esa-sceva/satcom-chunk-collection huggingface dataset:
# Recent Advances in Laser-Induced Surface Damage of KH\({}_{2}\)Po\({}_{4}\) [PERSON]
## 2 Mechanism of Pulsed Laser-Induced Surface Damage  
With the continuous upgrading of high-power laser systems, the interaction mechanism between laser and crystal materials is of interest. The interaction between laser and crystal materials is closely related to laser parameters such as pulse width, wavelength, and spot area.  
Figure 1[31, 32] shows the physical phenomenon on optical materials induced by lasers with different power densities. As a transparent optical material, KDP crystal has a relatively low absorption coefficient. When the laser is irradiated on the ideal KDP crystal, most of the energy of the laser will be transmitted, and only a small part of the laser energy will be absorbed.  
It is necessary to consider how to determine the occurrence of damage exactly, i.e., the determination of the damage initiated for understanding the LIDT of KDP crystal. It is generally believed that the LID is initiated when any permanent change occurs on the surface of the material. At present, the critical free electron density is used as the criterion of damage judgment to determine the LIDT. The critical free electron density \(n_{cr}\) of the material can be expressed as Equation (1) [33]:  
\[n_{cr}=\frac{\varepsilon_{0}m_{e}^{*}\omega^{2}}{e^{2}} \tag{1}\]  
where \(e\) is the electronic charge, \(\varepsilon_{0}\) represents the dielectric constant of electron space, \(m_{e}\) is the electron reduced mass, and \(\omega\) is the angular frequency of incident laser.
# Principle and Implementation of Stokes Vector Polarization Imaging Technology
);  
2  
[PERSON]
1 Laboratory of Optical Detection and Imaging, School of Science, Qingdao University of Technology, Qingdao 266033, China; [EMAIL_ADDRESS] (Y.S. ); [EMAIL_ADDRESS] (X.H. ); [EMAIL_ADDRESS] (Y.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (H.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (J.X. ); [EMAIL_ADDRESS] (J.T. ); [EMAIL_ADDRESS] (X.K. ); [EMAIL_ADDRESS] (Z.W. ); [EMAIL_ADDRESS] (J.Y. ); [EMAIL_ADDRESS] (Y.K. ); [EMAIL_ADDRESS] (Z.W. ); [EMAIL_ADDRESS] (J.Y. ); [EMAIL_ADDRESS] (J.Y. );  
2  
Yaqing [PERSON]
1 Laboratory of Optical Detection and Imaging, School of Science, Qingdao University of Technology, Qingdao 266033, China; [EMAIL_ADDRESS] (Y.S. ); [EMAIL_ADDRESS] (X.H. ); [EMAIL_ADDRESS] (Y.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (H.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (J.X. ); [EMAIL_ADDRESS] (J.T. ); [EMAIL_ADDRESS] (X.K. ); [EMAIL_ADDRESS] (Z.W. ); [EMAIL_ADDRESS] (J.Y. );  
2  
[PERSON]
1 Laboratory of Optical Detection and Imaging, School of Science, Qingdao University of Technology, Qingdao 266033, China; [EMAIL_ADDRESS] (Y.S. ); [EMAIL_ADDRESS] (X.H. ); [EMAIL_ADDRESS] (Y.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (H.L. ); [EMAIL_ADDRESS] (X.Z. ); [EMAIL_ADDRESS] (J.X. ); [EMAIL_ADDRESS] (J.T. ); kongxiaoefel20 imaging measurement objective reflection of the degree of polarization and polarization angle information, can effectively make up for the inadequacy of traditional imaging, solve traditional photometry is difficult to solve the problem in target recognition, and obtain the result of the high precision, and on the basis of traditional strength imaging increased the polarization information dimension, The polarization information of each point in the image can be obtained as well as the distribution of light intensity in two-dimensional space. At present, polarization imaging technology has shown potential application prospects in biomedicine [1,2], modern military [3,4], industrial detection [5, 6, 7], descattering imaging [8, 9, 10, 11], navigation [12] and other fields. Polarization imaging methods mainly include Stokes vector imaging and Muller matrix imaging, and Stokes vector can be expressed as 4 \(\times\) 1 column matrix or 1 \(\times\) 4 row matrix, which is more convenient in describing fully polarized light, natural light and partially polarized light. In this paper, the principle, realization, and applications of polarization imaging based on Stokes vector are reviewed. Firstly, Stokes vector and its measurement theory are introduced. Secondly, we introduce various polarization imaging techniques. Finally, the applications of polarization imaging based on Stokes vector are illustrated. The purpose of this paper is to promote the further development of polarization imaging technology and provide reference for applications in various fields.
