The Invariants of Optimal Synthesis

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₊ .