The solution of an optimal control problem using the DCNLP (direct collocation with nonlinear programming) method yields an open-loop controller. A feedback type of controller would be preferable. For example, an optimal trajectory for an interceptor missile is useless if the target's trajectory has changed since the optimal trajectory was determined. A feedback controller that can accommodate small changes in the system, e.g., changes in initial conditions or changes in terminal conditions, is being designed. We hope to find the most efficient way to determine the feedback gain time histories using the discrete solution of the DCNLP problem.

The spectacular collision of the Shoemaker-Levy 9 asteriod with Jupiter in July 1994 was a dramatic reminder of the fact that Earth has experienced many similar events and will continue to do so. A consensus is developing that while the probability of a large asteroid or comet colliding with the Earth is low, the potential for destruction is immense and thus some resources should be devoted to threat detection and possible interdiction. In this work optimal (minimum-time) trajectories are determined for the interception of asteroids that pose a threat of collision with Earth. An impulsive-thrust escape from Earth is used initially to reduce flight time but is followed with continuous low-thrust propulsion because of the significant propellant mass advantages of electric propulsion.

Strategies for discretizing optimal control problems are under investigation in an effort to develop methods of generating efficient discretizations that provide accurate numerical results. These problems are governed by a set of first order, nonlinear differential equations with associated boundary conditions. The continuous problem is discretized and cast into the form of a nonlinear programming (NLP) problem. Numerical solutions to the governing equations are computed using nonlinear constraints based on integration formulas. Formulas with smaller truncation errors, in comparison to the Simpson quadrature rule that is commonly used, are being studied. The objective is to reduce the size of a given problem (i.e., the number of NLP parameters required) by the use of improved integrators. The method is being applied to the optimization of low-thrust spacecraft trajectories.

The problem of optimal control for atmospheric reentry is examined with the objective of minimizing the number and size of the controllers. For example, the trajectory of a future Space Shuttle-type vehicle might be determined in such a way as to simultaneously mimimize the fuel required for reentry and the size of the flight control surfaces such as ailerons, elevators, and rudder needed for the vehicle to maneuver in the atmosphere. The DCNLP (direct collocation with nonlinear programming) method will be used to convert the continuous problem into a discrete problem.

Use of spacecraft in the 50- to 100-kg range, launched as secondary payloads, has enabled nontraditional users to take advantage of space resources. Future missions may require the docking of microsats for replenishment of mission consumables or the inclusion of a secondary structure such as an antenna. Docking mechanisms for large spacecraft historically include avionics packages weighing in the hundreds of kilograms. Mass and size restrictions on microsats necessitate a novel approach. An automated docking scheme adhering to these restrictions is being developed and ground tested.

Control algorithms that detumble and reorient underactuated spacecraft are investigated. An underactuated spacecraft is defined as having control torques supplied by external thrusters about only two of the principal axes. Variable structure control is used to detumble the spacecraft.

A genetic algorithm is used to generate a near-optimal solution and requires only the initial and final orbits to be supplied. The orbit transfer is broken into segments and a thrust direction is encoded over each of the segments. Both minimum-time and fixed-time transfers are determined. A thrust/no-thrust variable is encoded to allow the trajectory to exhibit coast arcs. Also, constant and variable thrust propulsion systems are modeled.

Optimal impulsive trajectories in an inverse-square gravity field are determined using a genetic algorithm. Genetic algorithms are robust parameter search techniques based on the Darwinian concept of natural selection. The ability of the genetic algorithm to search over a large-parameter space aids in identifying globally minimizing solutions. The genetic algorithm selects the optimal number, location, magnitude, and direction of the impulses that will result in the least amount of propellant consumption. The total time allowed for the rendezvous is fixed. The solutions are compared with results obtained using primer vector theory.

A direct method based on differential inclusion concepts is being developed to compute minimum propellant orbit transfers using variable and constant thrust propulsion systems. This new approach removes explicit control dependence from the problem thereby reducing the dimension of the parameter space for the nonlinear programming problem. Also, when compared to other direct methods, fewer nonlinear constraints are required to represent the dynamics of the problem.

Application of the simulated annealing algorithm to constrained optimization problems is being investigated. The search strategy is chosen based on an analogy to growing a single crystal from a melt. Genetic algorithms may be considered also. Specific applications include spacecraft trajectory optimization.

Infinite-dimensional optimization problems can be solved using the calculus of variations or optimal control theory. A set of first order necessary conditions is satisfied to obtain extremal solutions. The use of second order necessary and sufficient conditions is being investigated for several problems of interest including optimal spacecraft trajectories. Satisfaction of the sufficient conditions guarantees that the extremal solution is in fact optimal. In some problems, the extremal solution contains a conjugate point, which makes the solution nonoptimal.

A subset of optimal power-limited trajectories exists for which the optimal thrust acceleration is a certain time-varying scalar multiple of the velocity vector. Motion is planar, but this solution exists in an arbitrary gravitational field. The behavior of certain characteristics such as the orbital angular momentum can be obtained in closed form for the thrusted trajectory. Generalizations of this result and applications are being investigated.