## Appendix A  
Vector spherical harmonic functions in the IIM T-Matrix method are obtained by solving the vector Helmholtz equation by the separation of variables method, and their expressions can be written as follows [42]:  
\[Rg\mathbf{M}_{mn}(kr,\theta,\varphi)=\left(\frac{2n+1}{4\pi n(n+1)}\right)^{1/2} (-1)^{m}\exp(\mathrm{i}mq\varphi)\Big{(}\hat{\Theta}i\pi_{mn}(\theta)-\hat{ \varphi}\tau_{mn}(\theta)\Big{)}J_{n}(kr) \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**(a, b)**} & **Number of** & **Computational Time T\({}_{0}\)** & **Computational Time T\({}_{1}\)** & **(T\({}_{0}-\) T\({}_{1}\))/** \\  & **Threads** & **of the Model without** & **of the Model with** & **T\({}_{0}\times 100\%\)** \\  & **Lorenz–Mie Theory/s** & **Lorenz–Mie Theory/s** & **Lorenz–Mie Theory/s** & **75.10643** \\ \hline (1.5, 2.0) & 4 & 4463 & 1111 & 75.10643 \\ (1.0, 2.0) & 4 & 4438 & 2232 & 49.70708 \\ (0.7, 2.0) & 4 & 4485 & 2879 & 35.80825 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The computational time needed for the IIM T-Matrix model with and without the Lorenz–Mie theory.  
\[\begin{split} Rg\mathbf{N}_{mn}(kr,\theta,\varphi)&=\left( \frac{2n+1}{4\pi m(n+1)}\right)^{1/2}\left(-1\right)^{m}\exp\left(\mathrm{i}m \varphi\right)\\ &\times\left[\frac{n(n+1)}{kr}I_{n}(kr)\cdot d_{0m}^{n}\hat{ \mathbf{r}}+\frac{1}{kr}\frac{d}{d(kr)}(krJ_{n}(kr))\Big{(}\hat{\Phi}\tau_{mn} (\theta)+\hat{\varphi}i\pi_{mn}(\theta)\Big{)}\right]\end{split} \tag{10}\]  
\[\begin{split}\mathbf{M}_{mn}(kr,\theta,\varphi)&= \left(\frac{2n+1}{4\pi n(n+1)}\right)^{1/2}\left(-1\right)^{m}\exp\left( \mathrm{i}m\varphi\right)\left(\hat{\theta}i\pi_{mn}(\theta)-\hat{\varphi} \tau_{mn}(\theta)\right)H_{n}^{(1)}(kr)\end{split} \tag{11}\]  
\[\begin{split}\mathbf{N}_{mn}(kr,\theta,\varphi)=& \left(\frac{2n+1}{4\pi n(n+1)}\right)^{1/2}\left(-1\right)^{m}\exp\left( \mathrm{i}m\varphi\right)\\ &\times\left[\frac{n(n+1)}{kr}H_{n}^{(1)}(kr)\cdot d_{0m}^{n}\hat {\mathbf{r}}+\frac{1}{kr}\frac{d}{d(kr)}\Big{(}krH_{n}^{(1)}(kr)\Big{)}\Big{(} \hat{\theta}\tau_{mn}(\theta)+\hat{\varphi}i\pi_{mn}(\theta)\Big{)}\right] \end{split} \tag{12}\]  
From the equations above, it can be found that the vector spherical harmonic functions can be divided into 3 parts: the azimuth angle component, the radial component, and the scattering angle component. The azimuth angle component is denoted as \(\exp(\mathrm{i}m\varphi)\), the radial component is a combination of the spherical Bessel functions, where, in \(Rg\mathbf{M}_{mn}\) and \(Rg\mathbf{N}_{mn}\), the first kind of Bessel function \(J_{n}(kr)\) is used, while in \(\mathbf{M}_{mn}\) and \(\mathbf{N}_{mn}\), the Hankel function of the first kind is applied. The scattering angle component is a vector, which is constructed by the angular functions, i.e., \(\pi_{mn}(\theta)\) and \(\tau_{mn}(\theta)\), and the Wigner-d function \(d_{0m}^{n}(\theta)\).
