Control with Bounded Inputs

Abstract

We are concerned here with the linear control system

$$u'(t) = Au(t) + Bf(t)$$

(1.1)

where A is a (possibly unbounded) operator with domain D(A) in the Banach space E and range in E, B is a bounded operator from another complex Banach space F to E. As usual, we assume A to be the infinitesimal generator of a strongly continuous semigroup T(t), t > 0 ([4], Chapter VIII). If f( ∙) (the input or control) is any locally summable F-valued function and u is any element of E we define the “variation-of-constants” expression

$$ u(t) = T(t)u + \int_o^t T (t - s)Bf(s)ds $$

(1.2)

to be a solution of (1.1), where the integral^-on the right-hand side of (1.2) is a Bochner integral ([8], Chapter III). The solution (or trajectory or output) u( ∙) of (1.1) is always continuous and takes the value u for t = 0; if u, f( ∙) are “smooth” in one sense or another then u( ∙) is a genuine solution, i.e., it is continuously differentiable and satisfies (1.1) everywhere, while it is only a generalized solution in the general case. Solutions of (1.1) with given input and initial condition are unique. (See [2] for further discussion on these questions.)