@article{59792, abstract = {{Abstract Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem. }}, author = {{Flaßkamp, Kathrin and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Wembe Moafo, Boris Edgar}}, issn = {{0932-4194}}, journal = {{Mathematics of Control, Signals, and Systems}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Trim turnpikes for optimal control problems with symmetries}}}, doi = {{10.1007/s00498-025-00408-w}}, year = {{2025}}, } @unpublished{58544, abstract = {{We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes. }}, author = {{Kopylov, Denis and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina and Sperling, Jan}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits }}}, year = {{2025}}, } @unpublished{59794, abstract = {{The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover, deep networks are often unable to handle time-series data appearing at irregular intervals. These issues can be resolved by considering continuous-depth networks based on the neural ODE framework in combination with reversible integration methods that allow for variable time-steps. Reversibility of the method ensures that the memory requirement for training is independent of network depth, while variable time-steps are required for assimilating time-series data on irregular intervals. However, at present, there are no known higher-order reversible methods with this property. High-order methods are especially important when a high level of accuracy in learning is required or when small time-steps are necessary due to large errors in time integration of neural ODEs, for instance in context of complex dynamical systems such as Kepler systems and molecular dynamics. The requirement of small time-steps when using a low-order method can significantly increase the computational cost of training as well as inference. In this work, we present an approach for constructing high-order reversible methods that allow adaptive time-stepping. Our numerical tests show the advantages in computational speed when applied to the task of learning dynamical systems.}}, author = {{Maslovskaya, Sofya and Ober-Blöbaum, Sina and Offen, Christian and Singh, Pranav and Wembe Moafo, Boris Edgar}}, title = {{{Adaptive higher order reversible integrators for memory efficient deep learning}}}, year = {{2025}}, } @article{62980, abstract = {{We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which is in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes.}}, author = {{Kopylov, Denis A. and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina R. and Sperling, Jan}}, issn = {{2643-1564}}, journal = {{Physical Review Research}}, number = {{3}}, publisher = {{American Physical Society (APS)}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits}}}, doi = {{10.1103/sv6z-v1gk}}, volume = {{7}}, year = {{2025}}, } @unpublished{62979, abstract = {{We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes.}}, author = {{Meier, Torsten and Sharapova, Polina R. and Sperling, Jan and Ober-Blöbaum, Sina and Wembe Moafo, Boris Edgar and Offen, Christian}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits}}}, year = {{2025}}, } @article{59507, abstract = {{Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.}}, author = {{Wembe Moafo, Boris Edgar and Offen, Cristian and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Singh, Pranav}}, journal = {{J. Comput. Appl. Math}}, number = {{15}}, title = {{{Commutator-free Cayley methods}}}, doi = {{10.1016/j.cam.2025.117184}}, volume = {{477}}, year = {{2025}}, } @article{35206, author = {{Bonnard, Bernard and Rouot, Jérémy and Wembe Moafo, Boris Edgar}}, issn = {{2156-8472}}, journal = {{Mathematical Control and Related Fields}}, keywords = {{Applied Mathematics, Control and Optimization, General Medicine}}, pages = {{0--0}}, publisher = {{American Institute of Mathematical Sciences (AIMS)}}, title = {{{Accessibility properties of abnormal geodesics in optimal control illustrated by two case studies}}}, doi = {{10.3934/mcrf.2022052}}, year = {{2022}}, } @article{33869, author = {{Bonnard, B. and Cots, O. and Gergaud, J. and Wembe Moafo, Boris Edgar}}, issn = {{0167-6911}}, journal = {{Systems & Control Letters}}, keywords = {{Electrical and Electronic Engineering, Mechanical Engineering, General Computer Science, Control and Systems Engineering}}, publisher = {{Elsevier BV}}, title = {{{Abnormal geodesics in 2D-Zermelo navigation problems in the case of revolution and the fan shape of the small time balls}}}, doi = {{10.1016/j.sysconle.2022.105140}}, volume = {{161}}, year = {{2022}}, } @article{33871, abstract = {{The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.}}, author = {{Cots, Olivier and Gergaud, Joseph and Wembe Moafo, Boris Edgar}}, issn = {{2267-3059}}, journal = {{ESAIM: Proceedings and Surveys}}, keywords = {{Immunology}}, pages = {{43--53}}, publisher = {{EDP Sciences}}, title = {{{Homotopic approach for turnpike and singularly perturbed optimal control problems}}}, doi = {{10.1051/proc/202171105}}, volume = {{71}}, year = {{2021}}, } @phdthesis{33872, author = {{Wembe Moafo, Boris Edgar}}, pages = {{180}}, title = {{{Geometric and Numerical Methods in Optimal Control and Zermelo Problems on Revolution Surfaces - Applications}}}, year = {{2021}}, } @inbook{33870, author = {{Balsa, Carlos and Cots, Olivier and Gergaud, Joseph and Wembe Moafo, Boris Edgar}}, booktitle = {{Lecture Notes in Electrical Engineering}}, isbn = {{9783030586522}}, issn = {{1876-1100}}, publisher = {{Springer International Publishing}}, title = {{{Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow}}}, doi = {{10.1007/978-3-030-58653-9_22}}, year = {{2020}}, } @article{33866, abstract = {{Helhmoltz–Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. The analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. The explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.}}, author = {{Bonnard, Bernard and Cots, Olivier and Wembe Moafo, Boris Edgar}}, issn = {{1292-8119}}, journal = {{ESAIM: Control, Optimisation and Calculus of Variations}}, keywords = {{Computational Mathematics, Control and Optimization, Control and Systems Engineering}}, publisher = {{EDP Sciences}}, title = {{{A Zermelo navigation problem with a vortex singularity}}}, doi = {{10.1051/cocv/2020058}}, volume = {{27}}, year = {{2020}}, }