This thesis is dedicated to delay differential-algebraic equations (DDAEs), i.e., constraint dynamical systems where the rate of change depends on the current state and its past. Typical applications include - feedback control, where the delay is a direct consequence of the time required to measure the current state, compute the feedback, and implement the control action, - hybrid numerical-experimental testing environments, used for instance in earthquake engineering, - transmission and propagation delays, encountered for example in chemical reactions connected in series and wide-area power-networks, and - as a mathematical tool to analyze hyperbolic equations and time-integration schemes. The fact that algebraic equations can be included in the implicit system description fosters a rapid model development since complex models can be assembled from a library of existing models with well-defined interaction variables. From a mathematical point of view, DDAEs do not only feature difficulties already known from the theory of differential-algebraic equations (DAEs) and delay differential equations (DDEs) but pose additional challenges. For instance, initial trajectory problems for DDAEs may not be causal. Thus, even in a distributional solution space, they may not have a solution for all initial trajectories. This fact, combined with the infinite-dimensional character of delay equations and the high sensitivity to perturbation known from the theory of DAEs, renders DDAEs a challenging mathematical object. Consequently, the analysis of DDAEs is far from complete. The aim of this thesis is to address some of the many open problems. In the first part of the thesis, initial trajectory problems for linear time-invariant (LTI) and nonlinear DDAEs are discussed. We start our analysis with a distributional solution concept and establish the existence and uniqueness of solutions, whenever the DDAE is delay-regular. Jumps and Dirac-impulses in the solution can be avoided if the coefficient matrices of the LTI DDAE satisfy some algebraic conditions, which are obtained by tracking so-called primary discontinuities. We extend some of the results to the nonlinear setting resulting in existence and uniqueness results for a large class of nonlinear DDAEs. The second part of the thesis is dedicated to constructing a DDAE solely from a prescribed set of data points. Having a time-delay in the realization allows us to build an infinite-dimensional system from finitely many points capable of reproducing the transcendental character of the transfer function of a distributed parameter subsystem that models convection or transport. We construct a realization so that it interpolates the data set in the frequency domain and demonstrate its applicability with several numerical examples.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations arise naturally, if a feedback controller is applied to a descriptor system, since the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise smooth distributions and an algorithm to determine whether a DDAE is delay-regular.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov $n$-width.

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The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstraß form for regular matrix pencil, a complete characterization of the different propagation types is given and algebraic criteria in terms of the matrices are developed. The analysis, which is based on the method of steps, takes into account all possible inhomogeneities and history functions and thus serves as the worst-case scenario. Moreover, it reveals possible hidden delays in the DDAE. The new classification for DDAEs is compared to existing approaches in the literature and the impact of splicing conditions on the classification is studied.

We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.