
The Bulletin of Irkutsk State University. Series Mathematics, 2017, Volume 19, Pages 150–163
(Mi iigum294)




Nonlocal improvement of controls in nonlinear discrete systems
O. V. Morzhin^{} ^{} V. A. Trapeznikov Institute of Control Sciences RAS, 65, Profsoyuznaya st., Moscow, 117997
Abstract:
A nonlinear optimal control problem for discrete system with both control function and
control parameters (parameters are at the system's right side and at the initial condition)
is considered. For the given optimization problem, the problem of control's improvement
is studied. It's developed a known approach for nonlocal improvement of control based
on construction of the exact (without residual terms w.r.t. state and control variables)
formula for the cost functional's increment under some special conjugate system.
For the given optimization problem, it's considered the generalized Lagrangian following to
the theory by V. F. Krotov. The function $\varphi(t,x)$ which plays an important role
in the generalized Lagrangian is considered in this article in the linear w.r.t. $x$
form $\varphi(t,x) = \langle p(t), x \rangle$ where the function $p(t)$ is the solution of
the mentioned conjugate system. Thus, first of all, the exact formula of the cost functional's increment
is considered under the assumption on the solution $p(t)$ existence; and, secondly,
the linear function $\varphi(t,x)$ is used here in connection with creation of the mentioned
increment formula, and not for linear approximation of the generalized Lagrangian's increment.
The corresponding condition of control's improvement is formulated
in terms of the boundary value problem composed due to binding of the system
given in the optimization problem together with the conjugate system. The obtained increment condition
is similar to the increment conditions which were suggested before in the papers of the author
for discrete problems without control parameters.
There is an example of control's improvement in some problem where the control to be improved
gives the maximum of the Pontryagin's function for all values of $t$. The boundary value improvement problem
is solved with help of the shooting method, and the calculations are made analytically.
Keywords:
discrete systems; optimal control; control functions and parameters; nonlocal improvement.
DOI:
https://doi.org/10.26516/19977670.2017.19.150
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Bibliographic databases:
UDC:
517.977
MSC: 49J21, 93C10
Citation:
O. V. Morzhin, “Nonlocal improvement of controls in nonlinear discrete systems”, The Bulletin of Irkutsk State University. Series Mathematics, 19 (2017), 150–163
Citation in format AMSBIB
\Bibitem{Mor17}
\by O.~V.~Morzhin
\paper Nonlocal improvement of controls in nonlinear discrete systems
\jour The Bulletin of Irkutsk State University. Series Mathematics
\yr 2017
\vol 19
\pages 150163
\mathnet{http://mi.mathnet.ru/iigum294}
\crossref{https://doi.org/10.26516/19977670.2017.19.150}
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