Smoothing for time-varying systems using measurements containing colored noise
Optimal smoother derived for linear time-varying systems using measurements containing colored noise by means of calculus of variations
LINEAR SYSTEM↗
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Results for “Calculus of variations and optimization”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Optimum Interplanetary Rendezvous Trajectories With Powerlimited Vehicles
The optimum-thrust equations for both variable and constant thrust are presented. These thrust programs are used to generate rendezvous trajectories from the Earth to Mars for various flight times and launch dates during the years 1968-71. The manner in which the propulsion requirements vary with flight time and launch date are considered, and a comparison of vehicle performance using the variable- and constant-thrust programs is presented. The optimization of the pro- pulsion system parameters is discussed, and the existence of optimum launch dates is interpreted in terms of certain transversality conditions derivable from the calculus of variations. A brief comparison of the advanced propulsion vehicle and the ballistic vehicle propulsion requirements is made for Earth-Mars rendezvous trajectories. An appendix considering the analytical basis for this work is included.
INTERPLANETARY TRAJECTORY↗
Three-dimensional Wings of Maximum Lift-to- Drag Ratio in Hypersonic Flow
Optimizing lift-to-drag ratio of slender, flat-top wing of given planform in hypersonic flow by using variational calculus methods
LIFT-DRAG RATIO↗
Optimal control of a variable spin speed CMG system for space vehicles
Many future NASA programs require very high accurate pointing stability. These pointing requirements are well beyond anything attempted to date. This paper suggests a control system which has the capability of meeting these requirements. An optimal control law for the suggested system is specified. However, since no direct method of solution is known for this complicated system, a computation technique using successive approximations is used to develop the required solution. The method of calculus of variations is applied for estimating the changes of index of performance as well as those constraints of inequality of state variables and terminal conditions. Thus, an algorithm is obtained by the steepest descent method and/or conjugate gradient method. Numerical examples are given to show the optimal controls.
Liu, T. C.↗
Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations
Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.
Ibrahim, A. H.↗
Primer Vector Optimization: Survey of Theory, New Analysis and Applications
In this paper, a summary of primer vector theory is presented. The applicability of primer vector theory is examined in an effort to understand when and why the theory can fail. For example, since the Calculus of Variations is based on "small" variations, singularities in the linearized (variational) equations of motion along the arcs must be taken into account. These singularities are a recurring problem in analyse that employ small variations. Two examples, the initialization of an orbit and a line of apsides rotation, are presented. Recommendations, future work, and the possible addition of other optimization techniques are also discussed.
Guzman, J. J.↗
Optimal continuous low-thrust trajectories for co-altitude rendezvous.
Low thrust trajectories for minimum time rendezvous between continuous thrust interceptor and passive target vehicle, comparing calculus of variations and steepest ascent analyses
Burrows, R. R.↗
A comparison of two closely-related approaches to aerodynamic design optimization
Two related methods for aerodynamic design optimization are compared. The methods, called the implicit gradient approach and the variational (or optimal control) approach, both attempt to obtain gradients necessary for numerical optimization at a cost significantly less than that of the usual black-box approach that employs finite difference gradients. While the two methods are seemingly quite different, they are shown to differ (essentially) in that the order of discretizing the continuous problem, and of applying calculus, is interchanged. Under certain circumstances, the two methods turn out to be identical. We explore the relationship between these methods by applying them to a model problem for duct flow that has many features in common with transonic flow over an airfoil. We find that the gradients computed by the variational method can sometimes be sufficiently inaccurate to cause the optimization to fail.
Shubin, G. R.↗
Solution of an optimal control lifting body entry problem by an improved method of perturbation functions
This paper presents a solution to a complex lifting reentry three-degree-of-freedom problem by using the calculus of variations to minimize the integral of the sum of the aerodynamics loads and heat rate input to the vehicle. The entry problem considered does not have state and/or control constraints along the trajectory. The calculus of variations method applied to this problem gives rise to a set of necessary conditions which are used to formulate a two point boundary value (TPBV) problem. This TPBV problem is then numerically solved by an improved method of perturbation functions (IMPF) using several starting co-state vectors. These vectors were chosen so that each one had a larger norm with respect to show how the envelope of convergence is significantly increased using this method and cases are presented to point this out.
Garcia, F., Jr.↗
Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications
Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.
Ibrahim, A. H.↗
Sufficient conditions for a local minimum of the Bolza problem with multiple terminal point constraints
Sufficient conditions for a weak relative minimum are derived for a form of the Bolza problem of variational calculus. The derivation ties together first-order and second-order conditions in a unified consistent manner, addresses controllability considerations in some detail, and clarifies some small inconsistencies in earlier work. The resulting second-order conditions of optimality involve the integration of fewer backward-sweep matrix elements than the standard conditions in the literature for problems with an unspecified final time. As a result, the backward-sweep matrices have the same general structure and dimensionality whether the final time is specified or free, with the terminal values of two of three sweep matrices being more complicated in the latter case.
