Approximate Stabilizability via the Algebraic Riccati Equation

Weak and strong stabilization of linear systems

“…Moreover, if R > 0, then (18) weakly stabilizes (13) (see [13]). If in addition A has a compact resolvent, then the stability becomes a problem in the strong sense (see [1]).…”

“…2) There exists a "minimal" solution P * to (17) such that the corresponding feedback (18) is a solution to (16) and if P * > 0, then (18) regionally strongly stabilizes (13) in ω with a bounded state in the whole domain.…”

“…then the feedback control v * (t) = −B * P z(t) regionally strongly stabilizes (13) in ω, and we have the estimate…”

“…It is well known that the problem of stabilizing (13) in its geometrical domain can be linked to the following minimization problem…”

“…The projection operators P u and P s = I − P u commute with A, and we have A = A u + A s , with A u = P u AP u and A s = P s AP s . In this case, system (13) can be decomposed into the following:…”