ABSTRACT
Theorem 2.1.1. Assume (H .l), (H.2), and (H.4). With reference to the optimal control problem (1.2.3) for the ^-dynamics (1.1.1), we have
(i) For each x E [D((—A*)7)]' = Z, there exists a unique optimal pair denoted by {u°x(t),z^(t)} in X2(0, oo; U) x £ 2(0 ,oo;Z ), satisfying
z°x(t) = eAtx + {Lu°x}(t) - eAtx + B lU°x(t) + {L0u°x}(t), (2.1.1) and characterized by the optimality condition
u°x = -L*R*Rz°x e L 2(0 ,oo;£/), (2 .1.2)
which explicitly reads, via (1.3.12), (1.3.13), as
/ oo
eA'^ R * R z l { T ) d T . (2.1.3)
Thus, inserting (2.1.2) into (2.1.1), yields
{ / + LL*R*R]z°x}(t) = eAtx. (2.1.4)
(ii) The quantities ux and R z x are given explicitly in term s of the problem d a ta by the following formulas:
u°x = - [ I + L * R * R L ]-1L*R*R{eA' x ) e L 2(0 ,oc;U ) (2.1.5) Rz°x = [I + R L L * R * } - \ R e A x) e L 2(0 ,o c -Y ) . (2.1.6)
(iii) The optim al value J° of the cost functional J in (1.2.1) is given by /*00
J° = J(u°x ,z°x ) = [ I I ^ W I I ^ + I I ^ W I I ^ · a (2.1.7) Jo
A variation of the boo t-strap argument in the regular parabolic case, e.g., [L-T.4], [LT.5], [L-T.2, C hapter 2], [DaP-I.l], [F.l], will yield the following enhanced regularity of the optim al pair.
