An Optimal Control Problem with a Risk Zone

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10665)


We consider an optimal control problem for an autonomous differential inclusion with free terminal time in the situation when there is a set M (“risk zone”) in the state space \(\mathbb {R}^n\) which is unfavorable due to reasons of safety or instability of the system. Necessary optimality conditions in the form of Clarke’s Hamiltonian inclusion are developed when the risk zone M is an open set. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of problems with state constraints, this allows one to get conditions guaranteeing nondegeneracy of the developed necessary optimality conditions.


Risk zone State constraints Optimal control Differential inclusion Hamiltonian inclusion Stationarity condition 



This work is supported by the Russian Science Foundation under grant 14-50-00005.


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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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