A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L 2 input that produces an L 2 output. It is exponentially stabilizable if and only if for each initial state there exists at least one L 2 input that produces an L 2 state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in L 2 loc in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties. (2000). Primary 93D15, 49N10; Secondary 93C25. Mathematics Subject Classification

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We generalize the classical theory on algebraic Riccati equations and optimization to infinite-dimensional well-posed linear systems, thus completing the work of George Weiss, Olof Staffans and others. We show that the optimal control is given by the stabilizing solution of an integral Riccati equation. If the input operator is not maximally unbounded, then this integral Riccati equation is equivalent to the algebraic Riccati equation.Using the integral Riccati equation, we show that for (nonsingular) minimization problems the optimal state-feedback loop is always well-posed. In particular, the optimal state-feedback operator is admissible also for the original semigroup, not only for the closed-loop semigroup (as has been known in some cases); moreover, both settings are well-posed with respect to an external input. This leads to the positive solution of several central, previously open questions on exponential, output and dynamic (aka. "internal") stabilization and on coprime factorization of transfer functions.Our theory covers all quadratic (possibly indefinite) cost functions, but the optimal state feedback need not be well-posed (admissible) unless the cost function is uniformly positive or the system is sufficiently regular.

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We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (−∞, ∞) or on [0, ∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras 1 (N n ), 1 (Z n ), 1 (S) and 1 (S ∩ R + ), where S is an arbitrary subgroup of R, of the almost periodic algebra AP and of AP ∩H ∞ , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103 (3): [275][276][277][278][279][280][281] 1992). For the above algebras, the polydisc algebra A(D n ), the algebra C(T n ) of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier-Desoer algebra of causal exponentially decaying measures and L 1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bézout domains.Communicated by