Abstract
Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state \(q_i\) is subject to constraints \(q_i \in S_i \subseteq M\), for \(S_i\) a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.
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Notes
It can be shown through the implicit function theorem that for any compact neighborhood K of \(0 \in \mathbb {R}^3\) there exists \(\delta > 0\) such that F is well-defined and smooth for all \(0< h < \delta \) and \({\Pi } \in K\). It is standard to suppose that time step h is chosen small enough so that F is well-defined for \(({\Pi },h)\) occurring in the update scheme described above and we also make this assumption.
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Acknowledgements
The work presented in this paper was initiated, while the second author was a postdoctoral fellow at the Institute for Mathematics and its Applications (IMA) during the IMA’s annual program on “Control Theory and its Applications.”
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Kipka, R., Gupta, R. A geometric approach for the optimal control of difference inclusions. Math. Control Signals Syst. 31, 1–27 (2019). https://doi.org/10.1007/s00498-019-0231-y
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DOI: https://doi.org/10.1007/s00498-019-0231-y