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Numerical modeling has been instrumental in studying the behavior of continuous wave (CW) erbium-doped ring cavity fiber laser (EDFRL). This can be obtained by providing a customized solution for a large parameter. In this paper, erbium-doped ring cavity fiber laser model is presented. The proposed model is obtained using a simple application of a fast-shooting algorithm combined with a secant-method. The performance results show that this algorithm is a flexible tool that can be used in modeling and simulation. However, the solution can also be implemented in other types of fiber lasers including, ytterbium, and thulium as well as co-doped fiber lasers

We consider circuits driven by two or more signals with very different periods and propose a method to determine their steady state solution in the time domain. We present a new version of the shooting method based on the envelope following technique. We show how to use the envelope following method as the new engine to efficiently determine the trajectory of the circuit starting from the new guess of the initial conditions. It substitutes the less efficient time domain analysis used in the conventional implementation of the shooting method. We show that it is well suited to circuits where the ratio between the periods of the slow and fast behaviour is particularly high and characterised by strong non-linearities. The numerical properties at the basis of the proposed method are presented. Its features are shown by simulating different types of slow-fast circuits.

In this work, numerical techniques based on Shooting procedure, Relaxation scheme and Collocation technique have been used for recovering the profile of the membrane of a 1D electrostatic Micro-Electro-Mechanical-Systems (MEMS) device whose analytic model considers |E| proportional to the membrane curvature. The comparison among these numerical techniques has put in evidence the pros and cons of each numerical procedure. Furthermore, useful convergence conditions which ensure the absence of ghost solutions, and a new condition of existence and uniqueness for the solution of the considered differential MEMS model, are obtained and discussed.

The paper addresses a problem of the TE-wave propagation in the Goubau line. The propagation problem is reduced to a transmission eigenvalue problem for an ordinary differential equation. Spectral parameters of the problem are propagation constants of the waveguide. For the determination of approximate eigenvalues a version of the shooting method is proposed specifically developed for this class of problems. The method allows one to numerically find propagating, evanescent, and complex surface and leaky waves. Numerical results illustrating the method are presented.

The rapid charging of lithium ion battery cells while minimizing degradation is a key challenge in battery management. Offline optimal control frameworks employing physics-based models provide a first-principles approach to this problem that provides critical insights and benchmarks for real-time oriented online optimization algorithms. In this work, we approach the optimal control problem of minimizing both charging time and degradation of a NMC battery using the single shooting optimization framework, employing a coupled electrochemical- thermal-aging model. The aging mechanism used captures growth of the SEI layer, solvent diffusion. The effectiveness of the single shooting method using a piecewise polynomial control and the low sensitivity of the method to chosen numerical parameters in the multi-objective problem is illustrated. The tradeoff between charging time and degradation is investigated numerically.

When applying visual servo algorithms for unmanned aerial vehicles (UAVs) using visual registration targets, it is often desirable that the UAV would retrieve targets after the mission. In this study we plan an energy-optimal trajectory for a UAV to retrieve visual registration targets dropped on the initial flight and then construct a controller to track this trajectory in a stable way. The UAV kinematics are described as a Hamiltonian system on the Lie group SE(3); the boundary value problem (BVP) arising from the geometric framing of Pontryagin's Maximum Principle (PMP) is then applied. This BVP is solved using a numerical shooting method with a novel multi-arc interpolation approach determining simple optimal arcs between the targets using a semi-analytical Lie group integrator. A switching controller compensating for the switch between in-ground and out-of-ground dynamics is combined with a disturbance observer to create a controller to track the trajectory in a stable way under the ground effect and bounded disturbances. Simulation results complete the work.

Nonlinear coupled electromagnetic surface TE and leaky TM wave propagation in a cylindrical nonlinear inhomogeneous metal-dielectric waveguide (Goubau line) with circular cross section is considered. Physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. For the numerical solution, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. The coupled TE-TM waves propagating in GL are determined numerically.

This paper presents a problem of minimizing an aircraft energy flow, the model is obtained from the fundamental principle of dynamics. The Pontryagin’s Maximum Principle (PMP) is applied to determine the optimal control, then the Shooting method coupled with the Relaxation method for the numerical resolution.

