A bound on the solutions of a nonlinear volterra equation

AbstractWe study the scalar, nonlinear Volterra integrodifferential equation (∗), x′(t) + ∫[0,t] g(x(t − s)) dμ(s) = f(t) (t ⩾ 0). We let g be continuous, μ positive definite, and f integrable over (0, ∞). The standard assumption on g which yields boundedness of the solutions of (∗) prevents g(x) from growing faster than an exponential as x → ∞. Here we present a weaker condition on g, which does not restrict the growth rate of g(x) as x → ∞, but which still implies that the solutions of (∗) are bounded. In particular, when g is nondecreasing and either nonnegative or odd, we get bounds which are independent of g

On a nonconvolution Volterra resolvent

AbstractUnder fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝0t r(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L∞ space into another weighted L∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations

Spectral Decomposition and Invariant Manifolds for Some Functional Partial Differential Equations

AbstractWe study the integrodifferential convolution equationddt(x+μ∗x)−Ax−ν∗x=fon [0, +∞),x=φon (−∞, 0],as well as a nonlinear perturbation of the corresponding homogeneous equation. HereAis the generator of an analytic semigroup on a Hilbert spaceH, andμandνare operator-valued dominated measures with values inL(H) andL(D(A), H) respectively. Under the assumption that the operator given by the Laplace transform of the left-hand side of the equation is boundedly invertible on some right half-plane and on a line in the left half-plane, parallel to the imaginary axis, we decompose the solutions into components with different exponential growth rates. We construct projectors onto the stable and unstable subspaces, which are then used for the construction of stable and unstable manifolds for the nonlinear equation, which can have a fully nonlinear character. The results are applied to two equations of parabolic type. Moreover, the spectrum of the generator of the translation semigroup in various weighted spaces is determined, including the stable and unstable subspaces of our problem

De Branges-Rovnyak realizations of operator-valued Schur functions on the complex right half-plane

We give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization of an arbitrary given operator Schur function defined on the complex right-half plane. We work the theory out fully in the right-half plane, without using results for the disk case, in order to expose the technical details of continuous-time systems theory. At the end of the article, we make explicit the connection to the corresponding classical de Branges-Rovnyak realizations for Schur functions on the complex unit disk.Comment: 68 pages: General polishing; no essential change

Weak admissibility does not imply admissibility for analytic semigroups

Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator $B$ defined on an infinite-dimensional Hilbert space $U$ is an admissible control operator if for every element $u \in U$ the vector $Bu$ defines an admissible control operator. The other conjecture says that $B$ is an admissible control operator if a certain resolvent estimate is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we construct a semigroup example showing that the first estimate in the Hille-Yosida theorem is not sufficient to conclude boundedness of the semigroup

Coprime factorization and optimal control on the doubly infinite discrete time axis

We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic Riccati equation (of an arbitrary realization) have a solution (in general unbounded and even non densely defined) and that a coupling condition involving these solutions is satisfied

Functional model realizations for Schur functions on C+

For an arbitrary given operator Schur function defined on the complex right-half plane, we give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization. Topics appearing only in the right-half-plane setting, such as the extrapolation space, are also discussed

Well-posed linear systems

Indispensable to all working in systems theory, operator theory, delay equations and partial differential equations