Abstract
In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.
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Notes
Hereinafter it is considered that continuous functions or functions of bounded variation defined on \(T_i\) or \(T\) are extended onto the whole real line by constants, e.g.: \(x_i(t)=x_{1,i} \forall \, t<t_{1,i}, x_i(t)=x_{2,i} \forall \, t>t_{2,i}\). Measurable functions are extended out of \(T_i,T\) by the zero value.
Notation \(_{t\mathop {\longrightarrow }\limits ^{S}\tau }\) means that the limit is considered over the set \(S\).
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Acknowledgments
The authors are grateful to Professor E. R. Avakov for useful discussions. This research was supported by the Russian Foundation for Basic Research, grant numbers 13-01-00494, 15-01-04601 and by the Ministry of Education and Science of the Russian Federation, project no. 1.333.2014/K.
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Arutyunov, A.V., Karamzin, D.Y. Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints. J Glob Optim 64, 623–647 (2016). https://doi.org/10.1007/s10898-015-0272-9
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DOI: https://doi.org/10.1007/s10898-015-0272-9