## 3 Methods
### PSO-SVR Model for Estimating GCV  
As the primary goal of this study, the new hybrid artificial intelligence model PSO-SVR is described and offered in this section. It is a combination of the PSO and SVR algorithms for the generation of an optimal model in predicting GCV. Accordingly, SVR was determined as the primary model for predicting GCV. For the development of the SVR model, three kernel functions (i.e., radius basis function (RBF), Polynomial (P), and Linear (L)) were applied to map the GCV database. For each kernel function, hyper-parameter(s) were determined, as listed in Table 2. The primary purpose of the hyper-parameters was controlling the accuracy of the SVR model. Therefore, the selection of the optimal values of hyper-parameters was a complication, and it needed a solution for defining the optimal hyperparameters. In this case, the PSO algorithm was an optimal solution for searching optimal hyper-parameters of the SVR model. It performed a global search solution based on particles and their experiences. For each value of hyper-parameter searched, it calculated the performance of the SVR model through a fitness function (i.e., root-mean-squared error (RMSE)). Then, it checked the termination criteria. If the value of RMSE was satisfied (lowest RMSE), it stopped the searching process and gave the final PSO-SVR model. Otherwise, it continued searches until it was satisfied, or it reached the maximum of iterations. Figure 3 describes the framework of the proposed PSO-SVR model with three forms of kernel functions.
## 2 System Structure
### Passive Structure  
A passive structure [13; 14; 15; 16] has no converters in its power network. The fuel cell and battery are directly connected to the load. The absence of a converter yields higher efficiency, reduced cost, and a less complex structure. Despite these advantages, which are favorable for mobile robots, in reality, the passive structure is only marginally applicable. This is because with no converter in the power network, a voltage difference between the fuel cell, battery, and load is not allowed. Therefore, their voltage levels across the DC bus must be maintained at all times. However, it is difficult to acquire fuel cells that are dedicated to specific voltage requirements, particularly for prototyping. Batteries also share a similar problem, although to a lesser degree. In a passive structure, it is difficult to operate batteries within their safe voltage ranges and at desired operating points that produce the maximum efficiency or power. This would significantly dilute the benefits of the absence of a converter. Therefore, we consider the passive structure to be unsuitable for mobile robot applications.
## 2 Simulation Setup and Estimation
### Estimated Parameters
Overall, the estimation of the biases leads to a rank-deficient normal equation matrix in our simulations. Therefore, we introduced constraints for the bias estimation in different scenarios: constraints within the range of the expected precision of the optical links (1 mm) and a solution with five times looser constraints.
# A Multilevel Switched Capacitor Inverter with Reduced Components and Self-Balance
## 2 Design of Proposed SCMLI  
### Topology  
The proposed 13-level switched capacitor topology is depicted in Figure 1. It is made up of four switches (\(S_{1}\)-\(S_{4}\)), a switched capacitor unit (SCU), a L-type unit (LTU), two diodes, and three capacitors. The SCU contains a single DC source, five switches (\(S_{5}\)-\(S_{9}\)), a diode \(D_{1}\), as well as capacitors \(C_{1}\) and \(C_{2}\). Figure 2 shows the working states of the proposed switched capacitor basic unit. In Figure 2a, the capacitor voltage can be maintained at \(V_{\rm dc}\) when it is connected in parallel with the DC source. Figure 2b,c displays the discharge of capacitors connected in parallel/series. The SCU can achieve a triple voltage gain when the two capacitors are connected in parallel with the DC source. LTU includes the capacitor \(C_{3}\), three switches (\(S_{10}\)-\(S_{12}\)), and a diode \(D_{2}\), whilst the capacitor voltage of \(C_{3}\) can be maintained at 3 \(V_{\rm dc}\) through the L-type circuit loop.
## 5 Experiments
### Experimental Results and Analysis
#### 5.2.5 Robustness  
Regardless of the type of user data, such as movies, shopping, or advertising, which has an open nature, some businesses or individuals inject fake data into the recommendation system from the perspective of their own interests, attempting to alter the recommendation results. This is known as a recommendation attack. Common recommendation attacks include random attacks and average attacks. Random attacks involve adding 1%, 3%, 5%, and 10% of random entries to attack. Average attacks involve adding 1%, 3%, 5%, and 10% of entries to attack, with the mean of the dataset being added as one of the entries. Under attack conditions with 1%, 3%, and 5% data, the fluctuations of DG-MOSS remain within a small range. The attack with 10% data is relatively strong and has some impact on all models. Since this study focuses on maximizing rewards for recommendations, the robustness is demonstrated by achieving the maximum reward value even under different attack conditions, which helps in selecting the arms. As shown in Tables 4-6. DG-MOSS performs the best under all attack conditions, which indicates that this algorithm has good robustness.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithms** & **Original** & **1\% R** & **3\% R** & **5\% R** & **10\% R** & **1\% A** & **3\% A** & **5\% A** & **10\% A** \\ \hline \(\varepsilon\)-greedy & 0.1944 & 0.1932 & 0.2015 & 0.2070 & 0.2186 & 0.1969 & 0.2131 & 0.2291 & 0.2574 \\ Boltzmann & 0.0139 & 0.0192 & 0.0276 & 0.0368 & 0.0546 & 0.0143 & 0.0157 & 0.0181 & 0.0266 \\ CNAME & 0.2296 & 0.2266 & 0.1671 & 0.2496 & 0.2562 & 0.2420 & 0.2347 & 0.2696 & 0.2860 \\ UCB & 0.0778 & 0.0815 & 0.0870 & 0.0962 & 0.1092 & 0.0850 & 0.1022 & 0.1179 & 0.1504 \\ UCB1 & 0.0262 & 0.0357 & 0.0356 & 0.0415 & 0.0614 & 0.0310 & 0.0406 & 0.0411 & 0.0678 \\ FP-UCB & 0.0568 & 0.0421 & 0.0496 & 0.0546 & 0.0706 & 0.0636 & 0.0498 & 0.0547 & 0.0792 \\ BBANK & 0.1983 & 0.2379 & 0.2526 & 0.2562 & 0.2643 & 0.2421 & 0.2441 & 0.2494 & 0.2978 \\ MOSS & 0.0809 & 0.0847 & 0.0908 & 0.0983 & 0.1130 & 0.0887 & 0.1073 & 0.1220 & 0.1541 \\ R-MOSS & 0.0653 & 0.0621 & 0.0710 & 0.751 & 0.0908 & 0.0667 & 0.0752 & 0.0865 & 0.1430 \\
**DG-** & **0.2378** & **0.2396** & **0.2539** & **0.2585** & **0.2720** & **0.2422** & **0.2609** & **0.2729** & **0.2996** \\
**MOSS** & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Random attacks (R) and average attacks (A) on the movie dataset.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Datasets** & **Episode** \\ \hline Initial & 1000 \\ Random data-ND & 24 \\ Alibaba & 44 \\ Movie & 34 \\ Ads-CTR-Optimisation & 638 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Number of training episodes.
## 3 Results
### Mossbauer Spectroscopy  
The room-temperature Mossbauer spectra of the Fe-doped samples are shown in Figure 3, together with the corresponding fits.  
The parameters extracted from the fit procedure of each sub-spectrum are reported in Table 3, together with the relative weight calculated from the values of the effective thickness.  
