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We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.

We consider two problems in boundary controllability of coupled wave/Kirchoff systems. Let Ω be a bounded region in R n , n ⩾2, with Lipschitz continuous boundary Γ . In the motivating structural acoustics application, Ω represents an acoustic cavity. Let Γ 0 be a flat subset of Γ which represents a flexible wall of the cavity. Let z denote the acoustic velocity potential, which satisfies a wave equation in Ω , and let v denote the displacement on Γ 0 , which satisfies a Kirchoff plate equation on Γ 0 . These equations are coupled via ∂z / ∂ν = v t on Γ 0 (where ν is the exterior unit normal to Γ 0 ), and the backpressure— z t appears in the Kirchoff equation. In the first problem, we consider a control u 0 in the Kirchoff equation on Γ 0 , and an additional control u 1 in the Neumann conditions on a subset Γ 1 of Γ , where Γ ⧹ Γ 1 satisfies geometric conditions. Using both controls, we obtain exact controllability of the wave and plate components in the natural state space. In the second problem we consider only the control u 0 . Without geometric conditions, exact controllability is not possible, but we show that for any initial data, we can steer the plate component exactly, and the wave component approximately.

We derive necessary conditions for exact controllability of infinite-dimensional systems described by x=Ax+Bu, where the generator A has a Riesz basis of eigenvectors and the input space is finite-dimensional. These conditions are in terms of the eigenvalues of A and the degree of unboundedness of B, but no further information about B is needed. Our results easily imply lack of exact controllability for a wide range of distributed parameter systems. It is known from partial differential equation (PDE) examples that the addition of damping into an exactly controllable second order system can destroy exact controllability. We discuss the effect of damping on such systems using our general conditions. We also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension.

In the sampled-data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled-data control. For finite-dimensional systems the necessary conditions are also known to be sufficient. We show that this equivalence extends to the infinite-dimensional case if the underlying semigroup is analytic. However, for general systems, the necessary conditions are not sufficient, nor are the sufficient conditions necessary. We prove this by a single example with a free parameter – one choice of parameter shows that the necessary conditions are too weak, and another choice shows that the sufficient conditions are too strong.

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup SH(t) on X. Let λ0 be a real number in the resolvent set of A, and let ϱ ϵ [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ0 − A)ϱCH(λ0 − A)−ϱ also generates a strongly continuous semigroup SK(t) on X which has the same exponential growth rate as SH(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H02[0, 1] × L2[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A − 1 2 - bounded feedback, but we can find an A e− 1 4 - bounded feedback, for any e > 0, which does stabilize the beam.

In this paper we consider distributed parameter control systems of the form x′(t) = Ax(t) + Bu(t) , where x(t) is in the state space X, u(t) is in an infinite dimensional control space U , and B is an unbounded input operator which is admissible in a standard sense. We find necessary conditions for there to exist a compact operator K : X → U such that A + BK generates a C 0 - semigroup which is of exponential growth type μ. In particular, for every γ > μ , A must have at most finitely many eigenvalues of finite multiplicity in { Re (z) ≥ γ} . Furthermore, the part of A corresponding to the remaining spectrum must generate a semigroup with type at most γ. We apply this to a model for a clamped Euler-Bernoulli plate with Dirichlet control applied to the entire boundary. We show that a compact state feedback cannot exponentially stabilize the plate in the state space of maximum regularity.