Radiation can be important in deflagrations when the mixture contains particles. The effects of radiation on the well-known thermal-diffusive (Turing) instabilities of plane flames are being explored numerically. There is both experimental evidence and physical reasoning which suggests that a temperature-dependent Planck length can lead to a non-Turing instability, so that a formulation is adopted that could capture such a phenomenon.

A nonpremixed flame-sheet with an edge defines an edge-flame. Unbounded edge-flames have been studied, and the edge-speed has been calculated as a function of the Damkohler number. Positive speeds (ignition waves) are possible and negative speeds (quenching waves) are possible depending on the relative values of the applied Damköhler number and the quenching Damköhler number of the underlying diffusion flame. Stability results show that edge pulsations can occur when one of the Lewis numbers is sufficiently large, and edge corrugations can occur when one of the Lewis numbers is sufficiently small.

A stream of air and a stream of fuel separated by a semi-infinite splitter plate mix downstream of the plate trailing edge and can support a flame. This flame consists of a stream-wise sheet, a diffusion flame, and an edge structure lying off the plate edge where premixing occurs. A simple model is being used to describe the steady solution and its stability. There is a minimum Damköhler number below which the flame is blown off. Experiment suggests that, for some mixtures, pulsating instabilities will precede blow-off, and this is being explored theoretically.

Nonpremixed flames held by the rim of a tube burner experience nonuniform flows in the Navier-Stokes region and the Goldstein wake downstream of the rim. The response of flames to such nonuniformities is being examined, with particular attention to blow-off. For some Damköhler numbers there can be three solutions, one of which is unstable. Which of the other two solutions is achieved depends on the ignition history.

It is known from experiment that a diffusion flame can be established in a tube with the plane of the flame perpendicular to the tube axis. Quenching effects at the tube wall create an annular dead space between the flame and the wall. This project is concerned with the nature of this dead space and its size. It has been shown that total flame extinction will occur at a Damköhler number that is significantly larger than the classical quenching Damköhler number for a one-dimensional diffusion flame.

This project is concerned with the effects of dust, inert or reactive, on the structure of a 1-D detonation wave. The particles are large enough that the temperatures and the velocities of the two phases are different, and this can lead to quenching. An essential ingredient of the steady structure is the location of the sonic point in the region of nonvanishing reaction, a situation different from the classical Chapman-Jouget detonation. A numerical strategy and an asymptotic strategy are being used.

Deposition of energy in an explosive mixture (by a spark, for example) will only lead to an explosion if the energy is sufficient. Knowledge of the sufficiency is important for safety reasons. A combined numerical/asymptotic strategy is being used to calculate minimum initiation energies for explosive mixtures that contain substantial amounts of dust.

Smoldering combustion is the slow propagation of heterogeneous reaction through a porous solid. Transition to gas-phase combustion can occur (flaming) and this is responsible for a large number of accidental fires. Most t heoretical efforts have been concerned with idealized 1-D configurations, quite different from the real creatures. This project is concerned with 2-D structures that travel at well-defined speeds (smolder waves). The disparate phase den sities lead to shallow waves, waves that are long compared to their depth, and this permits an asymptotic approach that defines simple numerical problems.

It has long been known that reaction-diffusion flame-sheets that arise in premixed combustion when the activation energy is large can be unstable. This instability is characterized by the small time and length scales of the sheet, and has nothing to do with the Turing instabilities of the Darrieus-Landau instability of the outer structure. The effects of the Lewis number on this instability have not previously been examined, and this project is concerned with filling this lacuna.

The study of turbulent premixed flames is complicated by large density variations and the flow field induced because of these variations when the flame is wrinkled by turbulence. This project is concerned with theoretical treatments of premixed flame analogs that have been examined, by others, in experiments, and do not have these difficulties. In these analogs, reactive fronts are generated in liquids. These fronts travel at well-defined speeds, as do flames, and their response to an applied turbulent flow can be studied.

Experimental research is being conducted in a very high pressure combustion facility to gain an understanding of the high-pressure (300 MPa) surface and gas phase chemistry responsible for solid particulate boron combustion for use as a fuel additive in propulsion devices and explosives. Boron has the potential for twice the volumetric energy release than that of typical hydrocarbon fuels, but the combustion processes are slower than most high-speed applications will allow. It is the intent of this research to investigate several mechanisms to enhance the burning rate of boron, so that the full potential of the energy release can be achieved within high-speed combustion and explosive devices.

The energetic reaction of mixtures of hydrogen fluoride and boron particles with oxygen, water vapor, and light hydrocarbons is being studied using a 12-m shock tube. Measurements of boron ignition delay and combustion (burn) time, as well as measurements by emission spectroscopy of the transient reactive species, in the range of 0.5 to 5 MPa and temperature ranging from 1800K to 3000K, will test various theories of chemical processes for boron fuel utilization.

New energetic solid propellants will contain metals such as aluminum, magnesium, and boron. Experiments in a high-pressure shock tube measuring boron ignition delay and combustion (burn) time, as well as measurements by emission spectroscopy of the transient reactive species, in the range 0.5 to 5 MPa and temperatures ranging from 1800° to 3000°, will test various theories of chemical processes for boron fuel utilization in propellant combustion gases. The work will primarily impact chemical kinetic theories of reaction pathways for such two-phase mixtures.