The Mossbauer spectra of the two samples were very similar and can both be described by two doublets (namely, site 1 and site 2), one of which (site 1) is clearly prevailing. The isomer shift had approximately the same value for all the sub-spectra, i.e., about 0.35 mm/s, a value associated with Fe\({}^{3+}\) ions. Moreover, a similar value of the quadrupole splitting was found for both samples. However, the two sites showed a different value for the Gaussian broadening, indicating the presence of a distribution of hyperfine parameters; in particular,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Samples** & **Na/Mn Stoichiometric** & **Na/Mn from EDS** & **EDS Composition** \\ \hline
**NMO** & 0.74 & 0.78 & Na\({}_{0.77}\)Mn\({}_{0.98}\)O\({}_{2}\) \\ \hline
**Cu02** & 0.925 & 0.925 & Na\({}_{0.74}\)Mn\({}_{0.80}\)Cu\({}_{0.20}\)O\({}_{2}\) \\ \hline
**Fe02** & 0.925 & 0.88 & Na\({}_{0.73}\)Mn\({}_{0.83}\)Fe\({}_{0.17}\)O\({}_{2}\) \\ \hline
**NMO-Q** & 0.74 & 0.75 & Na\({}_{0.75}\)Mn\({}_{1}\)O\({}_{2}\) \\ \hline
**Cu02-Q** & 0.925 & 0.89 & Na\({}_{0.73}\)Mn\({}_{0.82}\)Cu\({}_{0.20}\)O\({}_{2}\) \\ \hline
**Fe02-Q** & 0.925 & 0.90 & Na\({}_{0.72}\)Mn\({}_{0.80}\)Fe\({}_{0.20}\)O\({}_{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Chemical compositions from the EDS analysis for all the synthesized samples.  
Figure 3: Mössbauer spectra of the Fe02 (**right**) and Fe02-Q (**left**) samples acquired at 293 K (black line) and the corresponding fit (red line). The absorption cross-section of each contribution is also shown.  
site 2 had a broader distribution. Consequently, the quadrupole splitting of all the sub-spectra of both samples can be considered consistent between them. The quadrupole splitting provides information about bond properties and the local symmetry of the Fe site [28] and can suggest the degree of distortion from the ideal octahedral coordination. The obtained values corroborated the suggestion of the octahedral coordination of Fe\({}^{3+}\), as expected for the transition metals located on the layers, and were in good agreement with the literature values of similar samples [21; 29].  
These results can suggest that due to the presence of only one crystallographic site for the transition metal in both P2 and P'2 lattices, the iron ions were distributed in both the layered structures, with a preference toward the more regular sites of the P2 phase.
## 2 Results and Discussion
### Effect of Salinity and Rhizobacteria on the Expression of BnRSH Genes
liquefaciens_ induced the expression of _BnRCSH_ in cotyledons (Figure 8). In response to salt stress, _BnRSH_ gene expression is elevated in _S. plymuthica_ and _S. liquefaciens_ inoculated plants, whereas _M. timonae_ inoculated plants did not show substantial changes as compared with control plants (without bacteria but treated with NaCl at the same concentration). For all _BnRSH_ genes the highest level of expression was observed in plants inoculated with _S. plymuthica_ (Figure 8). There is little data in the literature about the possible relation between (p)ppGpp and PGPR bacteria. [PERSON] et al. [5] showed changes in the expression of _BnRSH1_ and _BnRSH3_ in roots of oilseed rape growing in the presence of the halotolerant PGPR bacterium _Pseudomonas stutzeri_ ISE12 under salt stress. Increased expression of plant _RSH_ genes was also demonstrated in response to pathogen attack. It was found that the infection of tobacco plants with the bacterial _Erwinia carovova_ pathogen leads to a 10-fold increase in the _NtRSH2_ protein level [44]. _S. plymuthica_ used in this study is characterized by high metabolic activity; it is able to biodegrade plastic in compost and agricultural soil and stimulate the growth of _B. napus_, _Miscanthus x giganteus_, and _Salix viiminalis_[70; 71]. It was shown that several salt-tolerant strains of _S. plymuthica_ improved cucumber biomass and yield via synthesis of IAA [72]. _S. liquefaciens_ improved salt stress tolerance and plant growth in maize and rape [6]. _M. timonae_ colonizes the rhizosphere, roots and leaves, and is a growth promoter via the production of IAA and siderophores in various plant species [73]. Our research clearly showed changes in mRNA levels of _BnRSHs_ grown in the presence of the strains _S. liquefaciens_ and _S. plymuthica_, but not in the presence of _M. timonae_ which suggests that some PGPR bacteria might also improve plant growth under salt stress via the stringent response pathway.
