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Amol Sasane

Publications and source records attributed to Amol Sasane.

51 records · Page 3Linked to original sources
arXiv2013

A subalgebra of the Hardy algebra relevant in control theory and its algebraic-analytic properties

We denote by A_0+AP_+ the Banach algebra of all complex-valued functions f defined in the closed right half plane, such that f is the sum of a holomorphic function vanishing at infinity and a ``causal'' almost periodic function. We give a complete description of the maximum ideal space M(A_0+AP_+) of A_0+AP_+. Using this description, we also establish the following results: (1) The corona theorem for A_0+AP_+. (2) M(A_0+AP_+) is contractible (which implies that A_0+AP_+ is a projective free ring). (3) A_0+AP_+ is not a GCD domain. (4) A_0+AP_+ is not a pre-Bezout domain. (5) A_0+AP_+ is not a coherent ring. The study of the above algebraic-anlaytic properties is motivated by applications in the frequency domain approach to linear control theory, where they play an important role in the stabilization problem.

math.FA↗
arXiv2013

Shannon's sampling theorem in a distributional setting

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-π,π) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(π(t -n)))/(π(t-n)) (t in R). In this article we prove a generalization of this result under the assumption that F is a compactly supported distribution with its support contained in (-π,π).

math.FA↗
arXiv2013

Projective freeness of algebras of real symmetric functions

Let D^n be the closed unit polydisk in C^n. Consider the ring C_r of complex-valued continuous functions on D^n that are real symmetric, that is, f(z)=(f(z^*))^* for all z in D^n. It is shown that C_r is projective free, that is, finitely generated projective modules over C_r are free. We also show that several subalgebras of the real symmetric polydisc algebra are projective free.

math.KT↗
arXiv2013

Reformulation of the extension of the $ν$-metric for $H^{\infty}$

The classical nu-metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball and Sasane [Complex Analysis and Operator Theory; 2012]. In Sasane [Mathematics of Control and Related Fields; 2012], an extension of the classical nu-metric was given when the underlying ring of stable transfer functions is the Hardy algebra, H^infty. However, this particular extension to H^infty did not directly fit in the abstract framework given in Ball and Sasane [CAOT 2012]. In this paper we show that the case of H^infty also fits into the general abstract framework in [BS CAOT 2012] and that the nu-metric defined in this setting is identical to the extension of the nu-metric defined in Sasane [MCRF 2012]. This is done by introducing a particular Banach algebra, which is the inductive limit of certain C*-algebras.

math.OC↗
arXiv2012

A generalized chordal metric making strong stabilizability a robust property

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Jonathan Partington ("Robust control and approximation in the chordal metric", in Robust Control, LNCIS 183, Springer, 1992) for H^infty, it is shown that strong stabilizability is a robust property in this metric.

math.OC↗
arXiv2011

On two natural extensions of Vinnicombe's metric: their noncoincidence yet equivalence on stabilizable plants over A_+

Let A_+ be the ring of Laplace transforms of complex Borel measures on R with support in [0,+\infty) which do not have a singular nonatomic part. We compare the nu-metric d_{A_+} for stabilizable plants over A_+ given in the article by Ball and Sasane [2010], with yet another metric d_{H^\infty}|_{A_+}, namely the one induced by the metric d_{H^\infty} for the set of stabilizable plants over H^\infty given in teh article by Sasane in 2011. Both d_{A_+} and d_{H^\infty} coincide with the classical Vinnicombe metric defined for rational transfer functions, but we show here by means of an example that these two possible extensions of the classical nu-metric for plants over A_+ do not coincide on the set of stabilizable plants over A_+. We also prove that they nevertheless give rise to the same topology on stabilizable plants over A_+, which in turn coincides with the gap metric topology.

math.OC↗
arXiv2011

Extension of the $ν$-metric for stabilizable plants over $H^\infty$

An abstract $ν$-metric was introduced by Ball and Sasane, with a view towards extending the classical $ν$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an important concrete special instance of the abstract $ν$-metric, by verifying that all the assumptions demanded in the abstract set-up are satisfied when the ring of stable transfer functions is the Hardy algebra $H^\infty$. This settles the open question implicit in \cite{BalSas2}.

math.OC↗
arXiv2011

Solvability of the $H^\infty$ algebraic Riccati equation in Banach algebras

Let $R$ be a commutative complex unital semisimple Banach algebra with the involution $\cdot ^\star$. Sufficient conditions are given for the existence of a stabilizing solution to the $H^\infty$ Riccati equation when the matricial data has entries from $R$. Applications to spatially distributed systems are discussed.

math.OC↗
arXiv2010

The new $ν$-metric induces the classical gap topology

Let $\calA_+$ denote the set of Laplace transforms of complex Borel measures $μ$ on $[0,+\infty)$ such that $μ$ does not have a singular non-atomic part. In \cite{BalSas}, an extension of the classical $ν$-metric of Vinnicombe was given, which allowed one to address robust stabilization problems for unstable plants over $\calA_+$. In this article, we show that this new $ν$-metric gives a topology on unstable plants which coincides with the classical gap topology for unstable plants over $\calA_+$ with a single input and a single output.

math.OC↗
arXiv2010

Generators for rings of compactly supported distributions

Let $C$ denote a closed convex cone $C$ in $\mathbb{R}^d$ with apex at 0. We denote by $\mathcal{E}'(C)$ the set of distributions having compact support which is contained in $C$. Then $\mathcal{E}'(C)$ is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on $\hat{f}_1,..., \hat{f}_n$ for $f_1,...,f_n \in \mathcal{E}'(C)$ to generate the ring $\mathcal{E}'(C)$. (Here $\hat{\cdot}$ denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Hörmander. En route we answer an open question posed by Yutaka Yamamoto.

math.FA↗
arXiv2009

Topological Stable Rank of $H^\infty(Ω)$ for Circular Domains $Ω$

Let $Ω$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(Ω)$ the Banach algebra of all bounded holomorphic functions on $Ω$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(Ω)$ is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the unit disk. We also show that for domains symmetric to the real axis, the Bass and topological stable ranks of the real symmetric algebra $H^\infty_\R(Ω)$ are 2.

math.CV↗
arXiv2008

The Corona theorem and stable rank for the algebra $\mC +BH^\infty$

Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\infty_B:=\mC+BH^\infty$. That is, we show the equivalence of the classical {\em corona condition} on data $f_1, ..., f_n \in H^\infty_B$: \[ \forall z \in \mD, \sum_{k=1}^{n} |f_k(z)| \geq δ>0, \] and the {\em solvability of the Bezout equation} for $g_1, ..., g_n \in H^\infty_B$: \[ \forall z\in \mD, \sum_{k=1}^n g_k (z)f_k(z)=1. \] Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of $H^\infty_B$ is 1. Let $A(\mD)_B$ be the subalgebra of all elements from $H^\infty_B$ having a continuous extension to the closed unit disk $\bar{\mD}$. Analogous results are obtained also for $A(\mD)_B$.

math.CV↗
arXiv2007

Estimates in corona theorems for some subalgebras of $H^\infty$

We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such algebras is trivial and known, but corona theorem with estimates is new and much harder to prove. As an auxiliary result of independent interest we get the continuity of the best estimate in such algebras (including the classical case of the algebra $H^\infty$ of bounded analytic functions). We also show that for a fixed $n$ the best estimate is the same for all algebras we consider.

math.CA↗