Discretization methods for optimal control problems with state constraints

Chryssoverghi, I.; Coletsos, John; Kokkinis, B.

doi:10.1016/j.cam.2005.04.020

Cited by 18 publications

(7 citation statements)

References 7 publications

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“…Proposition A.1. The best piecewise linear approximation of a function Γ with nonzero constant curvature κ using three points is given by P T * I, 3 , where T * I, 3…”

Section: Discussionmentioning

confidence: 99%

Mesh Refinement Using Density Function for Solving Optimal Control Problems

Zhao

Tsiotras

2009

AIAA Infotech@Aerospace Conference

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This paper presents a general mesh refinement method based on a density function for distributing the grid points, which can be used for solving optimal control problems through a direct method. The algorithm captures any discontinuities and smoothness irregularities in the state and control variables with high resolution, thus helping with the convergence of the overall optimization algorithm. The algorithm also provides flexibility during the mesh refinement process, especially for problems with multiple control inputs. As a specific example of the general theory, a density function based on the best piecewise linear approximation of functions with curvature defined almost everywhere is proposed for refining the mesh. The technique is applied to solve the problem of the optimal powerlimited landing for a large commercial aircraft.

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“…Proposition A.1. The best piecewise linear approximation of a function Γ with nonzero constant curvature κ using three points is given by P T * I, 3 , where T * I, 3…”

Section: Discussionmentioning

confidence: 99%

Mesh Refinement Using Density Function for Solving Optimal Control Problems

Zhao

Tsiotras

2009

AIAA Infotech@Aerospace Conference

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“…Interconnections between direct and indirect methods were studied for some algorithms such as the covector mapping theorem of pseudospectral optimal control [17]. Without a thorough review of various direct methods, interested readers are referred to [2,6,10,12,13,18,14,16,17,26,25] and references therein. In [36], a direct method is applied to generate data for supervised learning.…”

Section: Direct Methods -Constrained Optimization Using Nonlinear Programmingmentioning

confidence: 99%

Algorithms of data generation for deep learning and feedback design: A survey

Kang

Gong

Nakamura-Zimmerer

et al. 2021

Physica D: Nonlinear Phenomena

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Recent research reveals that deep learning is an effective way of solving high dimensional Hamilton-Jacobi-Bellman equations. The resulting feedback control law in the form of a neural network is computationally efficient for real-time applications of optimal control. A critical part of this design method is to generate data for training the neural network and validating its accuracy. In this paper, we provide a survey of existing algorithms that can be used to generate data. All the algorithms surveyed in this paper are causality-free, i.e., the solution at a point is computed without using the value of the function at any other points. An illustrative example is given for the optimal feedback design using supervised learning in which the data is generated using causality-free algorithms.

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“…Many researchers have been interested to study the numerical solution of optimal control problems described by nonlinear elliptic PDEs [3,4], by semilinear parabolic PDEs [5][6][7], or by one dimensional linear hyperbolic PDEs with constant coefficients (LHPDES) [8]. The researchers also include two dimensional linear and nonlinear hyperbolic PDEs with constant coefficients [9][10][11][12], or by nonlinear ODEs [13]. These works attracted our attention to focus our interest on studying OCP described by LHPDES but with variable coefficients (LHBVPVC).…”

Section: Introductionmentioning

confidence: 99%

Mixed Implicit Galerkin â€“ Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

Al-Rawdanee

Al-Hawasy

2020

eijs

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This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.

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Discretization methods for optimal control problems with state constraints

Cited by 18 publications

References 7 publications

Mesh Refinement Using Density Function for Solving Optimal Control Problems

Mesh Refinement Using Density Function for Solving Optimal Control Problems

Algorithms of data generation for deep learning and feedback design: A survey

Mixed Implicit Galerkin â€“ Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

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