Optimizability and estimatability for infinite-dimensional linear systems

An infinite-dimensional linear system described by (x) over dot(t) = Ax(t) + Bu(t) (t greater than or equal to 0) is said to be optimizable if for eve ry initial state x(0), an input u is an element of L-2 can be found such th at x is an element of L-2. Here, A is the generator of a strongly continuou s semigroup on a Hilbert space and B is an admissible control operator for this semigroup. In this paper we investigate optimizability (also known as the finite cost condition) and its dual, estimatability. We explore the con nections with stabilizability and detectability. We give a very general the orem about the equivalence of input-output stability and exponential stabil ity of well-posed linear systems: the two are equivalent if the system is o ptimizable and estimatable. We conclude that a well-posed system is exponen tially stable if and only if it is dynamically stabilizable and input-outpu t stable. We illustrate the theory by two examples based on PDEs in two or more space dimensions: the wave equation and a structural acoustics model.