Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

  • Heinz H. Bauschke
  • Regina S. Burachik
  • C. Yalçın KayaEmail author


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.


Optimal control Dykstra projection method Douglas-Rachford method Aragón Artacho–Campoy algorithm Linear quadratic optimal control Control constraints Numerical methods 

AMS 2010 Subject Classification

49M27 65K10 90C20 


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Regina S. Burachik
    • 2
  • C. Yalçın Kaya
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.School of IT & Mathematical SciencesUniversity of South AustraliaMawson LakesAustralia
  3. 3.School of Information Technology and Mathematical SciencesUniversity of South AustraliaMawson Lakes, AdelaideAustralia

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