Existence and approximation of solutions of nonlinear boundary value problems

In chapter two, we establish new results for periodic solutions of some second order non-linear boundary value problems. We develop the upper and lower solutions method to show existence of solutions in the closed set defined by the well ordered lower and upper solutions. We develop the method of quasilinearization to approximate our problem by a sequence of solutions of linear problems that converges to the solution of the original problem quadratically. Finally, to show the applicability of our technique, we apply the theoretical results to a medical problem namely, a biomathematical model of blood flow in an intracranial aneurysm. In chapter three we study some nonlinear boundary value problems with nonlinear nonlocal three-point boundary conditions. We develop the method of upper and lower solutions to establish existence results. We show that our results hold for a wide range of nonlinear problems. We develop the method of quasilinearization and show that there exist monotone sequences of solutions of linear problems that converges to the unique solution of the nonlinear problems. We show that the sequences converge quadratically to the solutions of the problem in the C1 norm. We generalize the technique by introducing an auxiliary function to allow weaker hypotheses on the nonlinearity involved in the differential equations. In chapter four, we extend the results of chapter three to nonlinear problems with linear four point boundary conditions. We generalize previously existence results studied with constant lower and upper solutions. We show by an example that our results are more general. We develop the method of quasilinearization and its generalization for the four point problems which to the best of our knowledge is the first time the method has been applied to such problems. In chapter five, we extend the results to second order problems with nonlinear integral boundary conditions in two separate cases. In the first case we study the upper and lower solutions method and the generalized method of quasilinearization for the Integral boundary value problem with the nonlinearity independent of the derivative. While in the second case we show the nonlinearity to depend also on the first derivative. Finally, in chapter six, we study multiplicity results for three point nonlinear boundary value problems. We use the method of upper and lower solutions and degree arguments to show the existence of at least two solutions for certain range of a parameter r and no solution for other range of the parameter. We show by an example that our results are more general than the results studied previously. We also study existence of at least three solutions in the pressure of two lower and two upper solutions for some three-point boundary value problems. In one problem, we employ a condition weaker than the well known Nagumo condition which allows the nonlinearity f(t, x, x’) to grow faster than quadratically with respect to x’ in some cases

Use of the method of particular solutions in nonlinear, two-point boundary-value problems. Part 1 - Uncontrolled systems

Nonlinear two-point boundary-value problem solution by combined techniques of quasilinearization and method of particular solution

Dynamics and control of flexible spacecraft during and after slewing maneuvers

The dynamics and control of slewing maneuvers of NASA Spacecraft COntrol Laboratory Experiment (SCOLE) are analyzed. The control problem of slewing maneuvers of SCOLE is formulated in terms of an arbitrary maneuver about any given axis. The control system is developed for the combined problem of rigid-body slew maneuver and vibration suppression of the flexible appendage. The control problem formulation incorporates the nonlinear dynamical equations derived previously, and is expressed in terms of a two-point boundary value problem utilizing a quadratic type of performance index. The two-point boundary value problem is solved as a hierarchical control problem with the overall system being split in terms of two subsystems, namely the slewing of the entire assembly and the vibration suppression of the flexible antenna. The coupling variables between the two dynamical subsystems are identified and these two subsystems for control purposes are treated independently in parallel at the first level. Then the state-space trajectory of the combined problem is optimized at the second level

Some recent results in aerospace vehicle trajectory optimization techniques

Algorithms and computation techniques for solving trajectory optimization problem

Finite-thrust transfer in the two and three body problems Final report

Computer program for optimal finite thrust transfer between two orbit

Maneuver simulations of flexible spacecraft by solving TPBVP

The optimal control of large angle rapid maneuvers and vibrations of a Shuttle mast reflector system is considered. The nonlinear equations of motion are formulated by using Lagrange's formula, with the mast modeled as a continuous beam. The nonlinear terms in the equations come from the coupling between the angular velocities, the modal coordinates, and the modal rates. Pontryagin's Maximum Principle is applied to the slewing problem, to derive the necessary conditions for the optimal controls, which are bounded by given saturation levels. The resulting two point boundary value problem (TPBVP) is then solved by using the quasilinearization algorithm and the method of particular solutions. In the numerical simulations, the structural parameters and the control limits from the Spacecraft Control Lab Experiment (SCOLE) are used. In the 2-D case, only the motion in the plane of an Earth orbit or the single axis slewing motion is discussed. In the 3-D slewing, the mast is modeled as a continuous beam subjected to 3-D deformations. The numerical results for both the linearized system and the nonlinear system are presented to compare the differences in their time response

Identification of Systems

Quasilinearization for system identification and programming strategie

Quasilinearization and Boundary Value Problems at Resonance

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited