machine learning protocol to select the best
combination of preprocessing, feature selection and modelling algorithms, and
returns a predictive model and an estimate of its predictive performance. Any
features that could interfere with results (e.g., early dropping of
unpromising configurations [27]) were disabled. Also, in order to get accurate
timing results, only a single CPU core was used per configuration, and
analyses were not run in parallel.

Everything else was implemented in Python and is available on GitHub. For
machine model training, we used the scikit-learn package [18]. This includes
code for the implementation of SHSR, as well as code for producing all
results.

####  4.1.3 Algorithms and Configurations

For preprocessing, JADBio uses standardization and mean imputation for
continuous variables and mode imputation for the categorical ones. As feature
selection algorithms, the Statistical Equivalent Signatures (SES) algorithm
[12] and Lasso [25] were employed. Finally, for modelling JADBio uses
$L_{2}$-regularized linear and logistic regression [9], Support Vector
Machines (SVM) [2] and Support Vector Regression (SVR) with Linear,
Polynomial, and Gaussian kernels, Decision Trees [6], and Random Forests [5].

We used the default grid search parameters of JADBio with the tuning parameter
set to “extensive”: 6 hyper-parameter combinations for SES, 7 for Lasso, 7 for
linear and logistic regression, 150 for SVMs, 1500 for SVRs, 15 for Decision
Trees and 60 for Random Forests222The complete list of hyper-parameters is
available on GitHub.. Configurations were obtained by taking all combinations
of algorithms and hyper-parameter values, and constraining them to contain one
or multiple transformations (imputation is always included, if applicable, and
takes precedence over standardization), one feature selection algorithm, and
one modelling algorithm. The total number of configurations were 2983 and 8633
for classification and regression tasks, respectively.

Based on the above configurations, we created a total of 21 configuration
groups: (a) one group per feature selection algorithm (13 in total, one per
hyper-parameter combination), and (b) one group per modelling algorithm (8 in
total; SVMs with different kernels and polynomial kernels with different
degrees are considered separate algorithms). Each such group contains only the
subset of configurations for which the respective algorithm was used (e.g.,
the random forest group contains all results with random forest, irrespective
of the feature selection algorithm used).

For performance estimation, we ran JADBio with 10-fold cross-validation,
repeated at most 20 times (a stopping criterion is applied when a plateau is
reached during the repetitions) [27]. We note that JADBio returns unbiased
(and typically conservative) performance estimates, by applying the BBC-CV
algorithm [27] on the out-of-sample predictions from cross-validation, without
the need of using a test set or expensive techniques like nested cross-
validation.

At this point we note that the execution time of feature selection algorithms
was only counted once per fold and not per configuration, as in practice
feature selection has to be performed only once for a given dataset and hyper-
parameter combination, regardless of the number of subsequently trained
models.

####  4.1.4 Evaluation of SHSR

Table 1: Meta-features used in the experiments. Meta-feature | Description
---|---
n_samples | Number of samples
n_features | Number of features
samples_to_features | Number of samples to number of features ratio
total_missing | Number of missing values
total_missing_f | Proportion of missing values
samples_with_any_missing | Number of samples with at least one missing value
samples_with_any_missing_f | Proportion of samples with at least one missing value
categorical_features | Number of categorical features
numerical_features | Number of numerical features
target_majority_class_instances | Number of samples for majority class
target_majority_class_f | Proportion of samples for majority class
target_minority_class_instances | Number of samples for minority class
target_minority_class_f | Proportion of samples for minority class
categorical_to_numerical | Ratio of categorical to numerical features
silhouette_k | Silhouette index of k-means clustering
pca_p |  Number of components that explain $p\%$ of variance

Table 1 lists the meta-features used for the analysis [21]. We used 14 simple
measures, along with the silhouette index of k-means clustering for
$k\in\\{2,3,\dots,10\\}$ and the number of PCA components required to explain
$p\%$ of the variance of a dataset for $p\in\\{60,70,80,90\\}$, resulting in a
total of 27 meta-features. We note that, as the types of meta-features used
are not part of SHSR, we mainly chose simple measures because they are easy
and fast to compute.

To evaluate SHSR, we randomly keep out 10% of the datasets for testing, and
executed SHSR on the remaining 90%. For each held-out dataset, we applied the
model returned by SHSR to recommend a set of configurations to compute the
predictive performances and running times. The procedure was repeated 20 times
and averages are reported.

Finally, as a regression model in SHSR (see Line 3) we used regression trees,
and tuned them using 5-fold cross-validation, by optimizing over
min_samples_leaf $\in\\{3,5,7\\}$, as well as over the solution path returned
by the minimal cost-complexity pruning algorithm [18]. The reason we chose
decision trees is because they are interpretable, non-linear and have a low
computational cost.

###  4.2 Results

We ran experiments to investigate the effect of the threshold $T$ on
predictive performance and time savings when using SHSR to select
configurations. Next, we evaluate how SHSR performs when only a random subset
of the results is available. Finally, we compare SHSR to a trivial algorithm
which randomly removes a proportion of configurations. Results and additional
details about the experiments are presented below.

####  4.2.1 Effect of Threshold $T$ on Predictive Performance and Execution
Time

Figure 2: Effect of threshold $T$ on predictive performance and execution
time. The x-axis shows the threshold, and the y-axis shows the ratio between
the predictive performance (execution time) using the configurations returned
by SHSR, relative to using all configurations. Error bars show the 95%
Gaussian confidence intervals for the mean, resulting from 20 runs of the
experiment. We observe that SHSR leads to a significant reduction in execution
time, with minimal drop in predictive performance.

We varied the threshold hyper-parameter $T$ of SHSR to investigate the trade-
off between predictive performance and running time. Specifically, we
considered thresholds in $\\{0.95,0.97,0.99,0.999,0.9999\\}$. Lower values
were not tried, as they would lead to a significant drop in performance, with
minimal time savings.

The results are summarized in Fig. 2. As one would expect, with increasing
$T$, performance increases while execution time decreases. Furthermore, we
observe that for very low values of $T$ there is a negligible loss in
performance, while still providing significant time savings. For instance, a
threshold of $0.999$ leads to an average performance loss of $\sim 1.5\%$ for
regression and $\sim 0.1$ for classification problems, while saving $\sim
50\%$ and $\sim 40\%$ time respectively. On the other hand, if one was mainly
interested in execution time, they could use a lower threshold, sacrificing
predictive performance for a reduction in execution time. For example, a
threshold of $0.95$ reduces time by $\sim 95\%$ and $\sim 99\%$ for regression
and classification tasks respectively, while retaining $\sim 91\%$ and $\sim
95\%$ of the predictive performance.

####  4.2.2 Evaluation on Partial Results

Figure 3: Effect of running SHSR with partial results on predictive
performance and execution time. The x-axis shows the proportion of used
results, and the y-axis shows the ratio between the predictive performance
(execution time) using the configurations returned by SHSR, relative to using
all configurations. Error bars show the 95% Gaussian confidence intervals for
the mean, resulting from 20 runs of the experiment. We observe that SHSR is
able to perform well even with partial results, and that it performs better
the more results are available, as expected.

