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Convergence Rate of Proximal Inertial Algorithms Associated with Moreau Envelopes of Convex Functions Read chapter Read first chapter

2019 | OriginalPaper | Chapter

2. Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Authors : Heinz H. Bauschke, Regina S. Burachik, C. Yalçın Kaya

Published in: Splitting Algorithms, Modern Operator Theory, and Applications

Publisher: Springer International Publishing

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Abstract

We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra’s algorithm, (ii) the Douglas–Rachford (DR) method and (iii) the Aragón Artacho–Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.

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previous chapter Convergence Rate of Proximal Inertial Algorithms Associated with Moreau Envelopes of Convex Functions
next chapter Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets
Appendix
Available only for authorised users
Footnotes
1
In the general case, there is also an auxiliary sequence (p n) associated with A; however, because A is an affine subspace, it is not needed in our setting.
 
2
It appears that this constraint qualification is not easy to check in our setting.
 
3
Aragón Artacho and Campoy recommend α = 0.9 and β ∈ [0.7,  0.8]; see [3, End of Section 7].
 
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Metadata
Title
Constraint Splitting and Projection Methods for Optimal Control of Double Integrator
Authors
Heinz H. Bauschke
Regina S. Burachik
C. Yalçın Kaya
Copyright Year
2019
Book
Splitting Algorithms, Modern Operator Theory, and Applications

Print ISBN: 978-3-030-25938-9

Electronic ISBN: 978-3-030-25939-6

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-25939-6_2

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