# Therapeutic Applications of Halloysite
## 3 Halloysite Structure and Applications
### HNTs in Tissue Engineering  
Halloysite has a variety of applications in the field of tissue engineering. They are used in bone implants, dental fillings, and tissue scaffolds [63]. HNTs mixed with bone cement and used as a drug carrier and release system are promising applications. The research found HNTs loaded with the antibiotic gentamicin sulfate with a concentration of 5-8 wt% in the cement (PMMA) provide sustained release up to 300-400 h [40]. This PMMA/halloysite/gentamicin composite tensile strength does not deteriorate compared with pure cement, and its adhesion to bone is significantly increased [53]. HNTs resin- dentin bond is similar to halloysite-PMMA bone cement [64]. HNTs and functionalized HNTs improved mechanical properties significantly [64; 65; 66]. Silver nanoparticle immobilized HNT (HNT/Ag) fillers significantly improved mechanical properties [67]. This filler also showed a significant antibacterial activity observed on _S. mutans_[67]. [PERSON] et al., 2015 were the first to show that a nanoenhanced hydrogel could significantly enhance the biological activity of bone progenitor cells and achieve a sustained release of BMP-2 for over a week [49].  
Currently, hydrogel scaffolds are being applied to transplant cells and engineer nearly every tissue in the body, including cartilage, bone, and smooth muscle [68]. Compared to pure alginate scaffolds, alginate/halloysite nanotube (HNTs) composite scaffolds significantly enhance compressive strength and compressive modulus in dry and wet states [69]. Furthermore, HNTs increased the scaffold density, decreased the swelling ratio in water, and improved alginate's thermal stability [69]. In addition, the alginate/HNT composite scaffolds have better cytocompatibility [69]. Chitosan-halloysite nanotubes (HNTs) nanocomposite (NC) scaffolds have similar results as alginate HNTs composite scaffolds [70]. Compared to the pure chitosan scaffold, the NC scaffolds exhibited significantly improved compressive strength, compressive modulus, and thermal stability [70]. Furthermore, the chitosan-HNTs nanocomposites were cytocompatible even when the HNTs load was 80% [70].
## 2 Materials and Methods  
### Open-Loop Control Problem  
Let us consider a nonlinear discrete deterministic dynamical control system described by a state equation of the form  
\[x(t+1)=f(t,\ x(t),\ u(t)),\ t=0,1,\ldots,N-1, \tag{1}\]  
where \(t\) is a discrete time with number of stages \(N\); \(x\) is the \((n\times 1)\) state vector; \(u\) is the \((q\times 1)\) control vector, \(u\in U(t)=[a_{1}(t),\ b_{1}(t)]\times\ldots\times[a_{q}(t),\ b_{q}(t)]\); \(f_{i}(t,\ x,\ u)\), \(i=1,\ldots,n\) are a known continuous functions.  
Initial condition:  
\[x(0)=x_{0}. \tag{2}\]  
Let us define a set \(D(0,\ x_{0})\) of pairs \(d=(x(\cdot),\ u(\cdot))\), where \(x(\cdot)=\{x_{0},\ x(1),\ldots,x(N)\}\) is a trajectory, \(u(\cdot)=\{u(0),\ u(1),\ \ldots,\ u(N-1)\}\) is an open-loop control, satisfying the state equation (1) and initial condition (2).  
The performance index to be minimized is defined on the set \(D(0,x_{0})\) as  
\[I(d)=\sum_{t=0}^{N-1}f^{0}(t,\ x(t),\ u(t))+F(x(N)), \tag{3}\]  
or  
\[I(d)=F(x(0),\ldots,x(N);\ u(0),\ldots,u(N-1)), \tag{4}\]  
where \(f^{0}(t,\ x,\ u)\), \(F(x)\), \(F(x(\cdot)\), \(u(\cdot))\) are known continuous functions. The notation form (3) is typical for separable optimal control problems, and the form (4) is typical for nonseparable ones.  
It is required to find an optimal pair \(d^{*}=(x^{*}(\cdot),\ u^{*}(\cdot))\in D(0,\ x_{0})\) that minimizes the performance index, i.e.,  
\[I(d^{*})=\min_{d\in D(0,x_{0})}I(d). \tag{5}\]  
To solve the problem (5), an algorithm that imitates the behavior of a tomtits flock is proposed. This algorithm belongs to the nature-inspired (more precisely, bird-inspired) and swarm intelligence metaheuristic optimization algorithms [2,15,16,17]. This problem was solved using iterative dynamic programming and the Luus-Jaakola algorithm [1] and by the Perch School Search optimization algorithm [10]. The results of a comparative analysis of the obtained results are presented below in the solved examples.