In this experiment we evaluate SHSR when run with partial results (i.e., when
only a subset of configurations have been applied on each dataset). We
simulated that scenario by randomly sub-sampling $\\{20\%,40\%,60\%,80\%\\}$
of all results. Note that sub-sampling was performed on all results, not on
each dataset or configuration. The threshold $T$ was set to $0.999$ for this
experiment.

Fig. 3 shows the results. We can see that SHSR performs well, even if only 20%
of all results are available. Furthermore, we observe that execution time is
decreasing the more results are available to SHSR, especially for regression
tasks, where it drops from $\sim 75\%$ of the original running time when 20%
of the results are used, to $\sim 50\%$ when all results are used. This
confirms that SHSR is able to make better decisions the more data are
available, as one would expect.

####  4.2.3 Comparison against [4] and random elimination of configurations

As a final experiment, we compared SHSR to the KNN based meta-level algorithm
in [4], as well as a simple baseline which randomly eliminates a percentage of
configurations. The authors in [4] apply KNN on the meta-features of the
datasets in order to locate the $N$ most similar from a pool of already
evaluated datasets for a new incoming dataset. Subsequently they apply a
pairwise multi-criterion (performance/time) metric to compare all available
configurations on the training datasets and a configuration specific relative
score is being calculated. The latter can be used to rank configurations and
select the top ranking to run on the new dataset. Regarding the random
elimination, we considered randomly removing
$\\{50\%,60\%,70\%,80\%,90\%,95\%,97\%,99\%\\}$ of configurations. SHSR was
executed with $T$ taking the values $0.95,0.97,0.99,0.999,0.9999$. The KNN
based algorithm was applied with various hyper-parameters (see original paper
for a description of them):
$Nneighbors\in\\{1,3,10\\},AccD\in\\{0.001,0.01,0.1\\}$. The method was used
to rank configurations according to their performance and execution time on
test datasets and then the top $m$ ranked configurations were chosen to be
applied on the test datasets, where
$m\in\\{100,300,500,1000,1500,2000,2500\\}$ for classification and
$m\in\\{100,500,1000,2000,4000,6000\\}$ for regression. Each combination
setting was repeated 20 times.

The results are shown in Fig. 4. For regression tasks, random elimination of
less than $90\%$ of configuration leads to similar results as SHSR, both in
terms of predictive performance and running time. The same holds for
classification tasks, when fewer than $80\%$ of configurations are removed.
However, if more configurations are removed, performance of random elimination
drops abruptly, with minimal time savings. We note that the reason running
time does not drop linearly with configurations, which would be expected to
hold on average using random elimination, is that it is unlikely that a
feature selection algorithm instance is completely removed by chance, as this
can only happen if all configurations it participates in are removed. In
contrast, SHSR does not suffer from this, as it either keeps an algorithm
group or removes it completely.

The KNN based algorithm seems to be performing in-between the other two for
classification and slightly worse for regression for less than $90\%$ time
reduction. It does exceed random elimination’s performance even in regression
for heavily reduced configurations sets, though.

We further investigated why random elimination performs that well. We found
that often multiple configurations achieve maximum performance or close to it
(i.e., there is a lot of redundancy), making it likely to select a good
configuration by chance. This is explained by the fact that the hyper-
parameter configurations used by JADBio are curated and optimized to cover a
wide range of problems. Thus, we would expect random elimination’s performance
to drop if the proportion of “lower quality” configurations increases, while
SHSR wouldn’t be affected significantly, as it uses the maximum performance of
a group to determine its performance. To better understand that, consider an
example with 10000 configurations, 50 out of which are optimal or very close
to it, and assume that 9000 of the configurations are randomly removed. Then,
the probability of selecting any optimal configuration by chance is
$1-0.995^{1000}=99.33\%$. If we were to reduce the number of optimal
configurations to 25 (i.e., increase the proportion of bad configurations),
the probability of selecting any of them drops to $1-0.9975^{1000}=91.82\%$.
Thus, we argue that in practice SHSR is always preferable over random
elimination, especially when used for hyper-parameter configurations that have
not been fine-tuned. SHSR is also clearly superior to the KNN based algorithm
in all investigated scenarios.

Figure 4: Comparison between SHSR, KNN based algorithm, and random
elimination. The x-axis (y-axis) shows the ratio of execution time
(performance) using the configurations returned by the algorithms, relative to
using all configurations. Dotted lines show the 95% Gaussian confidence
intervals for the mean, resulting from 20 runs of the experiment. In the case
of the KNN based algorithm splines were used to smooth both the mean and the
CI lines. All three algorithms perform similarly when few configurations are
dropped, with SHSR being superior when many configurations are removed.

##  5 Discussion, Conclusions, and Future Work

SHSR introduces a recursive evaluation of choices and their filtering in meta-
level learning. It is evaluated on a large cohort of datasets, spanning a wide
range of dataset characteristics. It exhibits significant computational
savings for a simple grid search, with minimal relative drop in predictive
performance. The performance drop can be controlled by a tolerance threshold.
The algorithm can be coupled as a filtering step with any HPO strategy when
some hyper-parameter domains are discrete. Learning which values to filter in
SHSR is currently evaluated based on a simple Decision Tree algorithm. Any
modelling algorithm could be employed instead, of course. However, trees have
the advantage of being interpretable. Hence, the decision of the system to
drop some algorithmic choices from consideration based on the characteristics
of the dataset, can be explained to the user of the AutoML platform. In
addition, it can be used to provide intuition into the deficiencies of
algorithms in particular situations that may inspire new algorithms. There are
numerous future directions to explore, such as coupling SHSR with other HPO
algorithms than grid search and employing more powerful models than DTs for
filtering.

##  6 Limitations

This study contains some limitations inevitably. First of all, most of the
datasets for the classification experiments used in this study are molecular
(multi-omics) datasets which are high dimensional and typically contain
relatively few samples. Secondly, while the algorithm is general and can be
applied on any analysis task this study evaluates it only in binary
classification and regression tasks. Finally, we evaluated the SHSR employing
only Decision Trees for their rule generation feature without testing any
other modelling algorithm.

####  6.0.1 Acknowledgements

The research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Programme
(FP/2007-2013) / ERC Grant Agreement n. 617393, the Hellenic Foundation for
Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research
Projects to support Faculty members and Researchers and the procurement of
high-cost research equipment grant” (Project Number: 1941), and the METALASSO
project, which is co-financed by the European Union and Greek national funds
through the Operational Program Competitiveness, Entrepreneurship and
Innovation, under the call RESEARCH–CREATE– INNOVATE (project code:
T1EDK-04347).

