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On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints

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Abstract

We consider a class of optimal control problems with a state constraint and investigate a trajectory with a single boundary interval (subarc). Following R.V. Gamkrelidze, we differentiate the state constraint along the boundary subarc, thus reducing the original problem to a problem with mixed control-state constraints, and show that this way allows one to obtain the full system of stationarity conditions in the form of A.Ya. Dubovitskii and A.A. Milyutin, including the sign definiteness of the measure (state constraint multiplier), i.e., the nonnegativity of its density and atoms at junction points. The stationarity conditions are obtained by a two-stage variation approach, proposed in this paper. At the first stage, we consider only those variations, which do not affect the boundary interval, and obtain optimality conditions in the form of Gamkrelidze. At the second stage, the variations are concentrated on the boundary interval, thus making possible to specify the stationarity conditions and obtain the sign of density and atoms of the measure.

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Notes

  1. For a function \(\mu (t)\) of bounded variation, its generalized derivative \(\dot{\mu }(t)= \mathrm{{d}}\mu (t)/\mathrm{{d}}t\) is a generalized function in the sense that \(\dot{\mu }(t)\,\mathrm{{d}}t = \mathrm{{d}}\mu (t)\) is the Riemann–Stieltjes measure generated by the function \(\mu (t).\) If \(\mu (t)\) is absolute continuous, then \(\dot{\mu }(t)\) is a usual Lebesgue integrable function;  if \(\mu (t)\) is discontinuous at a point \(t_*\,,\) then \(\dot{\mu }(t)\) contains the Dirac \(\delta -\)function at \(t_*\,.\)

  2. If \(\psi (t)\) is the adjoint variable in the Dubovitskii–Milyutin form, \(\varPhi (t,x(t))\le 0\) is the state constraint, and a monotone function \(\mu (t)\) generates the corresponding measure, then \(\widetilde{\psi }(t) = \psi (t) -\mu (t)\,\varPhi '_x(t, x^0(t))\) is the adjoint variable in the Gamkrelidze form.

  3. This natural trick of replication of variables was first proposed, probably, in [9], and later was also used, may be independently, by many authors, e.g., in [8, 10,11,12,13,14,15].

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Acknowledgements

This research was partially supported by the Russian Foundation for Basic Research under Grant No. 16-01-00585.  The authors thank Nikolai Osmolovskii for useful discussions and the anonymous referees for valuable remarks.

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Authors and Affiliations

  1. Central Economics and Mathematics Institute, Russian Academy of Sciences, 47 Nakhimovsky Prospekt, Moscow, Russia, 117418

    Andrei Dmitruk

  2. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University (MSU), GSP-1, 1-52, Leninskiye Gory, Moscow, Russia, 119991

    Andrei Dmitruk & Ivan Samylovskiy

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  1. Andrei Dmitruk
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  2. Ivan Samylovskiy
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Correspondence to Ivan Samylovskiy.

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Communicated by Boris Vexler.

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Dmitruk, A., Samylovskiy, I. On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints. J Optim Theory Appl 173, 391–420 (2017). https://doi.org/10.1007/s10957-017-1089-0

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