The Invariants of Optimal Synthesis

  • L. F. Zelikina
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)


In this article we deal with the term “invariant of optimal synthesis.” By this term we mean geometrical rather than algebraic invariant. To explain, let us consider the simplest case
$$\begin{array}{l} T \to \inf ,\\ \left\{ {\begin{array}{*{20}{c}} {\mathop x\limits^. = {u_1}F(x,y,),x(0){x_{o,}}y(0 = yo,}\\ {\mathop y\limits^. = {u_2}F(x,y,)(x(T),y(T)) \in M,}\\ {{u_1} + {u_2} = 1,{u_i} \ge 0(i = 1,2);x > o,y > o.} \end{array}} \right. \end{array}$$
Here, F(x,y) > 0, \(\frac{{\partial F}}{{\partial x}}(x,y) > 0,\frac{{\partial F}}{{\partial y}}(x,y,) > 0\) is the smooth manifold, M ε R + 2 .


Optimal Control Problem Smooth Manifold Optimal Trajectory Optimal Synthesis Switching Surface 


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    Zelikina, L.F., High dimensional synthesis and turnpike theorems for optimal control problems, in V.I. Arkin, editor, Probasbilistic Control Problems in Economics, Nauka, Moscow, 1977, pp. 33–114 (in Russian).Google Scholar
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    Zelikina, L.F., On optimal control problems with nonregular synthesis, All-Union Conf. Dynamical Control, Abstracts of Reports, Sverdlovsk, 1979, p. 114 (in Russian).Google Scholar
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    Boltjanskii, V.G., Mathematical methods of optimal control,2nd rev., augm. ed., Nauka, Moscow, 1969. (English translation of 1st ed., Ilolt, Reinhart and Winston, 1971.)Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • L. F. Zelikina
    • 1

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