Wood, Lincoln J.↗
Optimum single modal and bimodal buckling design of symmetric laminates
Variational calculus is used to determine the design that maximizes the resistance of classical symmetric laminates against buckling. The orientations of the constituent orthotropic laminae with respect to the principal axes of the laminate are the design variables. It is shown that the optimal design may not be a point of analyticity of the buckling load. Local analytic extrema are obtained from the design derivatives of the buckling load. Nonanalytic extrema occur whenever the buckling load is a repeated eigenvalue. A novel approach, using a directional design derivative, is employed to determine nonanalytic extrema. Specific examples are presented for biaxial buckling for several different boundary conditions.
Qian, B.↗
Optimum configurations for bangless sonic booms.
A number of optimization problems are posed and solved for supersonic aircraft flight subject to the condition that a shock wave appears only incipiently in the sonic boom signal at a given point. The principal result is one giving the maximum effective gross weight of an aircraft of given effective length under given flight conditions. The calculus of variations with inequality constraints is used, with the novel features of a non-local isoperimetric relation and of only an upper bound on a control variable.
Hayes, W. D.↗
Frequency optimization of repetitive lattice beam-like structures using a continuum model
A new method for obtaining the maximum frequency design of a beam-like repetitive lattice structure is presented. Using existing techniques, the lattice is first modeled as an equivalent anisotropic Timoshenko beam. The computation of the stiffness and inertial properties of the beam, determined by matching the strain and kinetic energies of the beam with those of the lattice, is facilitated by the repetitive nature of the lattice. The optimum design is obtained by maximizing Rayleigh's quotient using methods of variational calculus. For the problem selected, results show excellent agreement with those obtained by traditional finite-element methods. Moreover, unlike FE methods, cpu time is relatively unaffected by the size of the truss.
Reiss, Robert↗
Sufficient conditions for a local minimum of the Bolza problem with a scalar terminal point constraint
Sufficient conditions for a weak relative minimum for a form of the Bolza problem of variational calculus are derived. Testing of the second-order conditions requires the backward integration of fewer matrix elements than in the case of most previously published sets of conditions. The derivation is felt to be more complete and straightforward than previous derivations. The variational problem considered is relatively simple, with just a scalar constraint to implicitly or explicitly determine the final time, in order to avoid the complexities associated with controllability considerations. Sufficient conditions for a local minimum for more general optimal control problems can be approached by building upon the derivation and results presented here.
Wood, Lincoln J.↗
DUKSUP - A high thrust trajectory optimization code
Designing missions on expendable launch vehicles (ELV's) includes determining launch vehicle performance capabilities and trajectory characteristics over the range of mission requirements for suitable launch periods. This analysis depends on mathematically modeling both the launch vehicle and the mission requirements. Generally the result is a mathematical model that is described by an objective function to be optimized subject to an assortment of algebraic and dynamic constraints. About 30 years ago, engineers at Lewis Research Center (LeRC) undertook the task of creating a software code to solve 3D versions of such problems and based it on the Calculus of Variations/Optimal Control Theory. One of these codes, DUKSUP, has been in use at LeRC for nearly 25 years, during which time it has played an important role in a large number of studies and actual missions. Currently it is being used by about 12 analysts in the Center's Advanced Space Analysis Office (ASAO) to do mission design, feasibility studies, corroboration of contractor data and planning studies for the Space Exploration Initiative (SEI). Today, it is one of the few ELV mission analysis production codes based on variational methods. With future ELV missions in mind, ASAO is presently creating a new code to upgrade DUKSUP's capabilities.
Balkanyi, Leslie R.↗
Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments
In this work, we introduce an efficient method, TTN-HEOM, for exactly calculating the open quantum dynamics for driven quantum systems interacting with highly structured bosonic baths by combining the tree tensor network (TTN) decomposition scheme with the bexcitonic generalization of the numerically exact hierarchical equations of motion (HEOM). The method yields a series of quantum master equations for all core tensors in the TTN that efficiently and accurately capture the open quantum dynamics for non-Markovian environments to all orders in the system–bath interaction. These master equations are constructed based on the time-dependent Dirac–Frenkel variational principle, which isolates the optimal dynamics for the core tensors given the TTN ansatz. The dynamics converges to the HEOM when increasing the rank of the core tensors, a limit in which the TTN ansatz becomes exact. We introduce TENSO, tensor equations for non-Markovian structured open systems, as a general-purpose Python code to propagate the TTN-HEOM dynamics. We implement three general propagators for the coupled master equations: two fixed-rank methods that require a constant memory footprint during the dynamics and one adaptive-rank method with a variable memory footprint controlled by the target level of computational error. We exemplify the utility of these methods by simulating a two-level system coupled to a structured bath containing one Drude–Lorentz component and eight Brownian oscillators, which is beyond what can presently be computed using the standard HEOM. Our results show that the TTN-HEOM is capable of simulating both dephasing and relaxation dynamics of driven quantum systems interacting with structured baths, even those of chemical complexity, with an affordable computational cost.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