]> 7.2.1 Admissible factor control vectors

7.2.1 Admissible factor control vectors

Assume additionally that the semigroup {S(t)}t0 is EXS.

Definition 7.2.1. A vector d H is called an admissible factor control vector if

0S(t)du(t)dt D(𝒜)u L2(0,) (7.1)

By EXS the operator W, Wu :=0S(t)du(t)dt belongs to L(L2(0,),H). Since 𝒜 is closed and (7.1) is equivalent to the inclusion R(W) D(𝒜) then applying the closed graph theorem we get Q = 𝒜W L(L2(0,),H). The operator Q is called the reachability map. Moreover, the following is known [44, Theorem 4.2] with the reflection operator Rt given by

(Rtu)(τ) := u(t τ),τ [0,t] 0, τ t ,u L2(0,) .

Theorem 7.2.1. Let d H be an admissible factor control vector and let u L2(0,). Then the function

x(t) := QRtu = 𝒜0tS(t τ)du(τ)dτ

is a weak solution of

(t) = 𝒜[x(t) + du(t)] x(0) = 0 (7.2)

i.e. it is a continuous H–valued function of t such that for y D(𝒜), tx(t),yH is absolutely continuous and for almost all t and all y D(𝒜) we have

d dt x(t),yH = x(t) + du(t),𝒜y H .

From [69, p. 258 - 259] we know that for almost all t 0

d dtS(t)x0,wH = S(t)x0,𝒜w Hx0 Hw D(𝒜) (7.3)

Now by Theorem 7.2.1 a unique weak solution of

(t) = 𝒜[x(t) + du(t)] x(0) = x0

takes the form

x(t) = S(t)x0 + QRtu,x0 H,u L2(0,) .