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# Exponential stability of Euler-Bernoulli beam under boundary controls in
rotation and angular velocity

Alemdar Hasanov111Department of Mathematics, Kocaeli University, Turkey
<EMAIL_ADDRESS>Department of Mathematics, Kocaeli University,
Turkey

###### Abstract

This paper addresses the analysis of a boundary feedback system involving a
non-homogeneous Euler-Bernoulli beam governed by the equation
$m(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the
initial $u(x,0)=u_{0}(x)$, $u_{t}(x,0)=v_{0}(x)$ and boundary conditions
$u(0,t)=0$,
$\left(-r(x)u_{xx}(x,t)\right)_{x=0}=-k^{-}_{r}u_{x}(0,t)-k^{-}_{a}u_{xt}(0,t)$,
$u(\ell,t)=0$,
$\left(-r(x)u_{xx}(x,t)\right)_{x=\ell}=-k^{+}_{r}u_{x}(\ell,t)-k^{+}_{a}u_{xt}(\ell,t)$,
with boundary control at both ends resulting from the rotation and angular
velocity. The approach proposed in this study relies on the utilization of
regular weak solutions, energy identity, and a physically motivated Lyapunov
function. By imposing natural assumptions concerning physical parameters and
other inputs, which ensure the existence of a regular weak solution, we
successfully derive a uniform exponential decay estimate for the system’s
energy. The decay rate constant featured in this estimate is solely dependent
on the physical and geometric properties of the beam. These properties
encompass crucial parameters such as the viscous external damping coefficient
$\mu(x)$, as well as the boundary springs $k^{-}_{r},k^{+}_{r}$ and dampers
$k^{-}_{a},k^{+}_{a}$. To illustrate the practical effectiveness of our
theoretical findings, numerical examples are provided. These examples serve to
demonstrate the applicability and relevance of our derived results in real-
world scenarios.

###### keywords:

Exponential stability, Euler-Bernoulli beam, boundary control, regular weak
solution, energy identity, Lyapunov function, decay rate.

††journal: arXiv

##  1 Introduction

Submarine pipelines and long bridges can be considered as an elastic beam with
both ends controlled by the boundary rotation and angular velocity [2, 18]. In
many studies related to pipeline modeling, the pipes are defined as beams
resting on a rigid seabed without any penetration (see [14] and references
therein). However, such hypotheses are not always satisfied in practice. An
analysis of the torsional effects on pipe lateral buckling was given in [9],
where essential influence of torsion under some specific boundary conditions
was demonstrated analytically. Similar situation arise in bridge models
governed by the Euler-Bernoulli beam. Namely, it is very important for the
sensitivity analysis of bridges to obtain a relationship between the rotation
spring constant and the bridge responses (deflections/slopes). This
relationship can then be used for evaluating the support condition of bridges
[19]. Furthermore, in modeling of long flexible structures through the Euler-
Bernoulli equation, the bending moment at the end of the beam is controlled by
the linear feedback of rotation angle and angular velocity, and the shear
force at the same end is controlled by the linear feedback of displacement and
velocity. We refer [12] and references therein, for the detailed description
of such models.

Considering the effect of the above factor on both models, there is a need for
a realistic model that will take into account the effects of both the rotation
spring and the angular velocity damper at both ends of the beam, within the
framework of the Euler-Bernoulli beam equation. In the most natural way, this
can be taken into account by the corresponding boundary conditions at both
ends of the beam, including a linear combinations of the rotation spring and
the angular velocity damper. This leads to the following mathematical model:

$\displaystyle\left\\{\begin{array}[]{ll}m(x)u_{tt}+\mu(x)u_{t}+\left(r(x)u_{xx}\right)_{xx}=0,\,(x,t)\in\Omega_{T},\\\\[4.0pt]
u(x,0)=u_{0}(x),\,u_{t}(x,0)=u_{1}(x),\,x\in(0,\ell),\\\\[4.0pt]
u(0,t)=0,\,\left(-r(x)u_{xx}(x,t)\right)_{x=0}=-k^{-}_{r}u_{x}(0,t)-k^{-}_{a}u_{xt}(0,t),\\\\[4.0pt]
\quad
u(\ell,t)=0,\,\left(-r(x)u_{xx}(x,t)\right)_{x=\ell}=k^{+}_{r}u_{x}(\ell,t)+k^{+}_{a}u_{xt}(\ell,t),\\\\[4.0pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~{}t\in[0,T],\end{array}\right.$
(6)

where $\Omega_{T}=(0,\ell)\times(0,T)$, $\ell>0$ is the length of the beam and
$T>0$ is the final time.

Here and below, $u(x,t)$ is the deflection, $u_{t}(x,t)$, $u_{x}(x,t)$,
$u_{xt}(x,t)$, $u_{xx}(x,t)$, $-\left(r(x)u_{xx}\right)$ and
$-\left(r(x)u_{xx}\right)_{x}$ are the velocity, rotation (or slope), angular
velocity, curvature, moment and shear force, respectively [6, 17]. Further,
$m(x)=\rho(x)S(x)>0$, while $\rho(x)$ is the mass density and $S(x)$ is the
cross section area of the beam, and $r(x):=E(x)I(x)>0$ represent the flexural
rigidity (or bending stiffness) of the beam, respectively, while $E(x)>0$ is
the elasticity modulus and $I(x)>0$ is the moment of inertia. The non-negative
coefficient $\mu(x):=\gamma\,m(x)$ of viscous resistance to transverse motion
of the beam represents the viscous external damping, while $\gamma\geq 0$ is
the damping constant of proportionality [1]. Furthermore, nonnegative
constants $k^{-}_{r},k^{-}_{a}\geq 0$ and $k^{+}_{r},k^{+}_{a}\geq 0$ are the
stiffness of the torsional springs and dampers on the left and right ends of
the beam, respectively.

$x$$u$$u(0,t)=0$$\left(-r(x)u_{xx}(x,t)\right)_{x=0}$$=-k^{-}_{r}u_{x}(0,t)-k^{-}_{a}u_{xt}(0,t)$$u(\ell,t)=0$$\left(-r(x)u_{xx}(x,t)\right)_{x=\ell}$$=k^{+}_{r}u_{x}(\ell,t)+k^{+}_{a}u_{xt}(\ell,t)$$k^{-}_{r}$$k^{-}_{a}$$k^{+}_{r}$$k^{+}_{a}$
Figure 1: Beam connected to torsional springs and dampers at both ends

The boundary conditions
$\left(-r(x)u_{xx}(x,t)\right)_{x=0}=-k^{-}_{r}u_{x}(0,t)-k^{-}_{a}u_{xt}(0,t)$
and
$\left(-r(x)u_{xx}(x,t)\right)_{x=\ell}=k^{+}_{r}u_{x}(\ell,t)+k^{+}_{a}u_{xt}(\ell,t)$
at the left and right ends of the beam, respectively, mean the controls
resulting from the linear combination of rotation and angular velocity. In
this context, the above parameters
$k^{-}_{r},\,k^{-}_{a},\,k^{+}_{r},\,k^{+}_{a}$ are defined also as the
boundary controls.

