OnJ-conservative scattering system realizations in several variables

We prove that an arbitrary function, which is holomorphic on some neighbourhood ofz=0 in ^N and vanishes atz=0, and whose values are bounded linear operators mapping one separable Hilbert space into another one, can be represented as the transfer function of some multi-parameter discrete time-invariant conservative scattering linear system whose state space is a Krein space.The author is thankful to Prof. D.Z. Arov for suggesting this problem. He wishes also to thank Leeds University, where the revised version of this paper was prepared, for its hospitality, and Dr. V.V. Kisil who organized his visit there under the International Short Visits Scheme of LMS (grant no. 5620).     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Some Local Properties of Spectrum of Linear Dynamical Systems in Hilbert Space

We prove some local properties of the spectrum of a linear dynamical system in Hilbert space. The semigroup generator, the control operator and the observation operator may be unbounded. We consider (i) the PBH test, (ii) the correspondence between the poles of the resolvent of the semigroup generator and the poles of the transfer function, and (iii) pole-zero cancellation between two transfer functions of the cascade connection of two dynamical systems. For our investigation we take well-posed linear systems and a subclass of them called weakly regular systems as the most general setting.     

Decoupling of Proper Transfer Functions

This paper studies the state space and feedback aspects of linear system decoupling. Given a minimal realization for a proper transfer function W (s), a general procedure is given for the parametrization of all the minimal decouplings of W (s) into two proper subsystems. This completes and unifies known results on factorization and cascade decomposition.     

State-Feedback Stabilization of Well-Posed Linear Systems

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 L² input that produces an L² output. It is exponentially stabilizable if and only if for each initial state there exists at least one L² input that produces an L² 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_loc² 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.     

Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

In the present paper we introduce the notion of dilation of a multiparametric linear stationary dynamical system (systems of this type, in particular dissipative, and conservative scattering ones were first introduced in [6]). We establish the criterion for existence of a conservative dilation of a multiparametric dissipative scattering system. This allows to distinguish the class of so-calledN-dissipative systems preserving the most important properties of one-parametric dissipative scattering systems.Research supported in part by the Ukrainian-Israeli project of scientific co-operation (contract no. 2M/1516-97).     

Sub-optimal Hankel norm approximation for the analytic class of infinite-dimensional systems

The sub-optimal Hankel norm approximation problem is solved under the assumptions that the system is given in terms of a triple of operators (–A, B, C), where–A is the infinitesimal generator of an exponentially stable, analytic semigroup on the Hilbert spaceZ,B L ( ^m ,Z where –1<0,C L is obtained in terms of the system parameters–A, B, C. (Z, ^p ), and the system is approximately controllable. An explicit parameterization of all solutions     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

The infinite-time admissibility of observation operators and operator Lyapunov equations

Let (A, –, C) be an abstract dynamical system withA being the generator of aC ₀-semigroup on a Hilbert spaceH, C:D(A)Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system (A, B, –), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province.     

Zero-pole interpolation for meromophic matrix functions on an algebraic curve and transfer functions of 2D systems

We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface X having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface X is assumed to be (birationally) embedded as an irreducible algebraic curve in ² and both input and output bundles are assumed to be equal to the kernels of determinantal representations for X. In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.     

Nonlinearity inH ∞-control theory,causality in the commutant lifting theorem,and extension of intertwining operators

The problems studied in this note have been motivated by our work in generalizing linearH control theory to nonlinear systems. These ideas have led to a design procedure applicable to analytic nonlinear plants. Our technique is a generalization of the linearH theory. In contrast to previous work on this topic ([9], [10]), we now are able to explicitly incorporate a causality constraint into the theory. In fact, we show that it is possible to reduce a causal optimal design problem (for nonlinear systems) to a classical interpolation problem solvable by the commutant lifting theorem [8]. Here we present the complete operator theoretical background of our research together with a short control theoretical motivation.This work was supported in part by grants from the Research Fund of Indiana University, the National Science Foundation DMS-8811084 and ECS-9122106, by the Air Force Office of Scientific Research F49620-94-1-0098DEF, and by the Army Research Office DAAL03-91-G-0019 and DAAH04-93-G-0332     

The Nehari problem for infinite-dimensional linear systems of parabolic type

A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR _d ,B=AB _D andB _D is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL ₂() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.     

Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales

The principal aim of this paper is to state and prove the so-called Reid roundabout theorem for the symplectic dynamic system (S) z ^Δ = \cal S _t z on an arbitrary time scale \Bbb T , so that the well known case of differential linear Hamiltonian systems ( \Bbb T = \Bbb R ) and the recently developed case of discrete symplectic systems ( \Bbb T = \Bbb Z ) are unified. We list conditions which are equivalent to the positivity of the quadratic functional associated with (S), e.g. disconjugacy (in terms of no focal points of a conjoined basis) of (S), no generalized zeros for vector solutions of (S), and the existence of a solution to the corresponding Riccati matrix equation. A certain normality assumption is employed. The result requires treatment of the quadratic functionals both with general and separated boundary conditions. Accepted 28 August 2000. Online publication 26 February 2001.     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

Impedance Passive and Conservative Boundary Control Systems

The external Cayley transform is used for the conversion between the linear dynamical systems in scattering form and in impedance form. We use this transform to define a class of formal impedance conservative boundary control systems (colligations), without assuming a priori that the associated Cauchy problems are solvable. We give sufficient and necessary conditions when impedance conservative colligations are internally well-posed boundary nodes; i.e., when the associated Cauchy problems are solvable and governed by C ₀ semigroups. We define a “strong” variant of such colligations, and we show that “strong” impedance conservative boundary colligation is a slight generalization of the “abstract boundary space” construction for a symmetric operator in the Russian literature. Many aspects of the theory is illustated by examples involving the transmission line and the wave equations. Received: August 21, 2006. Accepted: October 22, 2006.     

Transfer Functions of Regular Linear Systems Part III: Inversions and Duality

We study four transformations which lead from one well-posed linear system to another: time-inversion, flow^-inversion, time-flow-inversion and duality. Time-inversion means reversing the direction of time, flow-inversion means interchanging inputs with outputs, while time-flow-inversion means doing both of the inversions mentioned before. A well-posed linear system is time-invertible if and only if its operator semigroup extends to a group. The system is flow-invertible if and only if its input-output map has a bounded inverse on some (hence, on every) finite time interval [0, ] ( > 0). This is true if and only if the transfer function of has a uniformly bounded inverse on some right half-plane. The system is time-flow-invertible if and only if on some (hence, on every) finite time interval [0, ], the combined operator from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conservative, since then is unitary. Time-flow-inversion can sometimes, but not always, be reduced to a combination of time- and flow-inversion. We derive a surprising necessary and sufficient condition for to be time-flow-invertible: its system operator must have a uniformly bounded inverse on some left halfplane. Finally, the duality transformation is always possible.We show by some examples that none of these transformations preserves regularity in general. However, the duality transformation does preserve weak regularity. For all the transformed systems mentioned above, we give formulas for their system operators, transfer functions and, in the regular case and under additional assumptions, for their generating operators.