# The Invariants of Optimal Synthesis

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

## Abstract

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
$$\begin{array}{l} T \to \inf ,\\ \left\{ {\begin{array}{*{20}{c}} {\mathop x\limits^. = {u_1}F(x,y,),x(0){x_{o,}}y(0 = yo,}\\ {\mathop y\limits^. = {u_2}F(x,y,)(x(T),y(T)) \in M,}\\ {{u_1} + {u_2} = 1,{u_i} \ge 0(i = 1,2);x > o,y > o.} \end{array}} \right. \end{array}$$
(1)
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 .

## Keywords

Optimal Control Problem Smooth Manifold Optimal Trajectory Optimal Synthesis Switching Surface
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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Zelikina, L.F., Universal manifold and turnpike theorems for a class of optimal control problems, Dokl. Akad. Nauk SSSR,Vol. 224, No. 1, 1975. (English translation in Soviet Math. Dokl. 16 1975.)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