]> 7.5.2 The step response of the system: the case of distributed observation and control

7.5.2 The step response of the system: the case of distributed
   observation and control

The model with distributed observation and control has been already constructed in Example 1.1.1 (see (1.10)). It is a special case of the system (7.4), because we can take

d = 𝒜1b,h = 𝒜1c

with b and c given by (1.9). Moreover, d D(𝒜) and thus (7.5) holds. Since c#x = h𝒜x = cx for a dense subset D(𝒜) then the observation functional c# extends uniquely to the bounded everywhere defined functional c, cx = x,cH. This yields (P¯x)(t) = cet𝒜x and by EXS we have P¯ = P L(H,L2(0,)). Actually, P is a HS operator which follows from Theorem 2.4.4. All assumptions of Lemma 7.2.1 hold and from (7.8) we get

y(t) = y0y1(t),y1(t) = n=1eλnt 1 λn b,enHc,enH (7.26)

where

b,enH =0θ0 en(θ)dθ = 1 hn H R01 cos(μnθ0) μn2 + sin(μnθ0) μn (7.27)

and

c,enH =θ1θ2 en(θ)dθ = = 1 hn H R0cos(μnθ1) cos(μnθ2) μn2 + sin(μnθ2) sin(μnθ1) μn (7.28)