In this presentation we study robust output regulation for systems described by linear partial differential equations, such as wave equations, convection-diffusion equations, or hyperbolic systems. We focus on the "abstract approach", where the PDE model is first reformulated as an infinite-dimensional linear system, and the controller design utilises the internal model based controllers that are available for the relevant class of abstract systems. This approach is powerful and it especially allows us to invoke suitable versions of the Internal Model Principle as part of our controller design. However, the controller design in the abstract setting can be tedious and technically demanding -- especially in the case of PDE systems with boundary control and observation. In this talk we focus on illustrating and discussing the key steps in the internal model controller design for selected PDE models. In particular, we demonstrate how the questions regarding the tuning of the controller parameters can be converted from the operator-theoretic setting back into the PDE domain. We also discuss the various benefits and disadvantages of the abstract approach compared to other available options.

In this presentation we discuss the background theory and history of the theory of robust output regulation for control systems modelled with partial differential equations (PDEs). In particular, we focus on the Internal Model Principle and its influence on robust controller design. In the second part we present new(ish) results on finite-dimensional and low-order controller design for parabolic PDE models, and the application of these results in the robust output tracking and disturbance rejection for fluid flows and temperatures.
The research contains joint work with Konsta Huhtala, Weiwei Hu (University of Georgia, US), Jukka-Pekka Humaloja (University of Alberta, CA), and Duy Phan (University of Innsbruck, AT).

In this paper, we consider robust output tracking problem of an undamped Euler-Bernoulli beam with boundary control and boundary observation. In particular, we study a cantilever beam which has control and observation at the free end. As our main result, we construct a finite-dimensional, internal model based controller for the output tracking of the beam system. In addition, we consider a case where the controller achieves the robust output tracking for the cantilever beam with distributed control and observation. Numerical simulations demonstrating the effectiveness of the controller are presented.

We consider a PDE-ODE model of a satellite and robust output regulation of the corresponding model. The satellite is composed of two flexible solar panels and a rigid center body. Exponential stability of the model is proved using passivity and resolvent estimates in the port-Hamiltonian framework. In addition, we construct a simple low-gain controller for robust output regulation of the satellite model.

This paper addresses the problem of output regulation of infinite-dimensional linear systems subject to input saturation. We focus on strongly stabilizable linear dissipative systems with collocated actuators and sensors. We generalize the output regulation theory for finite- dimensional linear systems subject to input saturation to the class of considered infinite- dimensional linear systems. The theoretic results are illustrated with an example where we consider the output regulation of a flexible satellite model that is composed of two identical flexible solar panels and a center rigid body.

We consider output tracking for a class of viscous nonlinear fluid flows including the incompressible 2D Navier–Stokes equations. The fluid is subject to in-domain inputs and disturbances. We construct an error feedback controller which guarantees approximate local velocity output tracking for a class of reference outputs. The control solution covers point velocity observations and assumes a smooth enough state space. Efficacy of the control solution is illustrated through a numerical example.

This paper investigates an interval turnpike result for the adjoints/costates of finite- and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral de- composition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further provide a numerical example with a semilinear heat equation to illustrate the results.

This paper studies linear model predictive control of real matrix-valued single delay systems. The delay system is written as an abstract infinite- dimensional control system which is then mapped into an infinite-dimensional discrete-time control system using Cayley-Tustin discretization. A constrained model predictive control (MPC) problem is formulated for the discrete-time system where a terminal penalty function is utilized to cast the infinite-horizon optimization problem into a finite-horizon one. The proposed MPC design is demonstrated on an example of constrained stabilization of a 2 × 2 system. We will demonstrate that the proposed discrete-time MPC law not only stabilizes the discrete-time system but can be utilized in stabilizing the original continuous- time system as well, which is due to several favorable properties of the Cayley-Tustin discretization.

Polynomial stability of semigroups appears frequently in the study of coupled systems of linear partial differential equations. This is especially the case when PDEs of mixed types are coupled through a shared boundary or inside a common spatial domain. In this presentation we consider a selection of such PDE systems and discuss how many of them can be viewed as particular cases of composite systems consisting of an unstable and a stable abstract linear differential equation. As our main results we present general conditions for proving the polynomial stability of coupled PDE systems in this abstract setting. The results are motivated by and primarily applicable for coupled PDEs on one-dimensional spatial domains, and they can in particular be used in analysing stability of networks of PDEs of mixed types.

References:
S. Nicaise, L. Paunonen and D. Seifert. Stability of abstract coupled systems, arXiv preprint, March 2024 https://arxiv.org/abs/2403.15253.

In this presentation we study the stability properties of strongly continuous semigroups generated by operators of the form A-BB*, where A generates a unitary group or a contraction semigroup, and B is a possibly unbounded operator. Such semigroups are encountered in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. In the case of multidimensional wave equations with viscous damping, the associated semigroup is often not uniformly exponentially stable, but is instead only "polynomially" or "non-uniformly stable". Motivated by such situations, we present general sufficient conditions for polynomial and non-uniform stability of the semigroup generated by A-BB* in terms of generalised observability-type conditions of the pair (B*,A). The proofs in particular involve derivation of resolvent estimates for the operator A-BB* on the imaginary axis. In addition, we apply the results in studying the stability of hyperbolic PDEs with partial or weak dampings.
The research is joint work with R. Chill, D. Seifert, R. Stahn and Y. Tomilov.

References:
R. Chill, L. Paunonen, D. Seifert, R. Stahn, and Y. Tomilov. Non-uniform stability of damped contraction semigroups, Analysis & PDE, 2023. Preprint available at https://arxiv.org/abs/1911.04804.

In this presentation we study the stability properties of hyperbolic linear partial differential equations, especially damped wave equations. A large class of such equations can be represented as abstract differential equations on a Hilbert space. In this presentation we study so-called "non-uniform stability" and "polynomial stability". The characteristic feature of these stability types is that the solutions of the abstract differential equation may decay at sub-exponential rates with time, and these rates depend on the smoothness of the initial data. This kind of stability is especially encountered in multidimensional wave equations with partial or weak dampings. As our main results we introduce new sufficient conditions for the non-uniform stability of the equations based on observability properties of its operators. We also apply our results for particular partial differential equations and discuss the optimality of the obtained degrees of stability.
The research is joint work with R. Chill, D. Seifert, R. Stahn and Y. Tomilov.

The results of the article are based on the article:
R. Chill, L. Paunonen, D. Seifert, R. Stahn, and Y. Tomilov. Non-uniform stability of damped contraction semigroups, Analysis & PDE, accepted for publication. Preprint available at https://arxiv.org/abs/1911.04804.