Internally Well-Posed Systems

We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time-invariant well-posed system. The associated generator family has the form $$A+G(t)$$ , where $$G(t)$$ is a bounded operator on the state space and $$G(\cdot )$$ is strongly continuous. We show that the resulting time-varying system (the perturbed system) is well-posed and investigate its properties. In the particular case when the unperturbed system is scattering passive, we derive an energy balance inequality for the perturbed system. We illustrate this theory using it to formulate the system corresponding to an electrically conducting rigid body moving in vacuo in a bounded domain, with an electromagnetic field (both in the rigid body and in the vacuum) described by Maxwell’s equations.

An infinite-dimensional linear system described by $\dot{x}(t)=Ax(t)+Bu(t)$ $(t\geq 0)$ is said to be optimizable if for every initial state x(0), an input $u\in L^2$ can be found such that $x\in L^2$. Here, A is the generator of a strongly continuous semigroup on a Hilbert space and B is an admissible control operator for this semigroup. In this paper we investigate optimizability (also known as the finite cost condition) and its dual, estimatability. We explore the connections with stabilizability and detectability. We give a very general theorem about the equivalence of input-output stability and exponential stability of well-posed linear systems: the two are equivalent if the system is optimizable and estimatable. We conclude that a well-posed system is exponentially stable if and only if it is dynamically stabilizable and input-output stable. We illustrate the theory by two examples based on PDEs in two or more space dimensions: the wave equation and a structural acoustics model.

The class of well-posed systems includes many systems modeled by partial differential equations with boundary control and point sensing as well as many other systems with possibly unbounded control and observation. The closed-loop system created by applying state-feedback to any well-posed system is well-posed. A state-space realization of the closed loop is derived. A similar result holds for state estimation of a well-posed system. Also, the classical state-feedback/estimator structure extends to well-posed systems. In the final section state-space realizations for a doubly coprime factorization for well-posed systems are derived.

This chapter defines nominally well-posed hybrid systems and well-posed hybrid systems to be those hybrid systems, vaguely speaking, for which graphical limits of graphically convergent sequences of solutions, with no perturbations and with vanishing perturbations, respectively, are still solutions. In a classical setting, a well-posed problem is often defined as one in which a solution exists, is unique, and depends continuously on parameters. For hybrid dynamical systems, insisting on uniqueness of solutions and on their continuous dependence on initial conditions is very restrictive and, as it turns out, not necessary to develop a reasonable stability theory. In fact, stability theory results are possible for a quite general class of hybrid systems. The class of well-posed hybrid systems includes the Krasovskii regularization of a general hybrid system and, more generally, it includes every hybrid system meeting some mild regularity assumptions on the data.

We derive absolute stability results for well-posed innite-dimensional systems which, in a sense, extend the well-known circle criterion to the case that the underlying linear system is the series interconnection of an exponentially stable well-posed innite-dimensional system and an integrator and the nonlinearity satises a sector condition of the formh(u);(u) aui0 for some constant a> 0. These results are used to prove convergence and stability properties of low-gain integral feedback control applied to exponentially stable, linear, well-posed systems subject to actuator nonlinearities. The class of actuator nonlinearities under consideration contains standard nonlinearities which are important in control engineering such as saturation and deadzone.

Well-posed systems—The LTI case and beyond

The non-regular, incomplete and singular differential dynamical systems are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalised hypersurface) being the ambient space of the system. The system is the essentially non-regular differential dynamical system if there is observed reduction of the configuration space to the 'proper' configuration space of the system. The dynamical system, which is not observed by the solutions in the entire configuration space, is called the incomplete system. The system such that the dynamic space (the solution space) is not the topologically closed subspace of the ambient space of the system is called the singular dynamical system. The points in the topological closure of the configuration space that are not contained in the solution space of the dynamical system are called the impasse points of the system. The points in the topological closure of the solution space that are not contained in the solution space of the system are called the singularity points of the dynamical system. The systematic classification of the impasse points in dynamical systems is included which extends the classification proposed for the systems defined by the differential algebraic equations. The well posed and the relatively well-posed dynamical systems are defined. These are the systems, which are considered most often in the modelling of real systems. The classification which includes the non-regular, the incomplete, the well posed, and the relatively well-posed dynamical systems is proposed. The trajectory available impasse points have been extended for the general differential dynamical systems. The trajectory available impasse points are important notion in the modelling and in the simulation of real systems which exhibit the discontinuous trajectory behaviour. At the time when the trajectory of the system approached the singularity point, one usually observes the discontinuity in the dynamic behaviour of the system or the 'switching' of the trajectories which is possibly continuous takes place in the system. The examples are enclosed which illustrate the considerations.

