Relative Equilibrium Research Articles

Numerical Thermo-Hydraulic Simulation of Infiltration and Evaporation of Small-Scale Replica of Typical Dike Covers

Measurements taken on a historical dike in the Netherlands over one year showed that interaction with the atmosphere led to oscillation of the piezometric surface of about 0.7 m. The observation raised concerns about the long-term performance of similar dikes and promoted a deeper investigation of the response of the cover layer to increasing climatic stresses. An experimental and numerical study was undertaken, which included an investigation in the laboratory of the unsaturated behavior of a scaled replica of the field cover. A sample extracted from the top clayey layer in the dike was subjected to eight drying and wetting cycles in a HYPROP™ device. Data recorded during the test provide an indication of the delayed response with depth during evaporation and infiltration. The measurements taken during this continuous dynamic process were simulated by means of a finite element discretization of the time-dependent coupled thermohydraulic response. The results of the numerical simulations are affected by the way in which the environmental loads are translated into numerical boundary conditions. Here, it was chosen to model drying considering only the transport of water vapor after equilibrium with the room atmosphere, while water in the liquid phase was added upon wetting. The simulation was able to reproduce the water mass balance exchange observed during four complete drying–wetting cycles, although the simulated drying rate was faster than the observed one. The numerical curves describing suction, the amount of vapor and temperature are identical, confirming that vapor generation and its equilibrium is control the hydraulic response of the material. Vapor generation and diffusion depend on temperature; therefore, correct characterization of the thermal properties of the soil is of paramount importance when dealing with evaporation and related non-steady equilibrium states.

Adiposity and Mineral Balance in Chronic Kidney Disease.

Bone homeostasis is balanced between formation and resorption activities and remain in relative equilibrium. Under disease states this process is disrupted, favoring more resorption over formation, leading to significant bone loss and fracture incidence. This aspect is a hallmark for patients with chronic kidney disease mineral and bone disorder (CKD-MBD) affecting a significant portion of the population, both in the United States and worldwide. Further study into the underlying effects of the uremic microenvironment within bone during CKD-MBD are critical as fracture incidence in this patient population not only leads to increased morbidity, but also increased mortality. Lack of bone homeostasis also leads to mineral imbalance contributing to cardiovascular calcifications. One area understudied is the possible involvement of bone marrow adipose tissue (BMAT) during the progression of CKD-MBD. BMAT accumulation is found during aging and in several disease states, some of which overlap as CKD etiologies. Importantly, research has found presence of BMAT inversely correlates with bone density and volume. Understanding the underlying molecular mechanisms for BMAT formation and accumulation during CKD-MBD may offer a potential therapeutic avenue to improve bone homeostasis and ultimately mineral metabolism.

Ultrashort Oncologic Whole-Body [18F]FDG Patlak Imaging Using LAFOV PET.

Methods to shorten [18F]FDG Patlak PET imaging procedures ranging from 65-90 to 20-30 min after injection, using a population-averaged input function (PIF) scaled to patient-specific image-derived input function (IDIF) values, were recently evaluated. The aim of the present study was to explore the feasibility of ultrashort 10-min [18F]FDG Patlak imaging at 55-65 min after injection using a PIF combined with direct Patlak reconstructions to provide reliable quantitative accuracy of lung tumor uptake, compared with a full-duration 65-min acquisition using an IDIF. Methods: Patients underwent a 65-min dynamic PET acquisition on a long-axial-field-of-view (LAFOV) Biograph Vision Quadra PET/CT scanner. Subsequently, direct Patlak reconstructions and image-based (with reconstructed dynamic images) Patlak analyses were performed using both the IDIF (time to relative kinetic equilibrium between blood and tissue concentration (t*) = 30 min) and a scaled PIF at 30-60 min after injection. Next, direct Patlak reconstructions were performed on the system console using only the last 10 min of the acquisition, that is, from 55 to 65 min after injection, and a scaled PIF using maximum crystal ring difference settings of both 85 and 322. Tumor lesion and healthy-tissue uptake was quantified and compared between the differently obtained parametric images to assess quantitative accuracy. Results: Good agreement was obtained between direct- and image-based Patlak analyses using the IDIF (t* = 30 min) and scaled PIF at 30-60 min after injection, performed using the different approaches, with no more than 8.8% deviation in tumor influx rate value (Ki ) (mean difference ranging from -0.0022 to 0.0018 mL/[min × g]). When direct Patlak reconstruction was performed on the system console, excellent agreement was found between the use of a scaled PIF at 30-60 min after injection versus 55-65 min after injection, with 2.4% deviation in tumor Ki (median difference, -0.0018 mL/[min × g]; range, -0.0047 to 0.0036 mL/[min × g]). For different maximum crystal ring difference settings using the scan time interval of 55-65 min after injection, only a 0.5% difference (median difference, 0.0000 mL/[min × g]; range, -0.0004 to 0.0013 mL/[min × g]) in tumor Ki was found. Conclusion: Ultrashort whole-body [18F]FDG Patlak imaging is feasible on an LAFOV Biograph Vision Quadra PET/CT system without loss of quantitative accuracy to assess lung tumor uptake compared with a full-duration 65-min acquisition. The ultrashort 10-min direct Patlak reconstruction with PIF allows for its implementation in clinical practice.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Sediment dynamics of a major piedmont glacier: the Salzach Glacier in the North Alpine Foreland

