A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space
Integral Equations and Operator Theory, 2003
. We solve a bitangential interpolation problem for contractive multipliers on the Arveson spac... more . We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform.
We give a new proof on the disk that a Pick problem can be solved by a rational function that is ... more We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.
This paper is concerned with the design of a feedback controller to minimize accumulated queue le... more This paper is concerned with the design of a feedback controller to minimize accumulated queue lengths in the presence of unknown inflow disturbances at an isolated highway intersection and a simple network with two intersections. The admissible control set is taken to be a finite set at each point of time. The value function for the associated game is obtained
My first meeting with Israel Gohberg was in the early 1970s when Israel gave a colloquium lecture... more My first meeting with Israel Gohberg was in the early 1970s when Israel gave a colloquium lecture at the University of Virginia and I drove up from Blacksburg to hear his talk. As a student at U.Va. only a few years earlier, we had a student seminar working through parts of the first volume of Gohberg-Kreĭn and here was my
We formulate the L2-gain control problem for a general nonlinear, state-space system with project... more We formulate the L2-gain control problem for a general nonlinear, state-space system with projection dynamics in the state evolution and hard constraints on the set of admissible inputs. We develope specic results for an example motivated by a trac signal control problem. A state- feedback control with the desired properties is found in terms of the solution of an associated
We extend the $\nu$-metric introduced by Vinnicombe in robust control theory for rational plants ... more We extend the $\nu$-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.
Siam Journal on Control and Optimization, Jul 26, 2006
We obtain state-space formulas for the solution of the Nehari-Takagi/sub-optimal Hankel norm appr... more We obtain state-space formulas for the solution of the Nehari-Takagi/sub-optimal Hankel norm approximation problem for infinite-dimensional systems with a nonexponentially stable generator, via the method of J-spectral factorization. We make key use of a purely frequency domain solution of the problem.
The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurli... more The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot 0}$ Hilbert-space contraction operators. We discuss an analogue of all these ideas in the context of weighted Hardy spaces over the unit disk and an associated class of hypercontraction operators.
Lax–Phillips scattering and conservative linear system: a Cuntz-algebra multidimensional setting
Memoirs of the American Mathematical Society, 2005
Page 1. Number 837 Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multi... more Page 1. Number 837 Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting Joseph A. Ball Victor Vinnikov 'I— ., '\nn, x November 2005 • Volume 178 • Number 837 (first of 5 numbers ...
Page 1. Non-linear minimal square spectral factorization JOSEPH A. BALL{ and MARK A. PETERSEN{* R... more Page 1. Non-linear minimal square spectral factorization JOSEPH A. BALL{ and MARK A. PETERSEN{* Recent work has identified a correspondence between minimal square spectral factors of a given spectral density, invariant ...
An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the clas... more An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-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 additional concrete special instance of the abstract $\nu$-metric, by verifying all the assumptions demanded in the abstract set-up. This example links the abstract $\nu$-metric with the one proposed by Vinnicombe as a candidate for the $\nu$-metric for nonrational plants.
A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space
Integral Equations and Operator Theory, 2003
. We solve a bitangential interpolation problem for contractive multipliers on the Arveson spac... more . We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform.
We give a new proof on the disk that a Pick problem can be solved by a rational function that is ... more We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.
This paper is concerned with the design of a feedback controller to minimize accumulated queue le... more This paper is concerned with the design of a feedback controller to minimize accumulated queue lengths in the presence of unknown inflow disturbances at an isolated highway intersection and a simple network with two intersections. The admissible control set is taken to be a finite set at each point of time. The value function for the associated game is obtained
My first meeting with Israel Gohberg was in the early 1970s when Israel gave a colloquium lecture... more My first meeting with Israel Gohberg was in the early 1970s when Israel gave a colloquium lecture at the University of Virginia and I drove up from Blacksburg to hear his talk. As a student at U.Va. only a few years earlier, we had a student seminar working through parts of the first volume of Gohberg-Kreĭn and here was my
We formulate the L2-gain control problem for a general nonlinear, state-space system with project... more We formulate the L2-gain control problem for a general nonlinear, state-space system with projection dynamics in the state evolution and hard constraints on the set of admissible inputs. We develope specic results for an example motivated by a trac signal control problem. A state- feedback control with the desired properties is found in terms of the solution of an associated
We extend the $\nu$-metric introduced by Vinnicombe in robust control theory for rational plants ... more We extend the $\nu$-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.
Siam Journal on Control and Optimization, Jul 26, 2006
We obtain state-space formulas for the solution of the Nehari-Takagi/sub-optimal Hankel norm appr... more We obtain state-space formulas for the solution of the Nehari-Takagi/sub-optimal Hankel norm approximation problem for infinite-dimensional systems with a nonexponentially stable generator, via the method of J-spectral factorization. We make key use of a purely frequency domain solution of the problem.
The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurli... more The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot 0}$ Hilbert-space contraction operators. We discuss an analogue of all these ideas in the context of weighted Hardy spaces over the unit disk and an associated class of hypercontraction operators.
Lax–Phillips scattering and conservative linear system: a Cuntz-algebra multidimensional setting
Memoirs of the American Mathematical Society, 2005
Page 1. Number 837 Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multi... more Page 1. Number 837 Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting Joseph A. Ball Victor Vinnikov 'I— ., '\nn, x November 2005 • Volume 178 • Number 837 (first of 5 numbers ...
Page 1. Non-linear minimal square spectral factorization JOSEPH A. BALL{ and MARK A. PETERSEN{* R... more Page 1. Non-linear minimal square spectral factorization JOSEPH A. BALL{ and MARK A. PETERSEN{* Recent work has identified a correspondence between minimal square spectral factors of a given spectral density, invariant ...
An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the clas... more An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-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 additional concrete special instance of the abstract $\nu$-metric, by verifying all the assumptions demanded in the abstract set-up. This example links the abstract $\nu$-metric with the one proposed by Vinnicombe as a candidate for the $\nu$-metric for nonrational plants.
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