Multi-point Boundary Conditions Research Articles

Invariant imbedding and Riccati transformations

From its inception, the theory of invariant imbedding has been concerned with the study of various relations between the inputs and outputs of various physical processes. Where the processes could be modelled by differential or integro-differential equations, these ideas have led to the heuristic development of various functional relationships for the solutions of these equations. In this work, we show that for a general class of two point boundary value problems these relations can be obtained from mathematical arguments rather than physical ones. The principal result is the establishment of the equivalence of solving a family of two point boundary value problems and that of determining the existence of two transformations on the set of solutions of the given differential equations. We refer to these transformations as Riccati transformations. They are shown to be determined by a set of initial value problems which generalize the invariant imbedding equations obtained by previous authors. We work in the coordinate free setting of a Banach space. The usefulness of this approach is shown as we are able to readily extend our results to nonlocal and multipoint boundary conditions. An indication is made of how a similar theory applies to a class of problems for difference equations.

Orbit determination for a thrusted space vehicle

This paper is concerned with the problem of determining the time history of position, velocity and thrust acceleration of a space vehicle given only angular observations at discrete times and without prior knowledge of a nominal trajectory. It is shown that the motion, which involves unmeasurable state variables and partially unknown system dynamics, can be mathematically modeled by a system of ordinary differential equations subject to distributed boundary conditions. The resulting system is then solved by iteration with a modified quasilinearization technique. Results obtained from application of the method to a logarithmic spiral orbit will be examined. It will be shown that the method exhibits high precision and fast convergence even with poor estimates of the initial values for the state vector that is used to start the iterative process. ONTINUOUSLY thrusted interplanetary vehicles of the future will require new techniques for their guidance and navigation. At the present time much research has been done on trajectory analysis and optimal control for these missions, yet little work has been done on the critical problem of orbit determination. Jordan1 has shown that it is possible to solve the problem of determining small perturbations in the vehicle's position, velocity, and acceleration by applying filtering techniques to the data if one has prior knowledge of the vehicle's nominal trajectory. Apparently no one, however, has solved the more difficult problem of determining the completely unknown position, velocity, and thrust program of a powered vehicle from the types of observations which are now available; namely telescopic (angular observations) and radar (range and range-rate), without knowledge of a good approximate nominal. In this paper it will be shown that the motion of a thrusted vehicle, which involves unmeasurable state variables and partially unknown system dynamics can be mathematicall y modeled by a system of ordinary differential equations subject to multipoint boundary conditions. Prior knowledge of the vehicle's nominal trajectory is not necessary for a solution. Furthermore, the method of solution is not necessarily restricted to the class of problems where the thrust acceleration magnitude is small compared to the gravitational acceleration. The analytic method of differential approximation and the numerical technique of quasilineariz ation are the basic mathematical tools which will be used to solve the problem. First, the unknown portion of the system dynamics, that which describes the thrust program, is represented by the method of differential approximation which was originally introduced by Bellman 2 for the purpose of state estimation, and extended to the present application by Carpenter.3 Upon application of differential approximation the complete system dynamics can be described by a set of ordi