Abstract. The foreland basins of the European Alps were repeatedly covered by major piedmont glaciers during Quaternary glacial maxima. The Salzach Glacier was the easternmost of a series of large Pleistocene piedmont glaciers entering the North Alpine Foreland through major Alpine valleys. It covered an area of more than 1000 km2 during at least four glacial maxima. The slightly elevated molasse bedrock provides a high potential for preservation of glacial landforms and offers an ideal setting to explore the erosional and depositional dynamics of a major glacier piedmont lobe. This field excursion guide leads to some selected key sites representing deposits of glacial advance, relative equilibrium, and ice lobe collapse.

The Integral Manifolds of the 4 Body Problem with Equal Masses: Bifurcations at Relative Equilibria

The Integral Manifolds of the 4 Body Problem with Equal Masses: Bifurcations at Relative Equilibria

Distributed control of spacecraft formation under [formula omitted] perturbation in the port-Hamiltonian framework

To control the relative position and relative velocity of spacecraft formation, a time-varying nonlinear relative motion model under J2 perturbation in an elliptical orbit is established in the port-Hamiltonian (PH) framework, and a distributed control law for spacecraft formation is developed using the consensus algorithm for parameter estimation and the passivity-based control (PBC) method based on the state-error interconnection and damping assignment (IDA) technique. First, the influence of J2 perturbation on the potential energy and orbit parameters in the model is considered when the relative motion model is built in the PH frame, and the expression of the relative motion acceleration under J2 perturbation is given. Second, to solve the problem that the state of the chief spacecraft cannot be directly obtained from the deputy spacecraft under distributed communication, a consensus-based parameter estimation method is introduced, the estimated parameter of the chief spacecraft is applied to the relative motion model in the PH frame, and a state error model with the estimated parameters derived from the consistency algorithm is established. Then, after the stability analysis is conducted on the desired PH system containing the estimated parameter values, the desired Hamiltonian energy function with the estimated parameters is designed according to the time-varying errors of the relative equilibrium states of the deputy spacecraft and the errors between deputy spacecraft, and the distributed control law of spacecraft formation is derived based on the state-error IDA-PBC method. Finally, the expected relative motion trajectory designed based on the TH equation is used to simulate a spacecraft formation under J2 perturbation, and the results indicate that the spacecraft can quickly converge to the expected formation under the influence of the control law.

Transcriptomic analysis reveals bovine herpesvirus 1 infection regulates innate immune response resulted in restricted viral replication in neuronal cells

BackgroundBovine herpesvirus 1 (BoHV-1) is a major pathogen that affects the global bovine population, primarily inducing respiratory and reproductive disorders. Its ability to establish latent infections in neuronal cells and to reactivate under certain conditions poses a continual threat to uninfected hosts. In this study, we aimed to analyze the replication characteristics of BoHV-1 in neuronal cells, as well as the effects of viral replication on host cell immunity and physiology. MethodsUsing the Neuro-2a neuronal-origin cell line as a model, we explored the dynamics of BoHV-1 replication and analyzed differential gene expression profiles post-BoHV-1 infection using high-throughput RNA sequencing. ResultsBoHV-1 demonstrated restricted replication in Neuro-2a cells. BoHV-1 induced apoptotic pathways and enhanced the transcription of interferon-stimulated genes and interferon regulatory factors while suppressing the complement cascade in Neuro-2a cells. ConclusionsDifferent from BoHV-1 infection in other non-highly differentiated somatic cells result in viral dominance, BoHV-1 regulated the innate immune response in neuronal cells formed a “virus-nerve cell” relative equilibrium state, which may account for the restricted replication of BoHV-1 in neuronal cells, leading to a latent infection. These findings provide a foundation for further research into the mechanism underlying BoHV-1-induced latent infection in nerve cells.