Geometry of the problem (1) is given in Fig. 1.

This work is devoted to the systematic study of the following issues. _Under
what minimum conditions imposed on the input data is the energy of the system
governed by ( 6) exponentially stable?_ _If the system governed by ( 6) is
stable, how much does each damping parameter $\gamma$, $k^{-}_{a}$ and
$k^{+}_{a}$ contribute to this stability?_ It should be especially noted that
the nature of both the external and the boundary damping mechanisms greatly
changes the nature of the vibration, and hence controls the response of the
beam, as the experimental and theoretical results discussed in [1, 7] show.

Modeling of large flexible structures through a class of Euler-Bernoulli beams
with structural damping, has begun to be developed, starting with studies [3,
4, 21]. The exponential stability of distributed systems governed by Euler-
Bernoulli beam equation under classical boundary conditions has been discussed
starting from the work [4], and then more general results are obtained in [3,
15, 16]. Various methods have been developed in the literature for initial
boundary value problems for Euler-Bernoulli equations with a boundary feedback
systems. Among these methods, the spectral method turned out to be efficient
and useful since it allows to establish the Riesz basis property, which is the
most fundamental property of a linear vibrating system [5, 10, 11, 12]. In
turn, this property means that the generalized eigenvectors of the system form
an unconditional basis of the (state) Hilbert space. With semigroup approach,
this allows to derive the spectrum determined growth condition and the
exponential stability for a system.

In the exponential stability estimate $\mathcal{E}(t)\leq Me^{-\omega
t}\mathcal{E}(0)$ obtained in the studies listed above, the relationship of
the decay rate parameter $\omega>0$ with the physical and geometric parameters
of the beam, including the damping coefficient $\mu(x)\geq 0$ and the
stiffness $k^{-}_{a},k^{+}_{a}\geq 0$ of the torsional dampers, has not been
determined. Since the relationship of this decay rate parameter with the
damping parameters is not known, in concrete applications, such an evaluation
does not give a qualified result. In this paper, we develop the approach based
on the weak solution theory for the initial boundary value problem (6), energy
estimates and the Lyapunov method to establish an exponential stability
estimate for system (6) under minimum conditions imposed on the input data.
Furthermore, this approach allows us to derive the role of both types of
parameters in the exponential decay of the solution. To our knowledge, this
model, defined by the initial boundary value problem (6), in which the viscous
external and boundary (torsional) damping factors are considered together and
in the presence of torsional springs, is discussed for the first time in the
literature.

The rest of the paper is structured as follows. Energy identity and
dissipativity of system (6) are derived in Section 2. In Section 3, the
Lyapunov function is introduced and then energy decay estimate for system (6)
is derived. Numerical examples are presented in Section 4. Some concluding
remarks are given in the final Section 5.

##  2 Necessary estimates for the weak solution of problem (6)

We assume that the inputs in (6) satisfy the following basic conditions:

$\displaystyle\left\\{\begin{array}[]{ll}\rho_{S},\mu,r\in
L^{\infty}(0,\ell),\\\\[3.0pt] 0<m_{0}\leq m(x)\leq
m_{1},~{}0\leq\mu_{0}\leq\mu(x)\leq\mu_{1},\\\\[3.0pt] 0<r_{0}\leq r(x)\leq
r_{1},\,x\in(0,\ell),\\\\[3.0pt] u_{0}\in H^{2}(0,\ell),~{}u_{1}\in
L^{2}(0,\ell),\\\\[3.0pt] k^{-}_{r},k^{-}_{a},k^{+}_{r},k^{+}_{a}\geq
0,\\\\[3.0pt]
\gamma+k^{-}_{r}+k^{-}_{a}+k^{+}_{r}+k^{+}_{a}>0.\end{array}\right.$ (13)

For the case when all the parameters $k^{-}_{r},k^{-}_{a},k^{+}_{r},k^{+}_{a}$
are equal to zero, under conditions (13), the existence of the weak solution
$u\in L^{2}(0,T;\mathcal{V}^{2}(0,\ell))$, with $u_{t}\in
L^{2}(0,T;L^{2}(0,\ell))$ and $u_{tt}\in L^{2}(0,T;H^{-2}(0,\ell))$ of the
initial boundary value problem (6) was proved in [13]. Here and below,

$\displaystyle\mathcal{V}^{2}(0,\ell):=\\{v\in
H^{2}(0,\ell):\,v(0)=v(\ell)=0,\\},$

and $H^{2}(0,\ell)$ is the Sobolev space [8]. For system (6), with
$k^{-}_{r},k^{-}_{a},k^{+}_{r},k^{+}_{a}>0$, the existence of the weak
solution $u\in L^{2}(0,T;\mathcal{V}^{2}(0,\ell))$ can be proved in the
similar way. In this section we derive necessary energy identities and
estimates for the weak solution of problem (6).

###### Theorem 1

Assume that the inputs in (6) satisfy the basic conditions (13). Then the
following energy identity holds:

$\displaystyle\mathcal{E}(t)+\int_{0}^{t}\int_{0}^{\ell}\mu(x)u_{\tau}^{2}(x,\tau)dxd\tau\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad=\mathcal{E}(0)-k^{-}_{a}\int_{0}^{t}u_{x\tau}^{2}(0,\tau)d\tau-k^{+}_{a}\int_{0}^{t}u_{x\tau}^{2}(\ell,\tau)d\tau,\,t\in[0,T],$
(14)

where

$\displaystyle\mathcal{E}(t)=\frac{1}{2}\int_{0}^{\ell}\left[m(x)u^{2}_{t}(x,t)+r(x)u^{2}_{xx}(x,t)\right]dx\qquad\qquad\qquad$
$\displaystyle+\frac{1}{2}\,k^{-}_{r}u_{x}^{2}(0,t)+\frac{1}{2}\,k^{+}_{r}\,u_{x}^{2}(\ell,t),~{}t\in[0,T],$
(15)

is the total energy of system (6) and

$\displaystyle\mathcal{E}(0)=\frac{1}{2}\int_{0}^{\ell}\left[m(x)\left(u_{1}(x)\right)^{2}+r(x)\left(u^{\prime\prime}_{0}(x)\right)^{2}\right]dx\qquad\qquad\qquad$
$\displaystyle\qquad+\frac{1}{2}\,k^{-}_{r}\left(u^{\prime}_{0}(0)\right)^{2}+\frac{1}{2}\,k^{+}_{r}\left(u^{\prime}_{0}(\ell)\right)^{2}$
(16)

is the initial value of the total energy.

