Abstract
We formulate a controlled dynamic model with a boundary-value problem of minimizing a sensitivity function under constraints. Solution of boundary-value problem implicitly defines a terminal condition for the dynamic model. In the model, a unique trajectory corresponds to each control taken from a bounded set. The problem is to select the control such that the corresponding trajectory takes an object from an arbitrary initial state to the terminal state. In this paper, the dynamic model is treated as a problem of stabilization, and the terminal state of the object is interpreted as a state of equilibrium. If under the influence of external disturbances the object loses equilibrium then this object is returned back by selecting the appropriate control. A saddle-point method for solving problem is proposed. We prove its convergence to solution of the problem in all the variables.
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Notes
Hereinafter, the index “1” next to a variable means belonging to the finite-dimensional space of the right-hand end of a time interval \([t_0,t_1]\).
The method is called “dual”, since iterations run only in dual variables \(p_1\) and \(\psi (\cdot )\), while the optimization is done by primal variables \(y_1,x_1,x(\cdot )\).
\(|u(t)|=\sqrt{\sum \limits _{i=1}^r u_i^2(t)}= \sqrt{\langle u(t),u(t)\rangle },\; t \in [t_0,t_1], \quad \Vert u(\cdot )\Vert = \sqrt{\int _{t_0}^{t_1}|u(t)|^2dt}\).
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The reported study was supported by the Russian Science Foundation (Research Project 17-11-01353).
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Antipin, A., Khoroshilova, E. Controlled dynamic model with boundary-value problem of minimizing a sensitivity function. Optim Lett 13, 451–473 (2019). https://doi.org/10.1007/s11590-017-1216-8
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DOI: https://doi.org/10.1007/s11590-017-1216-8