К оценке геодинамической устойчивости геосфер Земли

The Earth, as a system, is characterized by certain average state parameters (density, temperature). These parameters cannot but depend on similar states of internal geospheres (subsystems), which follow the unified law of increasing entropy of the cooling Earth from the center in the direction of the Earth's radius. As a model for assessing the dynamic stability of the Earth's internal geospheres relative to each other, the principle of the golden proportion (golden section) was used with the given parameters of density, temperature, power of the geospheres to the extreme ratios of their states at points (0.618; 1.618; 2.618), which determine the conditions for the stable dynamic equilibrium of the compared parameters of geospheres. And if these relations satisfied the known theoretical, model, geophysical data, then the conclusion was made that the geospheres in the compared parameter relations were in a state close to a stable dynamic equilibrium of the exchange of matter and energy between them. Otherwise, it was recognized that the compared geospheres of the Earth were in a state of either unstable dynamic equilibrium with an assessment of the level of this deviation from the golden numbers, or the known values of the density and temperature of the geospheres, as well as their structure, should be subject to clarification. Average model estimates of the parameters of the Earth's internal geospheres reach a stable dynamic equilibrium at a level of deviation from the golden numbers of up to 2.72 %. This allows estimating the model duration of the modern geological activity of the Earth for another 124 million years until the formation of a new state of the existing continents. The instability of the dynamic equilibrium of density and temperature of matter is established at the boundaries: continental and oceanic crust; oceanic crust and Golitsin layer; at the boundary between the outer part of the Earth's core and the lower mantle; inside the Earth's core at the level of identified transition zones in the four-layer core model. The presence of spheres with a high core density from 16.05 to 26.45 g/cm3 suggests the stratification of its central part into essentially gold and platinoid. The presence of instability boundaries in the dynamic state of geospheres is in good agreement with the concept of plate tectonics and with the depths of plume origin. It can be assumed that the relative equilibrium of the internal structure of the Earth will continue for about 124 million years until the beginning of the formation of a new Pangea Proxima, which, according to Cristofor Scotese, will arise in 200 million years.

Relative equilibria of mechanical systems with rotational symmetry

We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class which we call normal, and a non-generic abnormal class. The eigenvalues of the locked inertia tensor descend to shape-space and endow it with the geometric structure of a three-web with the property that any normal relative equilibrium occurs as a critical point of the potential restricted to a leaf from the web. To demonstrate the utility of this web structure we show how the spherical three-body problem gives rise to a web of Cayley cubics on the three-sphere, and use this to fully classify the relative equilibria for the case of equal masses.

M6A Methylation of Transcription Leader Sequence of SARS-CoV-2 Impacts Discontinuous Transcription of Subgenomic mRNAs.

The SARS-CoV-2 genome has been shown to be m6A methylated at several positions in vivo. Strikingly, a DRACH motif, the recognition motif for adenosine methylation, resides in the core of the transcriptional regulatory leader sequence (TRS-L) at position A74, which is highly conserved and essential for viral discontinuous transcription. Methylation at position A74 correlates with viral pathogenicity. Discontinuous transcription produces a set of subgenomic mRNAs that function as templates for translation of all structural and accessory proteins. A74 is base-paired in the short stem-loop structure 5'SL3 that opens during discontinuous transcription to form long-range RNA-RNA interactions with nascent (-)-strand transcripts at complementary TRS-body sequences. A74 can be methylated by the human METTL3/METTL14 complex in vitro. Here, we investigate its impact on the structural stability of 5'SL3 and the long-range TRS-leader:TRS-body duplex formation necessary for synthesis of subgenomic mRNAs of all four viral structural proteins. Methylation uniformly destabilizes 5'SL3 and long-range duplexes and alters their relative equilibrium populations, suggesting that the m6A74 modification acts as a regulator for the abundance of viral structural proteins due to this destabilization.

On the computation of the gradient in implicit neural networks

Implicit neural networks and the related deep equilibrium models are investigated. To train these networks, the gradient of the corresponding loss function should be computed. Bypassing the implicit function theorem, we develop an explicit representation of this quantity, which leads to an easily accessible computational algorithm. The theoretical findings are also supported by numerical simulations.