Many infinite-dimensional linear systems can be modelled in a Hilbert space setting. Others, such as those dealing with heat transfer or population dynamics, need to be set more generally in Banach spaces. This is the first book dealing with well-posed infinite-dimensional linear systems with an input, a state, and an output in a Hilbert or Banach space setting. It is also the first to describe the class of non-well-posed systems induced by system nodes. The author shows how standard finite-dimensional results from systems theory can be extended to these more general classes of systems, and complements them with new results which have no finite-dimensional counterpart. Much of the material presented is original, and many results have never appeared in book form before. A comprehensive bibliography rounds off this work which will be indispensable to all working in systems theory, operator theory, delay equations and partial differential equations

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna-Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

Nominally well-posed hybrid inclusions are a class of hybrid dynamical systems characterized by outer/upper semicontinuous dependence of solutions on initial conditions. In the context of reachable sets of hybrid systems, this property guarantees outer/upper semicontinuous dependence with respect to both initial conditions and time. This article defines a counterpart to the notion of nominal well-posedness, referred to as nominal inner well-posedness, which ensures inner/lower semicontinuous dependence of solutions on initial conditions. Consequently, it is shown that reachable sets of nominally inner well-posed hybrid systems depend inner/lower semicontinuously on initial conditions and time under appropriate assumptions. Sufficient conditions guaranteeing nominal inner well-posedness are provided and demonstrated with an example.

A time delay present in the observation represents a mathematical challenge in an output feedback stabilization for linear infinite-dimensional systems. It is well known that for a linear hyperbolic system, a stabilizing output feedback control may become unstable when the observation has a time delay. For the fixed time delay in observation, the problem for one-dimensional partial differential equations (PDEs) has been solved by the observer-based feedback in the time interval where the observation is available and the predictor where the observation is not available. However, the generalization to multidimensional PDE systems has been a long-standing unsolved problem. In this paper, we investigate the problem from operator point of view for abstract first-order equation setting of infinite-dimensional systems. We formulate the problem in the framework of the well-posed and regular linear systems and solve it in the operator form. The result is then applied to the stabilization of a multidimensional Schrodinger equation.

An FFT-volume integral equation formulation based on equivalent polarization currents is presented for the analysis of inhomogeneous dielectric objects. The Galerkin discretization scheme used herein incorporates piecewise constant functions and results into well-posed systems even for scatterers of high contrast. Moreover, the original 6-D (volume-volume) Galerkin inner products are reduced to 4-D (surface-surface) integrals, allowing existing algorithms to compute with high accuracy the pertaining singular multidimensional integrals. Finally, the resulting numerical system is solved by means of iterative methods and the convergence is accelerated with the help of an analytical preconditioner.

In this chapter, we introduce a general class of hybrid dynamical systems that operate as follows. While the discrete state remains unchanged, the continuous one obeys a definite ordinary differential equation. Transition to another discrete state causes an alteration of this equation. The discrete state keeps its value, while the continuous one remains in a certain specific region and evolves as soon as this region is left. Furthermore, we introduce a number of basic notions related to the above class. Among them, there is, for example, that of the invariant domain, as well as of deterministic and well-posed systems. In particular, we provide mathematically rigorous definitions of some notions that were employed in Chapter 2.

This chapter defines local pre-asymptotic stability for a compact (closed and bounded) set and studies its properties for systems that are nominally well-posed or well-posed. While Chapter 3 dealt with global (and uniform) pre-asymptotic stability, the more general local pre-asymptotic stability is studied in this chapter, although for the more restrictive case of compact sets. Properties of the basin of attraction and uniformity of convergence are here analyzed, and some general examples of locally pre-asymptotically stable sets are given. The chapter reveals that, for nominally well-posed hybrid systems, pre-asymptotic stability turns out to be equivalent to uniform pre-asymptotic stability. For well-posed systems, pre-asymptotic stability turns out to be equivalent to uniform, robust pre-asymptotic stability and implies the existence of a Smooth Lyapunov function.

We introduce polynomial stabilizability and detectability of wellposed systems in the sense that a feedback produces a polynomially stable C 0-semigroup. Using these concepts, the polynomial stability of the given C 0-semigroup governing the state equation can be characterized via polynomial bounds on the transfer function. We further give sufficient conditions for polynomial stabilizability and detectability in terms of decompositions into a polynomial stable and an observable part. Our approach relies on a recent characterization of polynomially stable C 0-semigroups on a Hilbert space by resolvent estimates.