The work this paper describes identifies continuous natural relative trajectories near the collinear libration points numerically from the nonlinear CR3BP differential equations by utilizing a shooting method with a two-level differential corrector. The focus is on determining natural formations in the vicinity of the Earth-Moon $\mathrm{L}_{2}$ point when using the nonlinear equations of motion of the CR3BP. Upon identifying these relative trajectories analysis is conducted to determine drift rates of the uncontrolled trajectories compared to linearized formations near the collinear libration points. In addition, stationkeeping techniques are applied to determine the required maneuvers and frequencies and to assess $\triangle \mathcal{V}$ maintenance of the nominal trajectories.

An investigation is accomplished to scrutinize the outcome of energy activation on MHD flow of Prandtl Nano-liquid with chemical reaction. The governing PDEs are transmuted into ODEs using appropriate transformation, which are numerically solved via RK scheme along with shooting method. Graphical results explain the effect of velocity, temperature and nanoparticles concentration. Moreover, drag coefficient, Nusselt and Sherwood number are deliberated and offered via table. Results show that liquid velocity enhances by increasing Prandtl parameter as well as elastic parameter, whereas reverse activities are noted for temperature and nanoparticles concentration profiles.

Main objective for the orbit transfer problem is optimal minimization of time and fuel. In this research, 2-body system is represented by a nonlinear dynamic system. Theoretical analysis of Euler-Lagrange and Hamiltonian function is implemented for the application of optimal control. Certain assumptions are taken into account such as continuous thrust acceleration for the entire interval, unbounded control variables, and presence of certain state variables designated at terminal boundary conditions. These provide solution for the orbital transfer between 7000 km circular orbit and 8000 km circular orbit, which are coplanar in nature, by uniting a Linear Quadratic Regulator (LQR) and the Shooting method. For determining indirect optimal results about fuel or time, obtained results are compared with the Hohmann transfer. It is observed through the numerical simulations that the optimal theory applied coplanar rendezvous maneuver is optimally comparable to the minimum energy solution.

We consider propagation of leaky and surface TE waves in the rod filled with nonlinear metamaterial medium. The physical setting is reduced to a transmission eigenvalue problem for an ordinary differential equation. The eigenvalues of the problem are the wave propagation constants. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

In the present study, an electrically conducting Casson nanofluid flow along with heat and mass transport phenomenon towards the horizontally placed nonlinearly stretching sheet with the Joule heating, viscous dissipation and partial slip effects are examined. The mathematical formulation of the governing physical flow problem is carried out in Cartesian coordinate system. The nonlinear coupled differential equations are reduced to non-dimensional form by opting suitable transformations. Numerical procedure of shooting method is employed to attain approximate solutions. The effect of emerging physical parameters on velocity, temperature, and concentration fields are explored graphically. Moreover, the magnitude of shear stress, heat and mass fluxes are also computed at the surface. The key finding includes: prominent temperature drops for Casson nanofluid in comparison with nanofluid.

This work concerns a specific indirect method to solve an optimal control problem with state constraints in which the dynamics are driven by an ordinary differential equation involving a three-dimensional steady flow field. A moving object within this field is considered subject to a path constraint given by a cylinder. The given control system is linear, but the vector flow field may exhibit essential nonlinearity. An indirect numerical method is proposed based on the maximum principle in Gamkrelidze's form. This form of optimality conditions uses the concept of the extended Hamilton-Pontryagin function. The extended Hamilton-Pontryagin function differs from the conventional one in view of an additional term associated with the measure Lagrange multiplier. The proposed computational method essentially relies on the continuity of the measure multiplier, which is guaranteed under certain regularity conditions. Moreover, this property, together with the regularity condition, allows us to obtain an explicit expression for the measure multiplier. The application of the maximum principle yields a two-point boundary-value problem, which is solved by a variant of the shooting method.

In this work, we calculated the optimal heel angle trajectory of a rocket which have a constant mass and moves with a nonrectilinear motion from an initial point to a final one with a known altitude. The objective of the study is to maximize the lateral offset of the rocket. This problem is modelled as a nonlinear optimal control problem, where the control represents the heel angle of the rocket. In order to solve numerically the problem, we applied the shooting method which is based on the Pontryagin's maximum principle, then we developed an implementation with the MATLAB programming language. Finally, we presented simulation results which show the fast convergence and the accuracy of the shooting method.

We consider propagation of surface TE waves in a circular metal-dielectric waveguide filled with nonlinear (Kerr nonlinearity) metamaterial medium. Analysis is reduced to solving a nonlinear transmission eigenvalue problem for an ordinary differential equation; eigenvalues of the problem correspond to propagation constants of the waveguide. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

This paper presents an analytical expression that models the tilt angle of directors in a nematic liquid crystal (LC), depending on its elastic properties, its dielectric anisotropy, and the quasi-static electric field applied. The analytical solution obtained is fast and easily computable in comparison with numerical estimations and is of special interest in radiofrequency; for instance, for the LC modeling in full-wave electromagnetic simulators in the design process of electronically tunable devices, such as microwave phase shifters or electronically steerable antennas for satellite communications. Subsequently, a comparison is made between numerical approaches (self-implemented shooting method) and the analytical formulas when varying the parameters of the LC, being demonstrated its usefulness. The average LC director is then obtained and used to form the full permittivity tensor that completely characterizes the electrical properties of the material. Finally, an electromagnetic simulation is carried out to show the capabilities of the LC as a tunable phase shifter. It is shown that only 5 cm of a commercial 200-mm LC mixture is necessary to achieve 360 of the maximum variable phase shift at the 30-GHz band, This work was supported in part by the Spanish Research and Development National Program under Project TIN2016-75097-P, and in part by the Ministerio de Economía under Project TEC2017-85529-C3-1-R.

We consider propagation of surface TE waves in a circular metal-dielectric waveguide filled with nonlinear (Kerr nonlinearity) multilayered inhomogeneous medium. Analysis is reduced to solving a nonlinear transmission eigenvalue problem for an ordinary differential equation; eigenvalues of the problem correspond to propagation constants of the waveguide. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

In this work, we consider the diffraction of TM waves by a nonlinear inhomogeneous dielectric cylinder. Numerical experiments are carried out for the nonlinearity with saturation. The physical problem is reduced to solving a nonlinear boundary value problem for a system of ordinary differential equations. Numerical results are obtained using a modification of the shooting method which makes it possible to determine and plot the amplitude of the reflected field with respect to the amplitude of the incident field. Comparison between the nonlinear problem and the corresponding linear setting is performed.

The propagation of hybrid waves in a cylindrical anisotropic nonlinear metal-dielectric waveguide is considered. The physical setting is reduced to a transmission eigenvalue problem for a system of ordinary differential equations which is new type of nonlinear eigenvalue problem where spectral parameters are the wave propagation constants. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

Considering the problem of rapid solution of optimal ascent trajectory of multi-engined launch vehicle, a fast numerical solution algorithm is proposed. Firstly, the optimal ascent guidance problem of multi-engined launch vehicle subject to terminal orbital constraints is converted into nonlinear algebraic equations. During the specific implementation process of the algorithm, a simplified multiple shooting method and modified Newton algorithm is put forward to enhance the robustness of convergence in the internal layer, and then external layer uses line search method to iteratively solve the cut-off time of main engine to minimum the fuel consumption. Finally, simulation tests are conducted and results demonstrate the effectiveness and adaptability of the propose method.

The Pontryagin maximum principle is apllied to the Obukhov continuous hydrodynamic shell model in one space dimension restricted only to three shells. It is shown that many information can be extracted by simply reasoning with the conditions of the maximum principle, and, on the other hand, that this effort can become cumbersome. Thus, there are limits to what can be achieved in an efficient way in terms of computing the solution to the optimal control problem. The numerical computation of solutions to this optimal control problem based on a “brute-force” type of shooting method reveals the existence of switching points in the control.

International audience; An unmanned aerial vehicle (UAV) has grown rapidly the last years. They are being engaged in many types of missions, ranging from military to agriculture passing from entertainment and rescue or even delivery. This work consist to ensure the convergence of positions and yaw angle, to their desired trajectories while maintaining stability of roll and pitch angle. For this, formulate this problem to an optimal control problem which minimizes the distance between state and desired state in free final time. Our contribution is to use the Bocop software to solve this problem. And to implement Shooting method with Matlab software.

This paper introduces a real-time strategy to stabilize a falling humanoid robot by making hand contact with rails or walls in the environment. It uses an optimal control strategy with a simplified three-link model and finds an optimal hand contact using a direct shooting method. The objective function is designed to maintain stability while minimizing the probability of the contact slipping and minimizing impact that may damage the robot's arm. To achieve real-time performance, the method uses a precomputed database of necessary sticking friction coefficients at the contact points for all possible postimpact states. Validation is performed on a number of simulated falls in several rails and walls.