Assessment of cephalometric parameters and correlation with the severity of the obstructive sleep apnea syndrome.

In individuals diagnosed with obstructive sleep apnea syndrome (OSAS), variations in craniofacial structure have been inconsistently documented, showing differing degrees of alteration between obese and nonobese patients. In addition, sleep disturbance has also been shown to induce disequilibrium in this population of patients. This pilot observational study aimed to assess craniofacial values in obese and nonobese subpopulations of patients with OSAS and their correlation and association with the severity of OSAS. We also assessed whether OSAS patients are characterized by an impaired equilibrium in relation to and associated with the severity of OSAS. We included all consecutive adult patients with OSAS. Through cephalometry, we assessed the upper (UPa-UPp) and lower (LPa-LPp) pharynx diameters, superior anterior facial height (Sor-ANS), anterior facial height (ANS-Me), anterior vertical dimension (Sor-Me), posterior facial height (S-Go) and craniovertebral angle (CVA). Furthermore, we analyzed postural equilibrium through a stabilometric examination. Forty consecutive OSAS patients (45% female with a mean age of 56 ± 8.2years) were included. The subgroup of nonobese patients had a reduced UPa-UPp (p = 0.02). Cephalometric measurements were correlated with the severity of OSAS in nonobese patients, whereas only Sor-ANS was correlated with the severity of OSAS in the obese subpopulation. In the overall population, altered craniofacial values are associated with severe OSAS. Although there are differences in equilibrium between obese and nonobese OSAS patients, the stabilometric measurements were not correlated or associated with OSAS severity. Altered craniofacial values and compromised equilibrium in OSAS patients are linked to OSAS severity. Therefore, the management of OSAS should be tailored not only to weight management but also to craniofacial and postural rehabilitation to enhance patient outcomes.

Theoretical investigation on action mechanism and mollifying efficacy of propellant stabilizers.

This project performed quantum chemical computation, through kinetic and thermodynamic analyses to compare relative reactivity, reaction rate, and equilibrium composition from the possible pathways in connection with stabilizer-nitrodioxide reactions to determine the stability of the materials for practical application. Corresponding achievements have promoted the use of N-methyl-p-nitroaniline (MNA) and dinitrophenyl malonamide series (M3, M4, and M5) stabilizers as high priorities for selection. The Gaussian 09 program (G09) (Frisch et al 2009) and density functional theory (DFT) calculations with the B3LYP/6-31G(d,p) function were performed to obtain related geometric and thermodynamic energy data for the molecular systems in this study. The synchronous transit-guided quasi-Newton method (STQN) (Peng and Schlegel Isr J Chem 33:49, 1993) was applied through the QST3 procedure to identify single imaginary frequency-valued transition-state species. The related reaction rate constant (k) and pre-exponential factor (A) were obtained, based on transition state theory (Su 2008), using Eqs. 11 and 12.

On the motion stabilization of a three-link robotic manipulator with incomplete measurement

Abstract. This paper considers a mathematical model of a manipulator which consists of a vertical column, two links, connected to it in series, and a gripper with a load. The column resting on a fixed base can rotate around its vertical axis. The links are connected by cylindrical hinges allowing them to rotate in the same vertical plane. The column and the links are modeled as rigid bodies with the links having unequal principal moments of inertia. The position of the manipulator in space is determined by three rotation angles of the column and the links. The manipulator can have several types of steady-state program movements. When gravitational torques are compensated by control torques applied in the cylindrical hinges, the manipulator has a program equilibrium position. The manipulator can also have a program motion when the column rotates at a given constant angular velocity, and the links have given relative equilibrium positions in their plane. The stabilization problem of manipulator motion is investigated by means of control torques with feedback when only the rotation angles of the column and links are measured. The problem posed is solved in the form of a nonlinear proportional-integral controller taking into account the cylindrical phase space of the manipulator's mathematical model. The solution includes construction of a Lyapunov functional with a semi-definite derivative and application of the corresponding theorems on the asymptotic stability of non-autonomous functional differential retarded-type equations. The obtained conditions for the program motion stabilization are robust with respect to the mass-inertial parameters of the manipulator. The numerical simulation results demonstrate global attraction to its given position in cylindrical phase space.

