Abstract
The process considered in this paper is governed by the following two evolutional equations in Hilbert Spaces.
x1=A1x1+B1u∈X1 (1)
(dynamics of controlled object)
x2=A2x2+B2u∈X2, u=Hx2 (2)
(dynamics of actuator)
This system is called a coupled system. Here the variable u is the control input and the variable x1 is the final output of coupled system.
Now, combining tho above two equations, the coupled system can be described by the following one evolutional equation in Hilbert Space X1×X2.d/dt(x1x2)=(A10B1HA2)(x1x2)+(0B2)u (3)
while, it is very difficult to decide whether the unbounded operator (A1 0 B1H A2) in eq. (3) is the infinitesimal generator of the strongly continuous semi-group. Furthermore, even though this operator would be an infinitesimal generator of the semi-group, it can be never expected to represent this semi-group analytically by A1, A2 and B1H. By these reasons there are many obstacles in applying the results obtained in Ref. (1) directly to the coupled system problems.
In this paper the existence, uniqueness and representation of the time-optimal control of the coupled system are re-examined by making use of the geometrical properties of reachable outputs. In particular, concerning with the representation and the uniqueness of the time-optimal control, it is necessary for the coupled system to be output controllable for the output space X1. The output controllability is examined and the sufficient conditions are derived.