Proof. Multiply both sides of equation (6) by $u_{t}(x,t)$, integrate it over
$\Omega_{t}:=(0,\ell)\times(0,t)$, employ the identity

$\displaystyle\int_{0}^{t}\int_{0}^{\ell}(r(x)u_{xx})_{xx}u_{\tau}dxd\tau=\int_{0}^{t}\int_{0}^{\ell}[(r(x)u_{xx})_{x}u_{\tau}-r(x)u_{xx}u_{x\tau}]_{x}dxd\tau$
$\displaystyle+\,\frac{1}{2}\int_{0}^{t}\int_{0}^{\ell}\left(r(x)u_{xx}^{2}\right)_{\tau}dxd\tau,\quad$
(17)

$t\in(0,T]$. Then we obtain the following integral identity:

$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{0}^{\ell}\left(\rho_{S}(x)u_{\tau}^{2}\right)_{\tau}dx\,d\tau+\frac{1}{2}\int_{0}^{t}\int_{0}^{\ell}\left(r(x)u_{xx}^{2}\right)_{\tau}dx\,d\tau\qquad\qquad\qquad$
$\displaystyle\qquad+\int_{0}^{t}\left((r(x)u_{xx})_{x}u_{\tau}-r(x)u_{xx}u_{x\tau}\right)_{x=0}^{x=\ell}d\tau+\int_{0}^{t}\int_{0}^{\ell}\mu(x)u_{\tau}^{2}dxd\tau=0,$

for all $t\in(0,T]$. Using here the initial and boundary conditions (6), we
get:

$\displaystyle\frac{1}{2}\int_{0}^{\ell}\left[m(x)u^{2}_{t}+r(x)u_{xx}\right]dx+\frac{1}{2}\,k^{-}_{r}u_{x}^{2}(0,t)+\frac{1}{2}\,k^{+}_{r}\,u_{x}^{2}(\ell,t)\qquad\qquad\qquad$
$\displaystyle+\int_{0}^{t}\int_{0}^{\ell}\mu(x)u_{\tau}^{2}dxd\tau$
$\displaystyle=\frac{1}{2}\int_{0}^{\ell}\left[m(x)\left(u_{1}(x)\right)^{2}+r(x)\left(u^{\prime\prime}_{0}(x)\right)^{2}\right]dx+\frac{1}{2}\,k^{-}_{r}\left(u^{\prime}_{0}(0)\right)^{2}+\frac{1}{2}\,k^{+}_{r}\left(u^{\prime}_{0}(\ell)\right)^{2}$
$\displaystyle-k^{-}_{a}\int_{0}^{t}u_{x\tau}^{2}(0,\tau)d\tau-k^{+}_{a}\int_{0}^{t}u_{x\tau}^{2}(\ell,\tau)d\tau,\,t\in[0,T],$

for all $t\in(0,T]$. This leads to (1) with (1) and (1). $\Box$

###### Remark 1

The integral identity (1), with (1) and (1), clearly shows that the increase
in the stiffness of the torsional springs $k^{-}_{r}$ and $k^{+}_{r}$ leads to
an increase in the total energy $\mathcal{E}(t)$. Conversely, the increase in
the stiffness of the torsional dampers $k^{-}_{a}$ and $k^{+}_{a}$ leads to a
decrease in the total energy.

###### Remark 2

The sum

$\displaystyle\frac{1}{2}\,k^{-}_{r}u_{x}^{2}(0,t)+\frac{1}{2}\,k^{+}_{r}\,u_{x}^{2}(\ell,t),~{}t\in[0,T]$

in (1) represents the energy of the rigid motion of the elastic system (1),
generated by the spring constants $k^{-}_{r},k^{+}_{r}\geq 0$.

###### Lemma 1

Assume that the basic conditions (13) hold. Then for the decay rate of the
total energy the following integral formula is valid:

$\displaystyle\frac{d\mathcal{E}(t)}{dt}=-\int_{0}^{\ell}\mu(x)u^{2}_{t}dx-k^{-}_{a}u_{xt}^{2}(0,t)-k^{+}_{a}u_{xt}^{2}(\ell,t),\,t\in(0,T).$
(18)

Proof. From formula (1) for the total energy we deduce that

$\displaystyle\frac{d\mathcal{E}(t)}{dt}=\int_{0}^{\ell}\left[m(x)u_{t}u_{tt}+r(x)u_{xx}u_{xxt}\right]dx\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad+k^{-}_{r}u_{x}(0,t)u_{xt}(0,t)+k^{+}_{r}u_{x}(\ell,t)u_{xt}(\ell,t),~{}t\in[0,T].$

Using here the identities

$\displaystyle\int_{0}^{\ell}m(x)u_{t}u_{tt}dx=-\int_{0}^{\ell}\mu(x)u_{t}^{2}dx-\int_{0}^{\ell}\left(r(x)u_{xx}\right)_{xx}u_{t}dx,\qquad\qquad$
$\displaystyle\int_{0}^{\ell}\left(r(x)u_{xx}\right)_{xx}u_{t}dx=\int_{0}^{\ell}r(x)u_{xx}u_{xxt}dx+k^{-}_{r}u_{x}(0,t)u_{xt}(0,t)\qquad$
$\displaystyle+k^{-}_{a}u^{2}_{xt}(0,t)+k^{+}_{r}u_{x}(\ell,t)u_{xt}(\ell,t)+k^{+}_{a}u^{2}_{xt}(\ell,t),~{}t\in[0,T],$

we arrive at the required result (18). $\Box$

###### Corollary 1

Integrating (18) over $(0,t)$ we arrive at the energy identity introduced in
(1), that is

$\displaystyle\mathcal{E}(t)=\mathcal{E}(0)-\int_{0}^{t}\int_{0}^{\ell}\mu(x)u^{2}_{\tau}(x,\tau)dxd\tau\qquad\qquad\qquad\qquad$
$\displaystyle-\int_{0}^{t}\left[k^{-}_{a}u_{x\tau}^{2}(0,\tau)+k^{+}_{a}u_{x\tau}^{2}(\ell,t)\right]d\tau,~{}t\in[0,T].$
(19)

In particular,

$\displaystyle\mathcal{E}(t)\leq\mathcal{E}(0),~{}t\in[0,T],$

that is, the energy of the system (6) is dissipating.

##  3 Lyapunov function and exponential stability estimate

Introduce the auxiliary function:

$\displaystyle\mathcal{J}(t)=\int_{0}^{\ell}m(x)u\,u_{t}dx+\frac{1}{2}\int_{0}^{\ell}\mu(x)u^{2}dx+\frac{1}{2}\,k^{-}_{a}u_{x}^{2}(0,t)+\frac{1}{2}\,k^{+}_{a}u_{x}^{2}(\ell,t),$
(20)

$t\in[0,T]$, that includes all the damping parameters.

