Optimal control of differential-algebraic equations from an ordinary differential equation perspective

The aim of this paper is to study the distribution of colours, { X n }, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process { X n } is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process { X n }.

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Urn models and differential algebraic equations

Journal of Applied Probability ◽

10.1239/jap/1053003552 ◽

2003 ◽

Vol 40 (2) ◽

pp. 401-412 ◽

Cited By ~ 8

Author(s):

I. Higueras ◽

J. Moler ◽

F. Plo ◽

M. San Miguel

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Random Environment ◽

Differential Algebraic Equations ◽

Algebraic Equations ◽

Urn Model ◽

Recurrent Equation ◽

Strong Laws ◽

The Stability ◽

Generalized Pólya Urn

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

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Numerical Treatment of Optimal Control Problems with Differential-Algebraic Equations and Applications to Virtual Test-drives

Progress in Industrial Mathematics at ECMI 2000 - Mathematics in Industry ◽

10.1007/978-3-662-04784-2_74 ◽

2002 ◽

pp. 549-553

Author(s):

M. Gerdts

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Differential Algebraic Equations ◽

Control Problems ◽

Numerical Treatment ◽

Algebraic Equations ◽

Virtual Test

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GEKKO Optimization Suite

Processes ◽

10.3390/pr6080106 ◽

2018 ◽

Vol 6 (8) ◽

pp. 106 ◽

Cited By ~ 27

Author(s):

Logan Beal ◽

Daniel Hill ◽

R. Martin ◽

John Hedengren

Keyword(s):

Optimal Control ◽

Optimization Problems ◽

Object Oriented ◽

Differential Algebraic Equations ◽

Mixed Integer ◽

Modeling Languages ◽

Algebraic Equations ◽

Simulation And Optimization ◽

Algebraic Modeling Languages ◽

Real Time Optimization

This paper introduces GEKKO as an optimization suite for Python. GEKKO specializes in dynamic optimization problems for mixed-integer, nonlinear, and differential algebraic equations (DAE) problems. By blending the approaches of typical algebraic modeling languages (AML) and optimal control packages, GEKKO greatly facilitates the development and application of tools such as nonlinear model predicative control (NMPC), real-time optimization (RTO), moving horizon estimation (MHE), and dynamic simulation. GEKKO is an object-oriented Python library that offers model construction, analysis tools, and visualization of simulation and optimization. In a single package, GEKKO provides model reduction, an object-oriented library for data reconciliation/model predictive control, and integrated problem construction/solution/visualization. This paper introduces the GEKKO Optimization Suite, presents GEKKO’s approach and unique place among AMLs and optimal control packages, and cites several examples of problems that are enabled by the GEKKO library.

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On the Existence of Optimal Controls

Journal of Basic Engineering ◽

10.1115/1.3657236 ◽

1962 ◽

Vol 84 (1) ◽

pp. 13-20 ◽

Cited By ~ 23

Author(s):

L. Markus ◽

E. B. Lee

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Optimal Control ◽

Optimal Controls ◽

Optimum Control ◽

Existence Of Optimal Controls ◽

Differential Equation Models ◽

Controlling Processes

The problem of existence of various types of optimum controls for controlling processes which are described by ordinary differential equation models is considered. The results presented enable one to test if there does exist an optimum control in the class of controls under consideration before proceeding to the construction of an optimal control.

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