Conversion of Optimal Control Problems into Parameter Optimization Problems

1997 ◽  
Vol 20 (1) ◽  
pp. 57-60 ◽  
Author(s):  
David G. Hull
Keyword(s):  
Optimal Control ◽  
Parameter Optimization ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems

Shifted Chebyshev schemes for solving fractional optimal control problems

2019 ◽  
Vol 25 (15) ◽  
pp. 2143-2150 ◽  
Author(s):  
M Abdelhakem ◽  
H Moussa ◽  
D Baleanu ◽  
M El-Kady
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Numerical Examples ◽  
Fractional Optimal Control ◽  
Functional Approximation ◽  
Fractional Optimal Control Problems

Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes.

scholarly journals Solution to Singular Optimal Control by Backward Differential Flow

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Jinghao Zhu ◽  
Shangrui Zhao ◽  
Guohua Liu
Keyword(s):  
Optimal Control ◽  
Global Optimization ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Variational Problems ◽  
Equivalent Transformation ◽  
Control Problems ◽  
Singular Optimal Control ◽  
Differential Flow ◽  
The Cost

This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

scholarly journals Univariate Theory of Functional Connections Applied to Component Constraints

2021 ◽  
Vol 26 (1) ◽  
pp. 9
Author(s):  
Daniele Mortari ◽  
Roberto Furfaro
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Control Problems ◽  
Single Variable ◽  
Constrained Optimization Problems ◽  
Functional Connections ◽  
Mathematical Definitions ◽  
The Pontryagin Maximum Principle

This work presents a methodology to derive analytical functionals, with embedded linear constraints among the components of a vector (e.g., coordinates) that is a function a single variable (e.g., time). This work prepares the background necessary for the indirect solution of optimal control problems via the application of the Pontryagin Maximum Principle. The methodology presented is part of the univariate Theory of Functional Connections that has been developed to solve constrained optimization problems. To increase the clarity and practical aspects of the proposed method, the work is mostly presented via examples of applications rather than via rigorous mathematical definitions and proofs.

scholarly journals Applications of implicit parametrizations

2021 ◽  
Vol 66 (1) ◽  
pp. 5-15
Author(s):  
Dan Tiba
Keyword(s):  
Optimal Control ◽  
Nonlinear Programming ◽  
Shape Optimization ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Equality Constraints ◽  
Control Problems ◽  
Global Type ◽  
New Algorithms

We review several applications of the implicit parametrization theorem in optimization. In nonlinear programming, we discuss both new forms, with less multipliers, of the known optimality conditions, and new algorithms of global type. For optimal control problems, we analyze the case of mixed equality constraints and indicate an algorithm, while in shape optimization problems the emphasis is on the new penalization approach.

A Parameter Optimization Approach for the Optimal Control of Large-Scale Musculoskeletal Systems

1992 ◽  
Vol 114 (4) ◽  
pp. 450-460 ◽  
Author(s):  
M. G. Pandy ◽  
F. C. Anderson ◽  
D. G. Hull
Keyword(s):  
Optimal Control ◽  
Nonlinear Programming ◽  
Parameter Optimization ◽  
Large Scale ◽  
Optimization Problem ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Maximum Height ◽  
Optimization Approach ◽  
Point Boundary

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).

scholarly journals A Survey on Low-Thrust Trajectory Optimization Approaches

Aerospace ◽  
2021 ◽  
Vol 8 (3) ◽  
pp. 88
Author(s):  
David Morante ◽  
Manuel Sanjurjo Rivo ◽  
Manuel Soler
Keyword(s):  
Optimal Control ◽  
Objective Function ◽  
Trajectory Optimization ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Mathematical Framework ◽  
Low Thrust ◽  
Discrete State ◽  
Hybrid Optimal Control

In this paper, we provide a survey on available numerical approaches for solving low-thrust trajectory optimization problems. First, a general mathematical framework based on hybrid optimal control will be presented. This formulation and their elements, namely objective function, continuous and discrete state and controls, and discrete and continuous dynamics, will serve as a basis for discussion throughout the whole manuscript. Thereafter, solution approaches for classical continuous optimal control problems will be briefly introduced and their application to low-thrust trajectory optimization will be discussed. A special emphasis will be placed on the extension of the classical techniques to solve hybrid optimal control problems. Finally, an extensive review of traditional and state-of-the art methodologies and tools will be presented. They will be categorized regarding their solution approach, the objective function, the state variables, the dynamical model, and their application to planetocentric or interplanetary transfers.

scholarly journals Upper and Lower bounds for a certain class of constrained variational problems

1963 ◽  
Vol 3 (4) ◽  
pp. 449-453
Author(s):  
M. A. Hanson
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Constrained Variational Problems

Certain optimization problems involving inequality constraints, known as optimal control problems have been extensively studied during recent years especially in relation to the calculation of optimal rocket thrusts and trajectories. A summary of these works is given by Berkovitz [1] who also establishes necessary conditions for the existence of solutions for a wide class of such problems.

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