Geodesic
Refined Euler--Lagrange Inclusion for an Optimal Control Problem with Discontinuous Integrand
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 34-63

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We study a free-time optimal control problem for a differential inclusion with mixed-type functional in which the integral term contains the characteristic function of a given open set of “undesirable” states of the system. The statement of this problem can be viewed as a weakening of the statement of the classical optimal control problem with state constraints. Using the approximation method, we obtain first-order necessary optimality conditions in the form of the refined Euler–Lagrange inclusion. We also present sufficient conditions for their nondegeneracy and pointwise nontriviality and give an illustrative example.
Keywords: optimal control, differential inclusion, Pontryagin's maximum principle, refined Euler–Lagrange inclusion, state constraint, discontinuous integrand, risk zone. 
@article{TM_2021_315_a3,
     author = {S. M. Aseev},
     title = {Refined {Euler--Lagrange} {Inclusion} for an {Optimal} {Control} {Problem} with {Discontinuous} {Integrand}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {34--63},
     publisher = {mathdoc},
     volume = {315},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a3/}
}
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S. M. Aseev. Refined Euler--Lagrange Inclusion for an Optimal Control Problem with Discontinuous Integrand. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 34-63. http://geodesic.mathdoc.fr/item/TM_2021_315_a3/