## Isabelle Schneider and Babette de Wolff (FU Berlin)

#### Abstract

Pyragas control is a time delayed feedback control scheme designed to stabilize periodic orbits. From an implementation perspective, its main advantage is that the only knowledge required is the period of the target periodic orbit.

In situations where the uncontrolled dynamical system is equivariant with respect to some symmetry group, we can use equivariant Pyragas control to select and stabilize a specific spatio-temporal pattern.

In the first part of the talk, will review some results on Pyragas control and equivariant Pyragas control, with an emphasis on the dynamical properties of both the uncontrolled and the controlled dynamical system. We will then present results on the experimental implementation of equivariant Pyragas control in chemical oscillators.

## Wim Michiels (KU Leuven)

Slides

#### Abstract

In the presentation I will address algebraic decision problems on a finitely generated semigroup of matrices, which relate to the analysis of systems described by linear time-invariant delay-differential algebraic equations (DDAEs). The H2 norm of the input-output map of an exponentially stable system may not be finite (i.e. there exists a direct, static connection between input and output) even if there are seemingly no direct feedthrough terms in the DDAE model. It is shown that necessary and sufficient conditions for a finite H2 norm depend on the rational (in)dependence of the delay values, yet a system with rationally dependent delays can always be transformed into another system with rationally independent delays, without altering its H2 norm. The conditions for a finite H2 norm consist of an infinite number of linear equations and induce an algebraic decision problem, which bares similarities to the observability problem of a switched linear system.

As a core result, I will present a recursion formula for multi-variable powers of a set matrices, which can be interpreted as a natural generalization of the Cayley-Hamilton theorem for the single matrix case, and I will show how it allows us to turn the finiteness conditions for the H2 norm of a DDAE into a check of a finite number of algebraic equations. Since the number of equations to be checked (still) has a poor scalability with respect to the size of the problem (in particular, it exhibits exponential growth in the number of delays), I also propose a relaxed (i.e. sufficient) finiteness condition. The relaxed condition is shown to be computationally tractable and to imply the existence of both a joint triangularization of the matrices in the delay-difference equation associated to the DDAE, and a transformation to a neutral type system, without any need for differentiation of inputs and outputs. Finally, I will comment on the closeness of the finiteness conditions and discuss related open questions.

Even though quite technical, the presented results illustrate the richness of seemingly very simple continuous-time DDAEs, requiring an unexpected visit to the theory of multi-dimensional systems and tools from switched systems analysis.

This is joint work with Marco Antonio Gomez Alvarez (Universidad de Guanajuato) and Raphael Jungers (UC Louvain). The exact finiteness criterion and the generalization of the Cayley-Hamilton theorem have been published in (Gomez, Jungers, & Michiels, 2020). The other results concern work in progress.

1. Gomez, M. A., Jungers, R. M., & Michiels, W. (2020). On the m-dimensional Cayley-Hamilton theorem and its application to an algebraic decision problem inferred from the H2 norm analysis of delay systems. Automatica, 113, 108761. doi: 10.1016/j.automatica.2019.108761

## Jaqueline Mesquita (Brasília)

#### Abstract

In this talk, we will introduce the measure neutral functional differential equations with state-dependent delays and discuss their generality. Also, we will show that these equations encompass impulsive neutral FDEs with state-dependent delays and also, neutral functional dynamic equations on time scales. We will provide some results concerning existence and uniqueness of solutions, and present some examples.

## Alessia Andò (Udine)

Slides

#### Abstract

I will present an analysis of the convergence of (a variant of) the piecewise orthogonal collocation for periodic solutions of retarded functional differential equations defined by a generic right-hand side (Andò & Breda, 2020). Such analysis is highly based on (Maset, 2016) where a general framework for solving a certain class of boundary value problems (BVPs) is presented and accompanied by a rigorous proof of convergence of the corresponding iterative method. The novel contributions consist in the proofs of the validity of the assumptions required to apply the abstract approach of (Maset, 2016) in the case of periodic BVPs. Indeed, although the general BVP in (Maset, 2016) considers the presence of unknown parameters, it does not explicitly deal with the periodic case. In the presentation I will highlight the role of the period as the (main) unknown parameter of the problem, which leads to some effort in validating the required assumptions, being it directly linked to the course of time. It also affects the regularity that must be required from the functionals involved, as well as the choice of the relevant spaces where the solution, its derivative or the states must lie. I will conclude the presentation with some comments on the possibility of extending the convergence proof to the case of state-dependent delay differential equations and renewal equations.

1. Andò, A., & Breda, D. (2020). Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations. SIAM J. Numer. Anal. https://arxiv.org/abs/2008.07604
2. Maset, S. (2016). An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations. Numer. Math., 133(3), 525–555. doi: 10.1007/s00211-015-0754-1

## Sebastiaan Janssens (Utrecht)

Slides

#### Abstract

Using dual perturbation theory in a non-sun-reflexive context, in (Janssens, 2019) a correspondence was established between a class of abstract delay differential equations (DDEs) and a class of abstract $\text{weak}^{\star}$ integral equations involving the sun-star adjoint of a translation-like strongly continuous semigroup. For this purpose, the sun dual of the underlying state space was also characterized.

