Abstract
In a Hilbert space, the problem of terminal control with linear dynamics and fixed ends of the trajectory is considered. The integral objective functional has a quadratic form. In contrast to the traditional approach, the problem of terminal control is interpreted not as an optimization problem, but as a saddle-point problem. The solution to this problem is a saddle point of the Lagrange function with components in the form of controls, phase and conjugate trajectories. A saddle-point method is proposed, the convergence of the method in all components of the solution is proved.
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Notes
Scalar products and norms are defined, respectively, as
$$\begin{aligned}&\langle x(\cdot ),y(\cdot )\rangle = \int _{t_0}^{t_1}\langle x(t),y(t)\rangle dt,\; \Vert x(\cdot )\Vert ^2= \int _{t_0}^{t_1} |x(t)|^2dt,\\&\hbox {where }\;\langle x(t),y(t)\rangle = \sum \limits _{1}^{n} x_i(t)y_i(t), \; |x(t)|^2=\sum \limits _{1}^{n} x^2_i(t),\quad t_0\le t\le t_1,\\&x(t)=(x_1(t),\dots ,x_n(t))^\mathrm {T}, \; y(t)=(y_1(t),\dots ,y_n(t))^\mathrm {T} \end{aligned}$$.
See the proof below.
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The work was carried out with financial support from the Russian Foundation for Basic Research (Project No. 15–01–06045-a) and from the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST MISiS (Agreement 02.A03.21.0004).
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Antipin, A., Khoroshilova, E. Saddle point approach to solving problem of optimal control with fixed ends. J Glob Optim 65, 3–17 (2016). https://doi.org/10.1007/s10898-016-0414-8
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DOI: https://doi.org/10.1007/s10898-016-0414-8