Olof J. Staffans

Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

On a Nonlinear Hyperbolic Volterra Equation

We study questions of existence, boundedness and asymptotic behavior of the solutions of the initial value problem \[(*)\qquad \begin{array}{*{20}c} {u_t (t,x) - \int_0^t {a (t - s)\sigma (u_x (s,x))_x = f(t,x),\quad 0 < t < \infty ,\quad x \in R.} } \\ {u(0,x) = u_0 (x),\quad x \in R.} \\ \end{array} \] Here

Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems

a:R^ + = [0,\infty ) \to R,\sigma :R \to R,f:R^ + \times R \to R,u_0 :R \to R

Quadratic optimal control of stable well-posed linear systems

are given, sufficiently smooth functions, and the subscripts t and x denote partial derivatives. If

When is a linear system conservative

a(t) \equiv 1

Quadratic Optimal Control of Well-Posed Linear Systems

, then (*) reduces to a nonlinear wave equation, and it is well known that in this case classical solutions of (*) do not in general exist for all time. However, we show that for a large class of kernels of physical importance equation (*) has global classical solutions for small data. This class of kernels includes all those which are nonconstant, nonnegative, nonincreasing, convex and sufficiently smooth. We also analyze the asymptotic behavior of the solutions.

Coprime Factorizations and Well-Posed Linear Systems

Abstract. Let U be a Hilbert space. By an ℒ (U)-valued positive analytic function on the open right half-plane we mean an analytic function which satisfies the condition . This function need not be proper, i.e., it need not be bounded on any right half-plane. We study the question under what conditions such a function can be realized as the transfer function of an impedance passive system. By this we mean a continuous-time state space system whose control and observation operators are not more unbounded than the (main) semigroup generator of the system, and, in addition, there is a certain energy inequality relating the absorbed energy and the internal energy. The system is (impedance) energy preserving if this energy inequality is an equality, and it is conservative if both the system and its dual are energy preserving. A typical example of an impedance conservative system is a system of hyperbolic type with collocated sensors and actuators. We give several equivalent sets of conditions which characterize when a system is impedance passive, energy preserving, or conservative. We prove that a impedance passive system is well-posed if and only if it is proper. We furthermore show that the so-called diagonal transform (which may be regarded as a slightly modified feedback transform) maps a proper impedance passive (or energy preserving or conservative) system into a (well-posed) scattering passive (or energy preserving or conservative) system. This implies that, just as in the finite-dimensional case, if we apply negative output feedback to a proper impedance passive system, then the resulting system is (energy) stable. Finally, we show that every proper positive analytic function on the right half-plane has a (essentially unique) well-posed impedance conservative realization, and it also has a minimal impedance passive realization.

Quadratic optimal control of stable systems through spectral factorization

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

The four-block model matching problem in l 1 and infinite-dimensional linear programming

We consider infinite-dimensional linear systems without a-priori well-posedness assumptions, in a framework based on the works of M. Livsic, M. S. Brodskiǐ, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.

Mixed sensitivity minimization problems with rational l 1 -optimal solutions

We study the infinite horizon quadratic cost minimization problem for a well-posed linear system in the sense of Salamon and Weiss. The quadratic cost function that we seek to minimize need not be positive, but it is convex and bounded from below. We assume the system to be jointly stabilizable and detectable and give a feedback solution to the cost minimization problem. Moreover, we connect this solution to the computation of either a (J,S)-inner or an S-normalized coprime factorization of the transfer function, depending on how the problem is formulated. We apply the general theory to get factorization versions of the bounded and positive real lemmas. In the case where the system is regular it is possible to show that the feedback operator can be expressed in terms of the Riccati operator and that the Riccati operator is a stabilizing self-adjoint solution of an algebraic Riccati equation. This Riccati equation is nonstandard in the sense that the weighting operator in the quadratic term differs from the expected one, and the computation of the correct weighting operator is a nontrivial task.