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Optimal Control of the Double Integrator with Minimum Total Variation

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Abstract

We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly 2/3 of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.

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References

  1. Wellstead, P.E.: Introduction to Physical System Modelling. Control Systems Principles. Academic Press, London (2000)

    Google Scholar 

  2. Dontchev, A.L.: Best interpolation in a strip. J. Approx. Theory 73, 334–342 (1993)

    Article  MathSciNet  Google Scholar 

  3. Opfer, G., Oberle, H.J.: The derivation of cubic splines with obstacles by methods of optimization and optimal control. Numer. Math. 52, 17–31 (1988)

    Article  MathSciNet  Google Scholar 

  4. Bauschke, H.H., Burachik, R.S., Kaya, C.Y.: Constraint splitting and projection methods for optimal control of double integrator. In: Bauschke, H.H., Burachik, R.S., Luke, D.R. (eds.) Splitting Algorithms, Monotone Operator Theory, and Applications, pp. 45–68. Springer, Cham (2019)

    Chapter  Google Scholar 

  5. Rao, V.G., Bernstein, D.S.: Naive control of the double integrator. IEEE Control Syst. Mag. 21, 86–97 (2001)

    Google Scholar 

  6. Teo, K.L., Jennings, L.S.: Optimal control with a cost on changing control. J. Optim. Theory Appl. 68, 33–357 (1991)

    Article  MathSciNet  Google Scholar 

  7. Loxton, R., Lin, Q., Teo, K.L.: Minimizing control variation in nonlinear optimal control. Automatica 49, 2652–2664 (2013)

    Article  MathSciNet  Google Scholar 

  8. Wang, Y., Yu, C., Teo, K.L.: A new computational strategy for optimal control problem with a cost on changing control. Numer. Algebra Control Optim. 6, 339–364 (2016)

    Article  MathSciNet  Google Scholar 

  9. Gong, D., Tan, M.K., Shi, Q.F., van den Hengel, A., Zhang, Y.N.: MPTV: Matching pursuit-based total variation minimization for image deconvolution. IEEE Trans. Image Proc. 28, 1851–1865 (2019)

    Article  MathSciNet  Google Scholar 

  10. Kostyukova, O.I., Kurdina, M.A.: Parametric identification problem with a regularizer in the form of the total variation of the control. Differ. Equ. (Differentsial’nye Uravneniya) 49, 1056–1068 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Silin, D.B.: Total variation of optimal control in linear systems. Mat. Zametki 31, 761–772 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Kaya, C.Y.: URL: https://arxiv.org/src/1901.00491v1/anc/mintotalvar.avi. Ancillary file mintotalvar.avi for the preprint arXiv:1901.00491v1 (2018)

  13. Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966)

    MATH  Google Scholar 

  14. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)

    MATH  Google Scholar 

  15. Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)

    MATH  Google Scholar 

  16. Bonnel, H., Kaya, C.Y.: Optimization over the efficient set of multi-objective convex optimal control problems. J. Optim. Theory Appl. 147, 93–112 (2010)

    Article  MathSciNet  Google Scholar 

  17. Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57, 685–702 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to offer his warm thanks to two anonymous reviewers, whose comments and suggestions improved the paper.

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Authors and Affiliations

  1. Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, SA, 5095, Australia

    C. Yalçın Kaya

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  1. C. Yalçın Kaya
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Correspondence to C. Yalçın Kaya.

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Communicated by Enrique Zuazua.

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Kaya, C.Y. Optimal Control of the Double Integrator with Minimum Total Variation. J Optim Theory Appl 185, 966–981 (2020). https://doi.org/10.1007/s10957-020-01671-4

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  • DOI: https://doi.org/10.1007/s10957-020-01671-4

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