Sascha Trostorff

On evolutionary equations with material laws containing fractional integrals

A well-posedness result for a time-shift invariant class of evolutionary operator equations involving material laws with fractional time-integrals of order α ϵ ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time-)derivative established as a normal operator in a suitable L2 type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann-Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker-Planck equation, equations describing super-diffusion and sub-diffusion processes, and a Kelvin-Voigt type model in fractional visco-elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright

On a comprehensive class of linear control problems

We discuss a class of linear control problems in a Hilbert space setting. This class encompasses such diverse systems as port-Hamiltonian systems, Maxwells equations with boundary control or the acoustic equations with boundary control and boundary observation. The boundary control and observation acts on abstract boundary data spaces such that the only geometric constraint on the underlying domain stems from requiring a closed range constraint for the spatial operator part, a requirement which for the wave equation amounts to the validity of a Poincare–Wirtinger-type inequality. We also address the issue of conservativity of the control problems under consideration.

On integro‐differential inclusions with operator‐valued kernels

We study integro-differential inclusions in Hilbert spaces with operator-valued kernels and give sufficient conditions for the well-posedness. We show that several types of integro-differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well-posedness within this framework. As an example, we apply our results to the equations of visco-elasticity and to a class of nonlinear integro-differential inclusions describing phase transition phenomena in materials with memory. Copyright

Exponential Stability for Linear Evolutionary Equations.

We give an approach to exponential stability within the framework of evolutionary equations due to [R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci., 32(14):1768-1803,2009]. We derive sufficient conditions for exponential stability in terms of the material law operator, which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.

A note on a class of conservative, well-posed linear control systems

We discuss a class of linear control problems in a Hilbert space setting. The aim is to show that these control problems fit in a particular class of evolutionary equations such that the discussion of well-posedness becomes easily accessible. Furthermore, we study the notion of conservativity. For this purpose we require additional regularity properties of the solution operator in order to allow point-wise evaluations of the solution. We exemplify our findings by a system with unbounded control and observation operators.

On some models in linear thermo-elasticity with rational material laws

In the present work, we shall consider some common models in linear thermo-elasticity within a common structural framework. Due to the flexibility of the structural perspective we will obtain well-posedness results for a large class of generalized models allowing for more general material properties such as anisotropies, inhomogeneities, etc.

On higher index differential-algebraic equations in infinite dimensions

We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for arbitrary initial values in a distributional sense. Moreover, we construct a nested sequence of subspaces for initial values in order to obtain classical solutions.

A note on a two-temperature model in linear thermoelasticity

We discuss the so-called two-temperature model in linear thermoelasticity and provide a Hilbert space framework for proving well-posedness of the equations under consideration. With the abstract perspective of evolutionary equations, the two-temperature model turns out to be a coupled system of the elastic equations and an abstract ordinary differential equation (ODE). Following this line of reasoning, we propose another model which is entirely an abstract ODE. We also highlight an alternative method for a two-temperature model, which might be of independent interest.

Some remarks on local activity and local passivity

We study local activity and its contrary, local passivity, for linear systems and show that generically an eigenvalue of the system matrix with positive real part implies local activity. If all state variables are port variables we prove that the system is locally active if and only if the system matrix is not dissipative. Local activity was suggested by Leon Chua as an indicator for the emergence of complexity of nonlinear systems. We propose an abstract scheme which indicates how local activity could be applied to nonlinear systems and list open questions about possible consequences for complexity.

On a Class of Block Operator Matrices in System Theory

We consider a class of block operator matrices arising in the study of scattering passive systems, especially in the context of boundary control problems. We prove that these block operator matrices are indeed a subclass of block operator matrices considered in (Trostorff in J Funct Anal 267(8):2787–2822, 2014), which can be characterized in terms of an associated boundary relation.