ABSTRACT
Lemma 1 The function xq —► W + (xo) is a continuous quadratic form on X ; i.e. there exists W £ £ (X ) , W = W *f such that
W + (x 0) = (x0,W x 0) .
O
Moreover,
Lemma 2 If (x(-),u(·)) satisfies Eq. (1) the following inequality holds:
W +(*(*)) - W +(xo) + f F(x(s),u(s)) ds> 0 . (3) Jo
Moreover, if it happens that for some xo there exists an optimal control u+(·; Xo) then:
Wr+(*+(*; xo)) - W + (x0) + [ F {x+(s ; x0), «+(5; *0)) ds = 0. (4) Jo
for each t > 0. Here x+(-;xo) w the solution which corresponds to the optimal control u+ (']xo) ( optimal trajectory).