In the article a general simulation task of index-2 differential-algebraic equations (DAEs) is considered. To overcome difficulties connected with a nonlinear process dynamics, a new constraints type — the variability constraints — was introduced. The designed solution procedure is based on a multiple shooting method — it was assumed, that process dynamics in each shooting interval is constant. Therefore, the presented methodology can be treated as an extended direct shooting approach for simulation of the index-2 DAE equations with the variability constraints, which have an important practical application in various areas of applied informatics, e.g. manufacturing process modeling, simulation and optimization. Finally, the presented methodology was investigated on a two-phase reactor simulation task, which was described by the index-2 differential-algebraic equations.

In this work, we consider the diffraction of TM waves by an open metal-dielectric waveguide, a Goubau line (GL), with a nonlinear inhomogeneous dielectric cover. Numerical experiments are carried out for the nonlinearity with saturation. The physical problem is reduced to solving a nonlinear boundary value problem for a system of ordinary differential equations. Numerical results are obtained using a modification of the shooting method which makes it possible to determine and plot the amplitude of the reflected field with respect to the amplitude of the incident field. Comparison between the nonlinear problem and the corresponding linear setting is performed.

This paper investigates the problem of interplanetary transfer orbit optimization based on two intelligent optimization algorithms, particle swarm optimization (PSO) and biogeography-based optimization(BBO). In order to transfer from Earth orbit to Mars orbit, a continuous thrust model is established and normalized. The shooting method and the penalty function method are used to convert the small thrust space orbit problem into a nonlinear programming problem. By using two intelligent algorithms to optimize the Earth-Mars trajectory transfer problem, the feasibility of the two algorithms in the trajectory optimization problem is verified, and the optimization performance of the two algorithms is compared.

Attitude slew motions for spacecraft are usually undertaken using feedback control where only the desired final attitude is stated. In this paper attitude guidance is considered which could be used, in addition to feedback control, to enhance the efficiency of slew motions by pre-planning time-dependent attitude motions. This is achieved using a three-step method in which the angular velocities are expressed as analytic functions in terms of free parameters (on the virtual time domain), and the boundary conditions on the rotation are matched using a shooting method based on a discretized form of Rodrigue's formula. Following this, the virtual time is reparametrized. This is applied to design a rest-to-rest two-impulse slew manoeuver and a slew motion using only two reaction wheels.

In this paper a hierarchically structured framework is proposed for the time-optimal trajectory planning and control of autonomous electric vehicles with state and input constraints for fixed test paths. The time-optimal reference signals are obtained by solving a dynamic nonlinear optimal control problem associated with the kinematic model of the vehicle using direct discretization and multiple shooting method. The high-level control ensures that the deviations from the reference trajectories are minimized, while the low-level control computes the necessary torques required for the electric motors to produce the optimal motion. The efficiency of the control system is verified by simulation under realistic performance characteristics and accurate 16 degree-of-freedom nonlinear model of the electric vehicle.

We study the framework of the shooting method for nonlinear second-order ordinary differential equations with boundary conditions. We propose a new numerical scheme, based on the second degree F-transform for solving the corresponding initial value problem. We use the inverse F-transforms of the zero and first degrees to obtain approximate solutions to the initial second-order differential equation. We make experiments that demonstrate effectiveness of the proposed F-transform-based shooting method in the class of second order numerical methods.

The propagation of monochromatic nonlinear symmetric hybrid waves in a cylindrical nonlinear dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. Numerical results are found with the modification of the shooting method. The method allows us to determine approximate eigenvalues with any prescribed accuracy. The comparison with known exact solutions (for particular values of parameters) are made. The approach described in this paper can be applied to other problems, e.g., to multilayered inhomogeneous waveguides.

We consider propagation of surface TM waves in a circular dielectric waveguide filled with nonlinear (Kerr nonlinearity) multilayered inhomogeneous medium. Each layer is characterized by a specific value of the nonlinearity coefficient α. Analysis is reduced to solving a nonlinear transmission eigenvalue problem for an ordinary differential equation; eigenvalues of the problem correspond to propagation constants of the waveguide. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

The propagation of monochromatic electromagnetic waves in metal-dielectric waveguides of simple geometry (circular cylindrical) filled with nonlinear inhomogeneous medium is considered. The Kerr nonlinearity is studied. A physical problem is reduced to solving a nonlinear transmission eigenvalue problem for a system of ordinary differential equations. Eigenvalues of the problem correspond to propagation constants of the waveguide. A method is proposed for finding approximate eigenvalues of the nonlinear problem based on solving an auxiliary Cauchy problem (by the shooting method). The existence of eigenvalues that correspond to a new propagation regime is predicted. A comparison with the linear case is given.