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We investigate flow-driven amoeboid motility as exhibited by microplasmodia of Physarum polycephalum. A poroelastic two-phase model with rigid boundaries is extended to the case of free boundaries and substrate friction. The cytoskeleton is modeled as an active viscoelastic solid permeated by a fluid phase describing the cytosol. A feedback loop between a chemical regulator, active mechanical deformations, and induced flows gives rise to oscillatory and irregular motion accompanied by spatio-temporal contraction patterns. We cover extended parameter regimes of active tension and substrate friction by numerical simulations in one spatial dimension and reproduce experimentally observed oscillation periods and amplitudes. In line with experiments, the model predicts alternating forward and backward ectoplasmatic flow at the boundaries with reversed flow in the center. However, for all cases of periodic and irregular motion, we observe practically no net motion. A simple theoretical argument shows that directed motion is not possible with a spatially independent substrate friction.

Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations., Comment: 21 pages, 8 figures , 6 videos

We investigate optimal control of dynamical systems which are affine, i.e., linear in control, but nonlinear in state. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible, a task also known as optimal trajectory tracking. To obtain well-behaved solutions to optimal control, a regularization term with coefficient $\varepsilon$ must be included in the cost functional. Assuming $\varepsilon$ to be small, we reinterpret affine optimal control problems as singularly perturbed differential equations. Performing a singular perturbation expansion, approximations for the optimal tracking of arbitrary desired trajectories are derived. For $\varepsilon=0$, the state trajectory may become discontinuous, and the control may diverge. On the other hand, the analytical treatment becomes exact. We identify the conditions leading to linear evolution equations. These result in exact analytical solutions for an entire class of nonlinear trajectory tracking problems. The class comprises, among others, mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model with a control acting on the activator., Comment: 5 pages, 1 figure, 11 pages appendix

Trajectory tracking of nonlinear dynamical systems with affine open-loop controls is investigated. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. We introduce exactly realizable desired trajectories as these trajectories which can be tracked exactly by an appropriate control. Exactly realizable trajectories are characterized mathematically by means of Moore-Penrose projectors constructed from the input matrix. The approach leads to differential-algebraic systems of equations and is considerably simpler than the related concept of system inversion. Furthermore, we identify a particularly simple class of nonlinear affine control systems. Systems in this class satisfy the so-called linearizing assumption and share many properties with linear control systems. For example, conditions for controllability can be formulated in terms of a rank condition for a controllability matrix analogously to the Kalman rank condition for linear time-invariant systems., Comment: 10 pages, no figures

This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of so-called exactly realizable trajectories is proposed. For exactly realizable desired trajectories exists a control signal which enforces the state to exactly follow the desired trajectory. For a given affine control system, these trajectories are characterized by the so-called constraint equation. This approach does not only yield an explicit expression for the control signal in terms of the desired trajectory, but also identifies a particularly simple class of nonlinear control systems. Based on that insight, the regularization parameter is used as the small parameter for a perturbation expansion. This results in a reinterpretation of affine optimal control problems with small regularization term as singularly perturbed differential equations. The small parameter originates from the formulation of the control problem and does not involve simplifying assumptions about the system dynamics. Combining this approach with the linearizing assumption, approximate and partly linear equations for the optimal trajectory tracking of arbitrary desired trajectories are derived. For vanishing regularization parameter, the state trajectory becomes discontinuous and the control signal diverges. On the other hand, the analytical treatment becomes exact and the solutions are exclusively governed by linear differential equations. Thus, the possibility of linear structures underlying nonlinear optimal control is revealed. This fact enables the derivation of exact analytical solutions to an entire class of nonlinear trajectory tracking problems with affine controls. This class comprises mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model., Comment: 240 pages, 36 figures, PhD thesis

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schl\"{o}gl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions., Comment: 22 pages, 3 figures, 2 tables

In two-dimensional reaction-diffusion systems, local curvature perturbations in the shape of traveling waves are typically damped out and disappear in the course of time. If, however, the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated wave shapes and to spreading spiral turbulence. For experimentally relevant parameter values, the photosensitive Belousov-Zhabotinsky reaction (PBZR) does not exhibit transversal wave instabilities. Here, we propose a mechanism to artificially induce these instabilities via a wave shape dependent spatio-temporal feedback loop, and study the emerging wave patterns. In numerical simulations with the modified Oregonator model for the PBZR using experimentally realistic parameter values we demonstrate the feasibility of this control scheme. Conversely, in a piecewise-linear version of the FitzHugh-Nagumo model transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilising flat wave propagation., Comment: 22 pages, 12 figures