Let X, ∂X, and U be Banach spaces, called, respectively, the state, the boundary, and the control space. Assume that Amax is a linear, closed, densely defined operator on X; D is the generator of a strongly continuous semigroup on ∂X; and L:D(Amax)--->∂ X is a linear bounded surjective operator, the boundary operator. The authors consider the following abstract initial value problem, with dynamical boundary conditions and boundary control:
where B:U --->∂ X is bounded, and v ∈ L1(ℜ+,U) is the control function.
To apply the theory of semigroups of operators, the authors reformulate the above system on the product space X x ∂X. Then, they illustrate these abstract concepts by studying the one-dimensional heat equation, subject to boundary control and dynamic boundary conditions. The paper is very well written. The results are interesting, and can be applied to several problems.