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

On the linear stability of the Lamb–Chaplygin dipole

The Lamb–Chaplygin dipole (Lamb, Hydrodynamics, 2nd edn, 1895, Cambridge University Press; Lamb, Hydrodynamics, 3rd edn, 1906, Cambridge University Press; Chaplygin, Trudy Otd. Fiz. Nauk Imper. Mosk. Obshch. Lyub. Estest., vol. 11, 1903, pp. 11–14) is one of the few closed-form relative equilibrium solutions of the two-dimensional (2-D) Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2-D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time integration of the 2-D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2-D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

Cost-effectiveness analysis of amivantamab plus chemotherapy versus chemotherapy alone in NSCLC with EGFR Exon 20 insertions.

Amivantamab plus chemotherapy has been proved to be an efficient treatment strategy for non-small-cell lung cancer (NSCLC) with epidermal growth factor receptor (EGFR) exon 20 insertions. The aim of this study was to conduct the cost-effectiveness analysis of amivantamab-chemotherapy compared with chemotherapy alone in NSCLC harboring EGFR exon 20 insertion mutations. We constructed a Markov model based on the data derived from the PAPILLON trial. We evaluated the cost, quality-adjusted life years (QALYs), and incremental cost-effectiveness ratio (ICER). One-way and probabilistic sensitivity analyses were used to evaluate the influence of different parameters on this model. Compared with chemotherapy alone, amivantamab combined with chemotherapy treatment gained an incremental effectiveness of 0.473 QALYs and an incremental cost of $361,950.952, which resulted in an ICER of $765,224/QALY. The ICER was much higher than the willingness-to-pay threshold of 15,0000/QALY. One-way sensitivity analysis revealed that amivantamab cost was the leading influential factor in the model. Compared with chemotherapy alone, amivantamab plus chemotherapy is not a cost-effective first-line treatment choice for NSCLC patients with EGFR exon 20 insertions. The costly price of amivantamab is one of the major reasons for the high cost of this combined treatment strategy. Therefore, it is imperative to take into account the high cost of amivantamab in the subsequent clinical application and strive to attain a relative equilibrium between its significant clinical benefit and economic encumbrance.

Effect of Microclimate on the Mass Emergence of Hypothenemus hampei in Coffee Grown under Shade of Trees and in Full Sun Exposure.

The rainfall regime has a significant impact on the microclimate and mass emergence of the coffee berry borer, Hypothenemus hampei (Coleoptera: Curculionidae) (CBB). Little is known, however, about the shade tree-microclimate-CBB mass emergence interaction. The objective of the present study was to compare the effect of microclimate on the mass emergence of CBB in a full sun-exposed plot with a plot shaded by trees. The experiment was conducted on a Robusta coffee farm in southern Chiapas, Mexico. In each plot, 18 traps baited with an alcohol mixture were installed to capture flying females, collecting caught individuals every hour from 8:00 to 18:00 h. A meteorological station recorded several microclimatic variables on 13 weekly sampling dates from February to May 2022. Significantly more CBB females were captured in the shaded plot. The largest number of CBB captures was recorded between 14:00 and 16:00 h for the shade plot and between 15:00 and 17:00 h for the sun-exposed plot. The mass emergence of CBB showed a positive association with precipitation, dew point, and wind speed samples and a negative association with maximum air temperature, average relative humidity, ultraviolet radiation, wind speed, and equilibrium moisture content. Our observations show that the relationship between shade trees, microclimate, and mass emergence of CBB is complex and that its study helps us to gain deeper insight into CBB bioecology and advance control techniques against this important pest.

Learning-Based Reconfiguration of Charged Spacecraft Formation in Geomagnetic Field.

This article introduces a novel approach for spacecraft formation flying utilizing Lorentz-augmented techniques. It demonstrates that the relative motion among spacecraft, driven by the Lorentz force, possesses equilibrium states beneficial for formation maintenance. However, for effective formation reconfiguration, reliance solely on the Lorentz force is insufficient; low thrust is also necessary. To address this, this article proposes an optimal control framework based on reinforcement learning (RL). It derives the nonlinear dynamics of relative motion within the geomagnetic field, considering intersatellite Lorentz force, atmospheric drag, and Earth's gravitational harmonics. The study employs Lagrangian coherent structure analysis to identify relative equilibrium configurations and develops an RL-based optimal control strategy for real-time formation reconfiguration. By leveraging optimal demonstrations, the framework guides the agent's actions to match these demonstrations over time, especially when encountering out-of-distribution states. Numerical simulations confirm the method's optimality, robustness, and real-time performance, highlighting its potential in achieving optimal control and adapting to varying environment in future space missions.