Multiple Shooting Technique Research Articles

Numerical computation of optimal reduction of CO2—emissions for a simplified climate-economy model

In this paper a highly simplifed model is considered which describes the interaction of anthropogenic climate changes represented by the influences due to the enhanced emission of Co2 resulting in the increase of the averaged surface air temperature on one side, and the economical effects described by the abatement costs for the reduction of emissions on the other side The model is formulated as a linear-quadratic optimal control problem with a compact control region. By applying the standard necessary conditions, a multipoint-boundary-value problem is derived and its numerical solution obtained by multiple-shooting technique is presented. Special attention is paid to the computation of the reachable set of the system and to the dependence of the control structure on the final state prescribed In order to smooth a certain irregular behaviour of the solutions near the end of the arbitrarilly fixed time-interval, an additional monotonicity constraint for the CO2-concentration is introduced. Solutions of thi...

Improving the efficiency of the multiple shooting technique

In this paper, we consider the numerical solution of a boundary value problem (BVP) by the multiple shooting technique, using Runge-Kutta (RK) methods for the integration of the associated initial value problems (IVP). A special RK method, with minimum computational cost per step, is designed with the purpose of saving Jacobian evaluations in the integration of the variational equation. Finally, some numerical experiments are presented showing a substantial reduction in the number of Jacobian evaluations.

Trajectory optimization on a parallel processor

The design of a trajectory for an aerospace vehicle involves choosing a set of variables to optimally shape the path of the vehicle. Typically the trajectory is simulated by numerically solving the differential equations describing the dynamics of the vehicle. The optimal trajectory is usually determined by using a nonlinear programming (parameter optimization) algorithm to select the variables. Problems that require choosing control functions are usually reduced to choosing a finite set of parameters. The computational expense of a trajectory optimization is dominated by two factors: the cost of simulating a trajectory and the cost of computing gradient information for the optimization algorithm. This paper presents a technique for using a parallel processor to reduce the cost of these calculations. The trajectory is broken into phases, which can be simulated in parallel, thereby reducing the cost of an individual trajectory. This multiple shooting technique has been suggested by a number of authors. The nonlinear optimization problem that results from this formulation produces a Jacobian matrix that is sparse. The Jacobian is computed using sparse finite differencing, which is also performed in parallel, thereby reducing the cost of obtaining gradient information for the optimization algorithm. This paper describes the application of sparse finite differencing to a multiple shooting formulation of the two-point boundary-value problem, in a manner suitable for implementation on a parallel processor. Computational experience with the algorithm as implemented on the BBN GPIOOO (Butterfly) parallel processing computer is described.

The non-linear beam via optimal control with bounded state variables

The non-linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with jump conditions which are solved by multiple-shooting techniques. A branching analysis is performed which exhibits the different solution structures. In particular, the second bifurcation point is determined numerically. The bifurcation diagram reveals a hysteresis-like behaviour and explains the jumping to a different state at this bifurcation point.

Numerical computation of singular control functions in trajectory optimization problems

This paper considers the problem of maximizing the final velocity for a two-dimensional climb maneuver of a vehicle in atmospheric flight. The ascent of the vehicle is charaterized by a fixed initial position and a given propellant allocation. The aim is to determine the maneuver time, the throttle setting, the angle between thrust and velocity vector, and the angle of attack, such that a prescribed altitude is reached with zero path inclination and maximal final velocity. For this problem, the necessary conditions of optimal-contr ol theory are derived. It turns out that the throttle setting, which enters linearly into the state equations, in general consists of several bang-bang and singular subarcs. The problem is formulated as a boundary-value problem with switching conditions, which is solved numerically by multiple-shoo ting techniques. Solutions are presented for various values of the final altitude and various amounts of fuel. Special effort is made to give a survey of the structure of the solutions in dependence of these two parameters.

Rotating field current drive in spherical plasmas

The technique of driving a steady Hall current in plasmas using a rotating magnetic field is studied both numerically and analytically in the approximation of negligible ion flow. The authors consider a spherical plasma bounded by an insulating wall and immersed in a uniform magnetic field which has both a rotating component (for current drive) and a constant 'vertical' component (for MHD equilibrium). They formulate the problem in terms of an expansion of field quantities in vector spherical harmonics. The numerical code SPHERE solves the resulting pseudo-harmonic equations by a multiple shooting technique. The results presented, in addition to being relevant to non-inductive current drive generally, have a direct relevance to the rotamak experiments.

Bifurcation curves of the Hénon map determined by a multiple-shooting technique

A multiple shooting method combined with a Newton iteration for determining bifurcation curves of two-dimensional maps is presented. This method is so effective that the high period bifurcation curves which could not be estimated with any other methods can be determined. Some P8 bifurcation curves of the Henon map are calculated as an example.

The calculation of wave fields in a cylindrical plasma using a multiple shooting technique

A set of four, coupled, first-order ordinary linear differential equations is derived which gives, at low to ion cyclotron frequency, a realistic description of waves excited by an antenna which surrounds a cylindrical plasma that is non-uniform in the radial direction. The equations describe wave modes with very different wavelengths, and they have been solved using a multiple shooting technique, with automatic selection of the shooting points. The efficacy of this technique is discussed with particular application to the radio-frequency heating of plasmas using the spatial Alfvén resonance.

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.

