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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
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.
We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as “the simple method”) optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. In conclusion, to outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.
Control optimization of guidance systems and trajectories
Variational calculus used to describe optimum aircraft flight trajectories