###### Lemma 2

Assume that the basic conditions (13) are satisfied. Then between the
auxiliary function $\mathcal{J}(t)$ and the energy function $\mathcal{E}(t)$,
the following relationship holds:

$\displaystyle\frac{d\mathcal{J}(t)}{dt}=2\int_{0}^{\ell}m(x)u_{t}^{2}dx-2\mathcal{E}(t),~{}t\in[0,T].$
(21)

Proof. Taking the derivative of the function $\mathcal{J}(t)$ with respect to
the time variable and using then the equation (6) we find:

$\displaystyle\frac{d\mathcal{J}(t)}{dt}=\int_{0}^{\ell}m(x)u^{2}_{t}dx-\int_{0}^{\ell}\left(r(x)u_{xx}\right)_{xx}udx\qquad\qquad\qquad\qquad$
$\displaystyle\qquad+k^{-}_{a}u_{x}(0,t)u_{xt}(0,t)+k^{+}_{a}u_{x}(\ell,t)u_{xt}(\ell,t),~{}t\in[0,T].$

To transform the second right-hand side integral here, we employ the identity

$\displaystyle-\int_{0}^{\ell}\left(r(x)u_{xx}\right)_{xx}udx=-\int_{0}^{\ell}r(x)u^{2}_{xx}dx-k^{-}_{r}u^{2}_{x}(0,t)-k^{-}_{a}u_{x}(0,t)u_{xt}(0,t)$
$\displaystyle-k^{+}_{r}u^{2}_{x}(\ell,t)-k^{-}_{a}u_{x}(\ell,t)u_{xt}(\ell,t),~{}t\in[0,T].$

Then we get:

$\displaystyle\frac{d\mathcal{J}(t)}{dt}=\int_{0}^{\ell}m(x)u^{2}_{t}dx-\int_{0}^{\ell}r(x)u^{2}_{xx}dx-k^{-}_{r}u^{2}_{x}(0,t)-k^{+}_{r}u^{2}_{x}(\ell,t),$

for all $t\in[0,T]$. This leads to the required result (21). $\Box$

The next lemma shows another relationship between the auxiliary function
$\mathcal{J}(t)$ and the energy function $\mathcal{E}(t)$. Namely, it shows
that the energy function serves as lower and upper bounds to the auxiliary
function introduced in (20).

###### Lemma 3

Assume that in addition to the basic conditions (13), the coefficient $r(x)$
in (6) satisfies the regularity condition: $r\in H^{2}(0,\ell)$. Then the
following inequalities hold:

$\displaystyle-\beta_{0}\,\mathcal{E}(t)\leq\mathcal{J}(t)\leq\beta_{1}\,\mathcal{E}(t),~{}t\in[0,T],$
(22)

where

$\displaystyle\left.\begin{array}[]{ll}\displaystyle\beta_{0}=\frac{\ell^{2}}{2}\,\sqrt{\frac{m_{1}}{r_{0}}}\\\\[14.0pt]
\displaystyle\beta_{1}=\beta_{0}\left\\{1+\frac{1}{\sqrt{m_{1}r_{0}}}\left[\ell^{2}\mu_{1}+\frac{2}{\ell}\left(k_{a}^{-}+k_{a}^{+}\right)\right]\right\\}\,,\\\
\end{array}\right.$ (25)

and $m_{1},\,\mu_{1},\,r_{0}>0$ are the constants introduced in (13).

Proof. We estimate separately each term on the right hand side of formula
(20). For the first term we use the $\varepsilon$-inequality to get

$\displaystyle\left|\int_{0}^{\ell}m(x)uu_{t}dx\right|\leq\frac{\varepsilon}{2}\,\int_{0}^{\ell}m(x)u_{t}^{2}dx+\frac{1}{2\varepsilon}\,\int_{0}^{\ell}m(x)u^{2}dx.$
(26)

Under the condition $r\in H^{2}(0,\ell)$ the exists the regular weak solution
$u\in L^{2}(0,T;H^{4}(0,\ell))$, with $u_{t}\in
L^{2}(0,T;\mathcal{V}^{2}(0,\ell))$, $u_{tt}\in L^{2}(0,T;L^{2}(0,\ell))$ and
$u_{ttt}\in L^{2}(0,T;H^{-2}(0,\ell))$ of problem (6) [13]. For this solution
we employ the inequality

$\displaystyle\int_{0}^{\ell}u^{2}dx\leq\frac{\ell^{4}}{4}\int_{0}^{\ell}u_{xx}^{2}dx,~{}t\in[0,T],$
(27)

which can be easily proved due to the conditions $u(0,t)=u(\ell,t)=0$. This
yeilds:

$\displaystyle\int_{0}^{\ell}m(x)u^{2}dx\leq\frac{\ell^{4}\rho_{1}}{4r_{0}}\int_{0}^{\ell}r(x)u_{xx}^{2}dx,$

Substituting this in (26) we get:

$\displaystyle\left|\int_{0}^{\ell}m(x)uu_{t}dx\right|\leq\frac{\varepsilon}{2}\,\int_{0}^{\ell}m(x)u_{t}^{2}dx+\frac{\ell^{4}m_{1}}{8\varepsilon
r_{0}}\int_{0}^{\ell}r(x)u_{xx}^{2}dx.$

Choose here the parameter $\varepsilon>0$ from the condition
$\varepsilon/2=\ell^{4}m_{1}/(8r_{0}\,\varepsilon)$ as

$\displaystyle\varepsilon=\frac{\ell^{2}}{2}\,\sqrt{\frac{m_{1}}{r_{0}}}\,,$

we obtain the following estimate:

$\displaystyle\left|\int_{0}^{\ell}m(x)uu_{t}dx\right|\leq\frac{\ell^{2}}{4}\,\sqrt{\frac{m_{1}}{r_{0}}}\left[\int_{0}^{\ell}m(x)u_{t}^{2}dx+\int_{0}^{\ell}r(x)u_{xx}^{2}dx\right].$
(28)

Now, we estimate the second right hand side integral in formula (20), using
inequality (27). We have:

$\displaystyle\int_{0}^{\ell}\mu(x)u^{2}dx\leq\frac{\ell^{4}\mu_{1}}{4r_{0}}\int_{0}^{\ell}r(x)u_{xx}^{2}dx.$
(29)

Finally, to estimate the third and fourth terms on the right side of formula
(20), we use the same argument as above to conclude that

$\displaystyle
u^{2}_{x}(0,t)=\left(-\int_{0}^{\tilde{x}}u_{xx}(x,t)dx\right)^{2}\leq\tilde{x}\,\int_{0}^{\tilde{x}}u^{2}_{xx}(x,t)dx,$
$\displaystyle
u^{2}_{x}(\ell,t)=\left(\int_{\tilde{x}}^{\ell}u_{xx}(x,t)dx\right)^{2}\leq(\ell-\tilde{x})\,\int_{0}^{\tilde{x}}u^{2}_{xx}(x,t)dx.$

Hence,

$\displaystyle\left.\begin{array}[]{ll}\displaystyle\frac{1}{2}\,k^{-}_{a}u_{x}^{2}(0,t)\leq\frac{\ell}{2}\,\frac{k^{-}_{a}}{r_{0}}\int_{0}^{\ell}r(x)u^{2}_{xx}(x,t)dx,\\\\[9.0pt]
\displaystyle\frac{1}{2}\,k^{+}_{a}u_{x}^{2}(\ell,t)\leq\frac{\ell}{2}\,\frac{k^{+}_{a}}{r_{0}}\int_{0}^{\ell}r(x)u^{2}_{xx}(x,t)dx.\end{array}\right.$
(32)

In view of (28), (29) and (32) we obtain the following upper estimate for the
auxiliary function $\mathcal{J}(t)$:

$\displaystyle\mathcal{J}(t)\leq\frac{\ell^{2}}{4}\,\sqrt{\frac{m_{1}}{r_{0}}}\int_{0}^{\ell}m(x)u_{t}^{2}dx\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad+\left[\frac{\ell^{2}}{4}\,\sqrt{\frac{m_{1}}{r_{0}}}+\frac{\ell^{4}}{4r_{0}}\,\mu_{1}+\frac{\ell}{2r_{0}}\left(k^{-}_{a}+k^{+}_{a}\right)\right]\int_{0}^{\ell}r(x)u_{xx}^{2}dx,$

for all $t\in(0,T]$. This leads to the upper bound

$\displaystyle\mathcal{J}(t)\leq\beta_{1}\,\mathcal{E}(t),~{}t\in[0,T],$

in terms of the energy functional $\mathcal{E}(t)$ and the constant
$\beta_{1}>0$ introduced in (25).

The lower bound

$\displaystyle\mathcal{J}(t)\geq-\beta_{0}\,\mathcal{E}(t),~{}t\in[0,T]$

follows from the second part

$\displaystyle\int_{0}^{\ell}m(x)uu_{t}dx\geq-\,\frac{\ell^{2}}{4}\,\sqrt{\frac{m_{1}}{r_{0}}}\left[\int_{0}^{\ell}m(x)u_{t}^{2}dx+\int_{0}^{\ell}r(x)u_{xx}^{2}dx\right]$

of estimate (28). This leads to the required estimates (22). $\Box$

###### Remark 3

The constants $\beta_{0},\beta_{1}>0$ introduced in (25) depend only on the
geometric and physical parameters of a beam.

We introduce now the Lyapunov function

$\displaystyle\mathcal{L}(t)=\mathcal{E}(t)+\lambda\mathcal{J}(t),\,t\in[0,T]$
(33)

through the energy function $\mathcal{E}(t)$ and the auxiliary function
$\mathcal{J}(t)$, where $\lambda>0$ is the penalty term.

###### Theorem 2

Assume that the inputs in (6) satisfy the basic conditions (13) and the
regularity condition $r\in H^{2}(0,\ell)$. Suppose, in addition that the
damping constant of proportionality is positive,

$\displaystyle\gamma_{0}>0.$ (34)

Then system (6) is exponentially stable, that is,

$\displaystyle\mathcal{E}(t)\leq
M\,e^{-\sigma\,t}\,\mathcal{E}(0),~{}t\in[0,T],$ (35)

where

$\displaystyle\left.\begin{array}[]{ll}\displaystyle
M=\frac{1+\beta_{1}\lambda}{1-\beta_{0}\lambda}~{},~{}\sigma=\frac{2\lambda}{1+\beta_{1}\lambda}~{},\\\\[14.0pt]
0<\lambda<\min(1/\beta_{0},\,\gamma\,m_{0}/(2m_{1})),\end{array}\right.$ (38)

where $\mu_{0},m_{1}>0$ and $\beta_{0},\beta_{1}>0$ are the constants
introduced in (13) and (25), respectively, and $\mathcal{E}(0)>0$ is the
initial energy defined in (1).

Proof. Using estimates (22) in (33) we get:

$\displaystyle\left(1-\beta_{0}\lambda\right)\,\mathcal{E}(t)\leq\mathcal{L}(t)\leq\left(1+\beta_{1}\lambda\right)\,\mathcal{E}(t),~{}t\in[0,T].$

From the positivity requirement of the first left hand side multiplier, we
find that the penalty term should satisfiy the following condition:

$\displaystyle 0<\lambda<1/\beta_{0},~{}\beta_{0}>0.$ (39)

Differentiate now $\mathcal{L}(t)$ with respect to the variable $t\in(0,T)$
and use formulas (18) and (21). We have:

$\displaystyle\frac{d\mathcal{L}(t)}{dt}+2\lambda\mathcal{E}(t)=-\int_{0}^{\ell}\left[\mu(x)-2\lambda
m(x)\right]u_{t}^{2}dx\qquad\qquad\quad$
$\displaystyle\qquad\qquad\qquad-k^{-}_{a}u^{2}_{xt}(\ell,t)-k^{+}_{a}u^{2}_{xt}(\ell,t),~{}t\in[0,T].$
(40)

Assume that, in addition to (39), the penalty term satisfies also the
following condition:

$\displaystyle\lambda\leq\mu_{0}/(2m_{1})$

which guarantees positivity of the term in the square bracket under the right
hand side intagral in (3). In view of the relation $\mu_{0}=\gamma\,m_{0}$,
this condition implies

$\displaystyle\lambda\leq\gamma\,m_{0}/(2m_{1}).$ (41)

This leads to

$\displaystyle\frac{d\mathcal{L}(t)}{dt}+2\lambda\mathcal{E}(t)\leq
0,~{}t\in[0,T],$

or, with $\mathcal{E}(t)\geq\mathcal{L}(t)/(1+\lambda\gamma_{1})$, to the
inequality

$\displaystyle\frac{d\mathcal{L}(t)}{dt}+\frac{2\lambda}{1+\lambda\gamma_{1}}\,\mathcal{L}(t)\leq
0,~{}t\in[0,T].$

Solving this inequality we find:

$\displaystyle\mathcal{L}(t)\leq e^{-\sigma\,t}\,\mathcal{E}(0),~{}t\in[0,T]$

which implies the required estimate (35). $\Box$

###### Remark 4

The constant $\sigma>0$ in (38), called the decay rate parameter, depends only
on the geometric and physical parameters of the beam and also on the stiffness
of the torsional dampers introduced in (13), as formulas (25) show. Hence, the
uniform exponential stability estimate (38) can be applied to study
exponential stability for Euler-Bernoulli beams with various physical and
geometric properties, under boundary controls in rotation and angular
velocity. Furthermore, considering formula (25), estimate (38) also clearly
shows the contribution of each damping factor $\mu(x)$, $k_{a}^{-}$ and
$k_{a}^{-}$ to the energy decay rate.

##  4 Numerical results

Although there is an exponential function $e^{-\sigma\,t}$ on the right side
of the estimate (35), with the decay rate parameter $\sigma>0$ introduced in
(38), in some cases, this appearance can be misleading. Namely, $\sigma>0$ is
dependent on the positive parameters $\lambda$ and $\beta_{1}$. The specific
values of these parameters play a crucial role in determining the decay
behavior of the function $e^{-\sigma\,t}$. Depending on the values of
$\lambda$ and $\beta_{1}$, the decay of this function can exhibit
characteristics similar to the decay of a linear function. To see such cases,
it is necessary to study the dependence of the decay rate parameter on not
only the geometric and physical parameters of the beam, but also on the
viscous external damping parameter $\mu(x)$ and the torsional dampers
$k_{a}^{-},k_{a}^{-}$ separately.