In this talk we explain how the above work, as well as earlier work by Diekmann and Gyllenberg on other classes of delay equations, has motivated the general approach taken in (Janssens, 2020). We introduce the notion of an admissible range and an admissible perturbation for a given $\mathcal{C}_0$-semigroup $T_0$ on a Banach space $X$ that is not assumed to be sun-reflexive with respect to $T_0$. We investigate the relationship between admissible ranges for $T_0$ and the subspace $X^{\odot\times}$ of $X^{\odot\star}$ introduced by Van Neerven. We answer two questions about robustness of admissibility with respect to bounded linear perturbations and we use these answers to study the semilinear problem and its linearization.

Partly as an application, and partly as a justification of existing work on local bifurcations in models taking the form of abstract DDEs, we compare the construction of center manifolds in the non-sun-reflexive case with known results by Diekmann and Van Gils for the sun-reflexive case. We show that a systematic use of the space $X^{\odot\times}$ facilitates a generalization of the existing results with relatively little effort. In this context we also give sufficient conditions for the existence of appropriate spectral decompositions of $X$ and $X^{\odot\times}$ without assuming that the linearized semiflow is eventually compact. A center manifold theorem for the motivating class of abstract DDEs then follows as a particular case.

1. Janssens, S. G. (2019). A class of abstract delay differential equations in the light of suns and stars. Part I. Preprint. https://arxiv.org/abs/1901.11526
2. Janssens, S. G. (2020). A class of abstract delay differential equations in the light of suns and stars. Part II. Preprint. https://arxiv.org/abs/2003.13341

## Alejandro López Nieto (FU Berlin)

#### Abstract

One of the key ingredients in the description of the global dynamics of scalar delay equations with monotone feedback are periodic orbits. In this setting I will discuss a new tool that permits an extremely accurate description of the set of periodic solutions, under additional symmetry assumptions.

## Irene Seifert (Heidelberg)

Slides

#### Abstract

Polyfold theory was developed in the context of symplectic geometry. Although invented for the study of moduli spaces of J-holomorphic curves, it seems quite natural in the context of delay equations. It provides a new sense of differentiability, called sc-differentiability, and a whole corresponding calculus. In particular, there is an Implicit Function Theorem for sc-smooth sc-Fredholm maps. Since reparametrization maps are sc-smooth between appropriate Banach spaces, the IFT can be used in the context of delay differential equations.

In my talk, I want to explain the very basics of polyfold theory and then give an example of how these methods can be used to prove an existence result for periodic solutions of delay differential equations. If time permits, I will be happy to discuss possible further projects.

## Davide Liessi (Udine)

Slides

#### Abstract

I will present a method based on pseudospectral collocation for approximating the spectra of evolution operators of linear renewal equations (Breda & Liessi, 2018), with some hints to the cases of delay differential equations (Breda, Maset, & Vermiglio, 2012), (Breda, Maset, & Vermiglio, 2015) and coupled renewal and delay differential equations (Breda & Liessi, 2020).

The method allows to study the stability of periodic orbits via the principle of linearised stability and Floquet theory.

I will also present the recent application of the method to linear neutral renewal equations and I will briefly touch upon some complementary topics related to linearising nonlinear equations.

1. Breda, D., & Liessi, D. (2018). Approximation of eigenvalues of evolution operators for linear renewal equations. SIAM J. Numer. Anal., 56(3), 1456–1481. doi: 10.1137/17M1140534
2. Breda, D., Maset, S., & Vermiglio, R. (2012). Approximation of eigenvalues of evolution operators for linear retarded functional differential equations. SIAM J. Numer. Anal., 50(3), 1456–1483. doi: 10.1137/100815505
3. Breda, D., Maset, S., & Vermiglio, R. (2015). Stability of Linear Delay Differential Equations (p. xii+158). Springer, New York. doi: 10.1007/978-1-4939-2107-2
4. Breda, D., & Liessi, D. (2020). Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations. Ric. Mat. doi: 10.1007/s11587-019-00467-7

## Francesca Scarabel (York University)

Slides

#### Abstract

No software is currently available for the numerical bifurcation analysis of renewal equations. We propose a technique which consists in approximating the renewal equation via a system of ODE (via pseudospectral approximation), and then applying well-established numerical bifurcation software for ODE.

If applied directly to the formulation of the dynamical system in the space $L^1$, the approximation returns an algebraic condition that should be coupled with the ODE system (Breda, Diekmann, Gyllenberg, Scarabel, & Vermiglio, 2016). To avoid this condition and to improve the efficiency of the numerical computations, we first construct a dynamical system in the subspace of absolutely continuous functions (interpreted as subspace of $NBV$, the normalized bounded variation functions) by looking at the integrated version of the original state. In this way, we obtain a differential equation for the integrated state, and the discretized system takes the form of an ODE. The theoretical framework for interpreting the renewal equation as a dynamical system in a subspace of $NBV$ has been recently introduced in (Diekmann & Verduyn Lunel, 2020).

We show that the equilibria of the approximating ODE system are in one-to-one correspondence with those of the renewal equation, and the stability properties are approximated with exponential order of convergence as the dimension of the approximating system increases. Several examples illustrate the efficacy of the approach, and the substantial improvement in computation times compared to approximating the system directly in the state space $L^1$.

1. Breda, D., Diekmann, O., Gyllenberg, M., Scarabel, F., & Vermiglio, R. (2016). Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis. SIAM J. Appl. Dyn. Syst, 15(1), 1–23.
2. Diekmann, O., & Verduyn Lunel, S. (2020). Twin semigroups and delay equations. https://arxiv.org/abs/1906.03409