An analytical study of the Chemical Vapor Deposition (CVD) processes in a rotating pedestal reactor

The epitaxial growth and fluid dynamics characteristics of pedestal Chemical Vapor Deposition (CVD) reactors were studied analytically, with emphasis on the effects of susceptor rotation and thermal diffusion on the epitaxial growth rate. The coupled transport equations were solved by numerical integration and a multiple shooting technique. The pressure, velocity, temperature and concentration distributions were calculated and studied for various rotation speeds and operating conditions. The ranges of the process parameters in actual CVD reactors were estimated. It is found that different combinations of the external forced flow and the susceptor rotation have similar temperature and concentration distributions for the Schmidt number around unity when the results are presented in terms of a properly normalized vertical coordinate. The changes in susceptor rotation and external forced flow have nearly identical effects on the epitaxial growth rate for the Schmidt numbers greater than unity. At small Schmidt numbers, the epitaxial growth rate can be enhanced by increasing the forced flow velocity.

The effects of excited-state diffusion and saturation on the strong field reflection and transmission properties of materials

We propose a method for calculating the effects of saturation of optical transitions (and other nonlinear polarization mechanisms), and the effects of particle diffusion, on the reflection and transmission of light through a medium when surface reflections are important. We use a multiple shooting technique to solve the two-point boundary value problem specified by the nonlinear wave equation for the electric field and the rate equation for the excited state population density of the medium, and their boundary conditions. The method can be used to describe both saturable absorption of organic dyes in a Fabry–Perot cavity and strong field reflection and transmission of semiconductors. As an illustrative example, the intensity dependence of the reflection and transmission from a slab of InSb is calculated.

Numerical Treatment of Minimax Control Problems by Multiple Shooting

Abstract This paper presents the application of multiple shooting technique to minimax optimal control problems. A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary value problem with switching conditions, which can be treated numerically by a special version of a multiple shooting algorithm.

Numerical treatment of minimax control problems by multiple shooting

This paper presents the application of multiple shooting technique to minimax optimal control problems. A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary value problem with switching conditions, which can be treated numerically by a special version of a multiple shooting algorithm.

The Conditioning of Linear Boundary Value Problems

Investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition. We derive a useful quantity to decide for well or ill conditioning. From this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem. It is shown that this quantity can fruitfully be used to explain why, e.g., the multiple shooting technique is stable (in contrast to the single shooting one) for certain well posed problems having solutions with different growth behavior. The results are sustained by a number of examples.

Numerical solution of an elastic boundary layer problem using a multiple shooting technique

A model is described for large axisymmetric deformations of a linearly elastic spherical shell compressed between two rigid plates. The deformation of the free region may be determined from a two-point boundary value problem, but the presence of a boundary layer makes the solution difficult by standard techniques. Solutions found by a multiple shooting method with some special features are presented here for a number of cases. Numerical evidence suggests that the onset of buckling in the contact region is controlled by a single nondimensional parameter.

Modelling and control of dissolved oxygen in an estuary

The two-dimensional spatial distribution of dissolved oxygen in Jamaica Bay, New York, is formulated as a multipoint boundary value problem. The problem, which includes the interactions involving the carbonaceous biochemical demand and the total unoxidized nitrogen, was solved by a multiple shooting technique. The resulting model is used to determine optimal economic nitrification treatment levels at each of the polluting sources discharging into the bay in order that a minimum dissolved oxygen concentration level of 5 mg 1 −1 is achieved throughout the bay.

Numerical solution of a kidney model by multiple shooting

The mathematical modeling of the renal counterflow system involves a two-point boundary-value problem for a system of nonlinear differential equations. In this paper the multiple shooting technique is used for the numerical solution of this problem. This method ensures highly precise and reliable computations. Special consideration is given to the treatment of the border between outer and inner medulla for various widths of the outer zone.

An efficient method for a finite-difference solution of the poisson equation on the surface of a sphere

An efficient shooting method is presented for the numerical solution of a discrete Poisson equation on the surface of the sphere. The solution is computed via two-dimensional shooting in the physical domain while the “missing initial conditions” needed to start the shooting are obtained in a one-dimensional setting in the Fourier domain. For a computational grid with J × J grid points, the operational count of this method is of O(( J 2/ m)log 2 J), where m is the number of grid points within each shooting subrange. The actual count is, for most practical purposes, less than 26 arithmetic operations per grid point. Stability of the method is a problem; this problem, however, can be overcome by the use of the multiple shooting technique. Numerical examples are given to demonstrate the applicability of the procedure.

A Direct Solution of the Spherical Harmonies Approximation to the Radiative Transfer Equation for an Arbitrary Solar Elevation. Part II: Results

Abstract The spherical harmonics approximation to the transfer equation for an azimuth-dependent component of intensity of scattered radiation is reduced by finite differences to a block algebraic system of particularly simple structure. This algebraic system can be solved numerically for homogenous or nonhomogenous models of a plane-parallel atmosphere, using the finite-differences analogue of the simple-shooting technique based on initial-value problems or the multiple-shooting technique for the solution of two-point boundary-value problems.

A Direct Solution of the Spherical Harmonics Approximation to the Radiative Transfer Equation for an Arbitrary Solar Elevation. Part I: Theory

The spherical harmonics approximation to the transfer equation for an azimuth-dependent component of intensity of scattered radiation is reduced by finite differences to a block algebraic system of particularly simple structure. This algebraic system can be solved numerically for homogenous or nonhomogenous models of a plane-parallel atmosphere, using the finite-differences analogue of the simple-shooting technique based on initial-value problems or the multiple-shooting technique for the solution of two-point boundary-value problems.