Abstract
We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows us to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton–Jacobi–Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.
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Acknowledgements
This work was partly supported by project 16-19526S of the Grant Agency of the Czech Republic, project ERC-ADG TAMING 666981,ERC-Advanced Grant of the European Research Council and grant number FA9550-15-1-0500 from the Air Force Office of Scientific Research, Air Force Material Command.
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Pauwels, E., Henrion, D., Lasserre, JB. (2017). Positivity Certificates in Optimal Control. In: Laumond, JP., Mansard, N., Lasserre, JB. (eds) Geometric and Numerical Foundations of Movements . Springer Tracts in Advanced Robotics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-51547-2_6
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DOI: https://doi.org/10.1007/978-3-319-51547-2_6
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