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Dynamics and Control Under State Constraints

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Dynamics and Control of Trajectory Tubes

Part of the book series: Systems & Control: Foundations & Applications ((SCFA,volume 85))

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Abstract

The topics of this chapter are problems of reachability and system dynamics under state constraints in the form of reach tubes. Indicated are general approaches based on the Hamiltonian formalism and a related Comparison Principle. Further emphasis is on the dynamics of linear systems under hard bounds on the controls and system trajectories. A detailed solution is presented based on ellipsoidal approximations of bounded trajectory tubes.

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Notes

  1. 1.

    The boundary \(\partial \mathcal{X}[\uptau ]\) of \(\mathcal{X}[\uptau ]\) may be defined as the set \(\partial \mathcal{X}[\uptau ] = \mathcal{X}[\uptau ]\setminus \mathrm{int}\mathcal{X}[\uptau ]\). Under the controllability assumption, \(\mathcal{X}[\uptau ]\) has a nonempty interior, \(\mathrm{int}\mathcal{X}[\uptau ]\not =\varnothing,\;\uptau> t_{0}\).

  2. 2.

    In the general case, under Assumption 7.2.1, the optimal trajectory may visit the smooth boundary of the state constraint for a countable set of closed intervals, and function L ∗ 0(⋅ ) allows not more than a countable set of discontinuities of the first order.

  3. 3.

    The same minimum value is also attained here in classes of functions broader than \(\mathcal{C}_{00}\).

  4. 4.

    The computer illustrations for this subsection were made by M.N. Kirilin.

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State (Lomonosov) University, Moscow, Russia

    Alexander B. Kurzhanski

  2. Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA

    Alexander B. Kurzhanski & Pravin Varaiya

Authors
  1. Alexander B. Kurzhanski
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  2. Pravin Varaiya
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Kurzhanski, A.B., Varaiya, P. (2014). Dynamics and Control Under State Constraints. In: Dynamics and Control of Trajectory Tubes. Systems & Control: Foundations & Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10277-1_7

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