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We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems "out of thin air" introduced by Tucsnak and Weiss in 2003. The modified differential equation contains a new nonlinear damping term that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall--Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax--Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation. [ABSTRACT FROM AUTHOR]

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in L2[0;1). Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded. [ABSTRACT FROM AUTHOR]

This survey is an introduction to well-posed linear time-invariant (LTI) systems for non-specialists. We recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems, some of the available linear feedback theory. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, conservative systems and systems with a special structure that belong to these classes. We illustrate this theory by examples of systems governed by heat and wave equations. We develop local and global well-posedness results for LTI systems with nonlinear (in particular, bilinear) feedback, by extracting the abstract idea behind various proofs in the literature. We apply these abstract results to derive well-posedness results for the Burgers and Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

We consider well-posed linear systems whose state trajectories satisfy ẋ = Ax + Bu, where u is the input and A is an essentially skew-adjoint and dissipative operator on the Hilbert space X. This means that the domains of A* and A are equal and A* +A = -Q, where Q ≥ 0 is bounded on X. The control operator B is possibly unbounded, but admissible and the observation operator of the system is B*. Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u = -κy+v, with κ > 0, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if B is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if κ is sufficiently small, then the closed-loop system is exponentially stable. [ABSTRACT FROM AUTHOR]

Let A0 a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A0½) (with the norm ||z||½2 = ⟨A0z, z⟩) to another Hilbert space U. In Part I of this work we have proved that the system of equations z(t) + A0z(t) + ½C0∗C0ż(t) = C0∗u(t) = y(t) = - C0ż(t) + u(t) determines a well-posed linear system Σ with input n and output y, input and output space U, and state space X = D(A0½) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we show that Σ is exactly controllable if and only if it is exactly observable, if and only if it is exponentially stable. Moreover, if we denote by A the generator of the contraction semigroup associated with Σ (which acts on X), then Σ is exponentially stable if and only if one of the entries in the second column of (iωI - A)-1 is uniformly hounded as a function of ω ∈ ℝ. We also show that, under a mild assumption, Σ is approximately controllable if and only if it is approximately observable, if and only if it is strongly stable, if and only if the dual system is strongly stable. We prove many related results and we give examples based on wave and beam equations. [ABSTRACT FROM AUTHOR]

We study the exact controllability of two systems by means of a common finite- dimensional input function, a property called simultaneous exact controllability. Most of the time we consider one system to be infinite-dimensional and the other finite-dimensional. In this case we show that if both systems are exactly controllable in time T0 and the generators have no common eigenvalues, then they are simultaneously exactly controllable in any time T > T[SUB0]. Moreover, we show that similar results hold for approximate controllability. For exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class H[SUP1] and H[SUP2]. We apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end. Finally, we consider an example of two infinite-dimensional systems: we characterize the simultaneously reachable subspace for two strings controlled from a common end. The result is obtained using a recent generalization of a classical inequality of Ingham. [ABSTRACT FROM AUTHOR]

Abstract: Let be the generator of a strongly continuous semigroup on the Hilbert space , and let be a linear operator from to another Hilbert space (possibly unbounded with respect to , not necessarily admissible). We consider the problem of estimating the initial state (with respect to the norm of ) from the output function , given for all in a bounded interval . We introduce the concepts of estimatability and backward estimatability for (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating from . This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell’s principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations. [Copyright &y& Elsevier]