On application of the implemented semigroup to a problem arising in optimal control

Remarks on the existence and uniqueness of solutions to the infinite dimensional sylvester equations

“…For more results on the properties of the implemented semigroup and its generator see e.g. [8], [5], [6] and references cited therein.…”

“…In this paper we use the implemented semigroup concept (see, e.g., [8], [5]) to study the infinite-dimensional differential Sylvester equation (DSE) of the forṁ…”

Infinite-dimensional Sylvester equations: Basic theory and application to observer design

“…In this section we introduce and characterize the notion of an implemented semigroup (Alber, 2001;Emirsajłow and Townley, 2005) as the main tool in the analysis of infinite-dimensional Sylvester equations. In order to state our results, we need to introduce the following notation and assumptions: • A is a linear, unbounded operator on H A generating a strongly continuous semigroup of operators…”

“…Motivated by applications of (29) in infinite-dimensional systems and control theory, we are interested in solutions which satisfy the condition (Z(t)) t≥0 ⊂ H. For an example of application of the inhomogeneous differential Lyapunov equation (Sylvester equation with E = A * ) in an optimal control problem, see the work of Emirsajłow and Townley (2005). From some previous results it follows that for Z ∈ H we have A ∼ Z = A −1 Z, and hence the differential equation (29) becomeṡ (30) and can be written explicitly aṡ…”

“…It turns out that in this case the situation becomes extremely complicated and requires the development of a suitable mathematical framework within which the problem can be solved. For this purpose it is useful to introduce the implemented semigroup concept (see, e.g., Alber, 2001;Emirsajłow and Townley, 2005;Phong, 1991) since it is a natural tool to study the infinite-dimensional algebraic Sylvester equation. In fact, we will start our considerations with the differential Sylvester equation of the forṁ…”