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Transformation of Lagrange multipliers and Euler equations to obtain control angle solutions by variational calculus
Processes and applications of calculus of variations in optimal control theory
Variational calculus methods used to maximize payload capability for multistage launch vehicles
Calculus of variation methods used for trajectory optimization and guidance of rocket propelled vehicles
Necessary conditions for optimal lunar trajectories with discontinuous state variables and intermediate point constraints
Guidance, flight mechanics, and trajectory optimization - calculus of variations and modern applications
The indirect method of the calculus of variations is used to optimize interplanetary round-trip trajectories for the case of a single, central, attracting body. The method of solution makes use of certain partial derivative properties of the Lagrangian multipliers associated with the Mayer formulation of the variational problem. This property of the multipliers allows the construction of mathematical expressions for certain other partial derivatives that must vanish when an optimum round trip has been found. These expressions are developed for the cases of propulsion systems using (1) fixed thrust and specific impulse or (2) variable thrust and constant exhaust jet power. Two numerical examples demonstrate how the analytical results may be applied to the solution of round-trip problems including (1) actual three-dimensional planetary positions and (2) planetocentric maneuvers.
Calculus of variations method used to maximize payload capability for multistage launch vehicles
Variational calculus, gradient methods, maximum principle, and dynamic programing methods of optimal control theory of space vehicles
Linear time optimal control problem and calculus of variations
Caratheodory unified approach to Hamilton-Jacobi theory in variational calculus problems of optimal control
Spacecraft guidance - multistage rocket trajectories
Definitions of extremaloids in relation to optimal control problems and Lagrange problems based on Pontryagin principles
Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrainted optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.
Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrained optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.
An optimal control problem with bounded state variables is transformed into a Lagrange problem by means of differentiable mappings which take some Euclidean space onto the control and state regions. Whereas all such mappings lead to a Lagrange problem, it is shown that only those which are defined as acceptable pairs of transformations are suitable in the sense that solutions to the transformed Lagrange problem will lead to solutions to the original bounded state problem and vice versa. In particular, an acceptable pair of transformations is exhibited for the case when the control and state regions are right parallelepipeds. Finally, a description of the necessary conditions for the bounded state problem which were obtained by this method is given.
An original technique for determining the optimal magnetic torque strategy for control of the attitude of spin stabilized spacecraft is presented. By employing Lagrange multipliers and the Calculus of Variations, optimal control equations are derived which define minimum time and minimum energy attitude maneuvers. Computer program algorithms to numerically solve these optimal control equations are also described. The performance of this technique is compared with a commonly employed planning method.
The detailed mission and system optimization of low thrust electric propulsion missions is a complex, iterative process involving interaction between orbital mechanics and system performance. Through the use of appropriate approximations, initial system optimization and analysis can be performed for a range of missions. The intent of these calculations is to provide system and mission designers with simple methods to assess system design without requiring access or detailed knowledge of numerical calculus of variations optimizations codes and methods. Approximations for the mission/system optimization of Earth orbital transfer and Mars mission have been derived. Analyses include the variation of thruster efficiency with specific impulse. Optimum specific impulse, payload fraction, and power/payload ratios are calculated. The accuracy of these methods is tested and found to be reasonable for initial scoping studies. Results of optimization for Space Exploration Initiative lunar cargo and Mars missions are presented for a range of power system and thruster options.