New Riccati equations for well-posed linear systems

Abstract

We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative selfadjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the infinite-dimensional generalization of this Riccati equation is not always well-defined. We show that X always satisfies alternative Riccati equations, which are more suitable for algebraic and numerical computations.

Hamiltonians and Riccati Equations for Linear Systems with Unbounded Control and Observation Operators

“…and hence the inclusion "⊃" in (a) holds. The other inclusion is clear by (15). If now S is also strictly dichotomous, then P ± are bounded.…”

“…This has been done for different kinds of linear systems, e.g. in [6,15,16,17,20]. On the other hand, the Riccati equation is closely connected to the so-called Hamiltonian operator matrix…”

Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations

“…Riccati equations of this type, and in particular their nonnegative solutions, are of central importance in systems theory, see e.g. [13,22] and the references therein; recently, the case of unbounded B and C has gained much attention [26,31,32,39]. The existence of solutions X of the Riccati equation ( 1) is intimately related to the existence of graph subspaces G(X) = {(u, Xu) | u ∈ D(X)} that are invariant under the associated Hamiltonian…”

Diffusive realization of operator solutions of certain operational partial differential equations

“…The optimal control theories, including the LQR, H 2 or H ∞ control, are now well established for linear problems even in infinite-dimension, see [5,6,8]. However, their use for real complex applications, as for real systems governed by partial differential equations, remains a difficult problem due to computation and communication complexity.…”