The examples below are provided to illustrate these situations and their
causes. Without loss of generality, here we consider the constant coefficient
beam equation

$\displaystyle mu_{tt}+\mu u_{t}+ru_{xxxx}=0,\,(x,t)\in\Omega_{T},$ (42)

where

$\displaystyle m=\rho\,S,~{}\mu=\gamma m,~{}r=EI,$

in accordance with the above notation. For this constant coefficients
equation, formulas (25) and (38) for the parameters
$\beta_{0},\,\beta_{1},\,M_{1},\,\sigma>0$ and conditions are as follow:

$\displaystyle\left.\begin{array}[]{ll}\displaystyle\beta_{0}=\frac{\ell^{2}}{2}\,\sqrt{\frac{m}{r}}\\\\[14.0pt]
\displaystyle\beta_{1}=\beta_{0}\left[1+\ell^{2}\sqrt{\frac{m}{r}}\,\gamma\right]+\frac{\ell}{2r}\left(k_{a}^{-}+k_{a}^{+}\right)\,,\\\\[14.0pt]
\displaystyle
M=\frac{1+\beta_{1}\lambda}{1-\beta_{0}\lambda}~{},~{}\sigma=\frac{2\lambda}{1+\beta_{1}\lambda}~{}.\end{array}\right.$
(46)

Here, the beam with the rectangular cross section $S=b\,h$, where $b>0$ and
$h>0$ are the width height, with the following numerical values of the
geometric and physical parameters are examined [20]:

$\displaystyle\left.\begin{array}[]{ll}\ell=0.502\,\mbox{m},~{}b=1.7\times
10^{-3}\,\mbox{m},~{}h=0.89\times 10^{-3}\,\mbox{m},\\\\[4.0pt]
\rho=1.42\times 10^{3}\,\mbox{Kg\,m}^{-3},~{}E=3.1\times
10^{9}\,\mbox{N/m}^{2},~{}\gamma\in[0.01,\,10]\,\mbox{s}^{-1}.\end{array}\right.$
(49)

With the numerical values in (49) we have:

$\displaystyle\left.\begin{array}[]{ll}S=1.51\times
10^{-6}\,\mbox{m}^{2},~{}I:=bh^{3}/12=0.1\times
10^{-12}\,\mbox{m}^{3},\\\\[4.0pt] m=2.14\times
10^{-3}\,\mbox{Kg\,m}^{-1},~{}r=0.31\times
10^{-3}\,\mbox{N\,m}^{2},~{}\mu=0.22\,\mbox{Kg\,m}^{-1}.\end{array}\right.$

We consider three-level, weak, medium, and high damping cases corresponding to
the values $\gamma=0.1$, $\gamma=1.0$ and $\gamma=5.0$ of the damping constant
of proportionality, using the following values $\langle
k^{-}_{a},k^{+}_{a}\rangle=\langle 0,\,0\rangle$ and $\langle
k^{-}_{a},k^{+}_{a}\rangle=\langle 0.01,\,0.01\rangle$ of the stiffness of the
torsional dampers.

The calculated by formulas given in (46) values of the decay rate parameter
$\sigma>0$ are listed in Table 1. The values of the penalty term $\lambda>0$
are set according to the requirement $0<\lambda<\min(1/\beta_{0},\,\gamma/2)$.

From the last column of Table 1 it can be seen that, in absence of the
torsional dampers ($k^{-}_{a}=k^{+}_{a}=0$), the increase in the value of the
damping constant from $\gamma=0.1$ to $\gamma=5.0$, leads to the increase of
the decay parameter $\sigma>0$. Thus, for the weak damping case $\gamma=0.01$
the value of the decay parameter is $\sigma=0.08$, and the energy decay is
only exponential in appearance, in fact, it is linear (Figure 1 on the left).

Table 1. The decay rate parameters corresponding to the geometric and physical
parameters given in (49).

Damping constant
---
$\gamma=0.1$
$\gamma=1.0$
$\gamma=5.0$
$\langle k^{-}_{a},k^{+}_{a}\rangle$ | $\langle\beta_{0},\beta_{1}\rangle$ | $\lambda$ | $M$ | $\sigma$
---|---|---|---|---
$\langle 0,\,0\rangle$ | $\langle 0.33,\,0.35\rangle$ | $0.04$ | $1.03$ | $0.08$
$\langle 0.01,\,0.01\rangle$ | $\langle 0.33,\,16.55\rangle$ | $0.04$ | $1.68$ | $0.05$
$\langle 0,\,0\rangle$ | $\langle 0.33,\,0.55\rangle$ | $0.4$ | $1.41$ | $0.66$
$\langle 0.01,\,0.01\rangle$ | $\langle 0.33,\,16.75\rangle$ | $0.4$ | $8.87$ | $0.10$
$\langle 0,\,0\rangle$ | $\langle 0.33,\,1.42\rangle$ | $2.4$ | $21.19$ | $1.09$
$\langle 0.01,\,0.01\rangle$ | $\langle 0.33,\,17.62\rangle$ | $2.4$ | $208.12$ | $0.11$

Figure 2: Behaviour of the function $\exp(-\sigma t)$: with
$k^{-}_{a}=k^{+}_{a}=0$ (left) and with with $k^{-}_{a}=k^{+}_{a}=0.01$
(right).

Comparing the values of the decay rate parameter, in the last column of Table
1, corresponding to zero and non-zero values of the stiffness of the torsional
dampers, we can observe the role of these boundary controls (Figure 1 on the
right).

##  5 Conclusions

This study proposes an approach for the exponential stability analysis of
Euler-Bernoulli beams under boundary controls in rotation and angular
velocity. By employing the regular weak solution, energy identity, and
Lyapunov function, we are able to derive a uniform exponential decay estimate
for the system’s energy.

Our approach is grounded in natural assumptions concerning physical parameters
and other inputs, ensuring the existence of a regular weak solution. The decay
rate constant in the derived estimate relies solely on the physical and
geometric parameters of the beam, which include the viscous external damping
coefficient, as well as the boundary springs and dampers. This feature enables
straightforward utilization of decay rate estimation in practical engineering
applications.

Furthermore, we have provided preliminary numerical examples that shed light
on the role of damping parameters. However, a more detailed analysis, focusing
on the individual contributions of each damping parameter to the overall
damping behavior, will be pursued in future research.

## Acknowledgments

The research has been supported by the Scientific and Technological Research
Council of Turkey (TUBITAK) through the Incentive Program for International
Scientific Publications (UBYT). The research of the author has also been
supported by FAPESP, through the Visiting Researcher Program, proc.
2021/08936-1, in Escola Politécnica, University of São Paulo, Brazil, during
the period November 02 - December 18, 2022.

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