We consider forced Lur'e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay-and partial-differential equations are known to belong to this class of infinitedimensional systems. We investigate input-to-state stability (ISS) and incremental ISS properties: our results are reminiscent of well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion. The incremental ISS results are used to derive certain convergence properties, namely the convergentinput convergent-state (CICS) property and asymptotic periodicity of the state and output under periodic forcing. In particular, we provide sufficient conditions for ISS and incremental ISS. The theory is illustrated with examples.

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input-state-output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.

Incremental stability and convergence properties for forced, infinite-dimensional, discretetime Lur'e systems are addressed. Lur'e systems have a linear and nonlinear component and arise as the feedback interconnection of a linear control system and a static nonlinearity. Discrete-time Lur'e systems arise in, for example, sampled-data control and integro-difference models. We provide conditions, reminiscent of classical absolute stability criteria, which are sufficient for a range of incremental stability properties and input-to-state stability (ISS). Consequences of our results include sufficient conditions for the converging-input convergingstate (CICS) property, and convergence to periodic solutions under periodic forcing.

We consider a general class of operator-valued irrational positive-real functions with an emphasis on their frequency-domain properties and the relation with stabilization by output feedback. Such functions arise naturally as the transfer functions of numerous infinite-dimensional control systems, including examples specified by PDEs. Our results include characterizations of positive realness in terms of imaginary axis conditions, as well as characterizations in terms of stabilizing output feedback, where both static and dynamic output feedback are considered. In particular, it is shown that stabilizability by all static output feedback operators belonging to a sector can be characterized in terms of a natural positive-real condition and, furthermore, we derive a characterization of positive realness in terms of a mixture of imaginary axis and stabilization conditions. Finally, we introduce concepts of strict and strong positive realness, prove results which relate these notions and analyse the relationship between the strong positive realness property and stabilization by feedback. The theory is illustrated by examples, some arising from controlled and observed partial differential equations.

Abstract.A stability/instability trichotomy for a class of nonnegative continuous-time Lur'e systems is derived. Asymptotic, exponential, and input-to-state stability concepts are considered. The presented trichotomy rests on Perron-Frobenius theory, absolute stability theory, and recent input-to-state stability results for Lur'e systems. Applications of the results derived arise in various fields, including density-dependent population dynamics, and two examples are discussed in detail.

We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an Euler-Bernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable.

We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur'e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur'e system and, in this context, the multiplicative forcing may correspond to harvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control.

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26 (1998), pp. 863-898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the Hankel operator in a Sobolev space. We also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation.