Cayley Commutator-free Methods for Krotov-Type Algorithms in Quantum Optimal Control - Wembe Moafo, Boris Edgar; Ali, Usman; Meier, Torsten; Ober-Blöbaum, Sina
This paper presents a class of structure-preserving numerical methods for quantum optimal control problems, based on commutator-free Cayley integrators. Starting from the Krotov framework, we reformulate the forward and backward propagation steps using Cayley-type schemes that preserve unitarity and symmetry at the discrete level. This approach eliminates the need for matrix exponentials and commutators, leading to significant computational savings while maintaining higher-order accuracy. We first recall the standard linear setting and then extend the formulation to nonlinear Schrödinger and Gross-Pitaevskii equations using a Cayley-polynomial interpolation strategy. Numerical experiments on state-transfer problems illustrate that the CF-Cayley method achieves the same accuracy as high-order exponential or Cayley-Magnus schemes at substantially lower cost, especially for longtime or highly oscillatory dynamics. In the nonlinear regime, the structure-preserving properties of the method ensure stability and norm conservation, making it a robust tool for large-scale quantum control simulations. The proposed framework thus bridges geometric integration and optimal control, offering an efficient and reliable alternative to existing exponential-based propagators.

Non static exponential turnpike property for optimal control problems with symmetries and boundary conditions - Maslovskaya, Sofya; Ober-Blöbaum, Sina; Wembe Moafo, Boris Edgar
Optimal control problems with symmetries often admit a non stationary turnpike property called trim turnpike, which characterizes the convergence of optimal solutions to certain symmetry induced trajectories called trim primitives. In this paper we establish an exponential trim turnpike property for a class of optimal control problems with structural properties related to Abelian Lie group symmetries. The key ingredient of our approach is the introduction of an appropriate reduced optimal control problem. We show that extremals of the original problem can be characterized through a reduced Hamiltonian boundary value problem that coincides with the optimality system of the reduced problem. Under a hyperbolicity assumption on the equilibrium of the corresponding reduced Hamiltonian system we prove that optimal trajectories remain exponentially close, up to boundary layers near the endpoints, to a trim primitive defined by the static reduced problem. The theoretical results are illustrated on three representative examples: linear and nonlinear problems with quadratic cost and the Kepler orbital transfer problem.

Numerical approaches to entangling dynamics from variational principles - Offen, Christian; Wembe Moafo, Boris Edgar; Ares, Laura; Sperling, Jan; Ober-Blöbaum, Sina (Journal of Physics A: Mathematical and Theoretical)
Abstract In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states, together with the discretization of the space of variables for numerical computations. As a first approach, we apply linear splitting methods to the restricted, continuous equations of motion derived from variational principles. We utilize an exchange interaction Hamiltonian to confirm that the numerical and analytical solutions coincide in the limit of small time steps. The application to different Hamiltonians shows the wide applicability of the method to detect dynamical entanglement. To avoid the derivation of analytical solutions for complex dynamics, we consider variational, numerical integration schemes, introducing a variational discretization for Lagrangians linear in velocities. Here, we examine and compare two approaches: one in which the system is discretized before the restriction is applied, and another in which the restriction precedes the discretization. We find that the ‘first-discretize-then-restrict’ method becomes numerically unstable, already for the example of an exchange-interaction Hamiltonian, which can be an important consideration for the numerical analysis of constrained quantum dynamics. Thereby, broadly applicable numerical tools, including their limitations, for studying entanglement over time are established for assessing the entangling power of processes that are used in quantum information theory.

Trim turnpikes for optimal control problems with symmetries - Flaßkamp, Kathrin; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Wembe Moafo, Boris Edgar (Mathematics of Control, Signals, and Systems)
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.

Adaptive higher order reversible integrators for memory efficient deep learning - Maslovskaya, Sofya; Ober-Blöbaum, Sina; Offen, Christian; Singh, Pranav; Wembe Moafo, Boris Edgar
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.

Multiphoton, multimode state classification for nonlinear optical circuits - Kopylov, Denis A.; Offen, Christian; Ares, Laura; Wembe Moafo, Boris Edgar; Ober-Blöbaum, Sina; Meier, Torsten; Sharapova, Polina R.; Sperling, Jan (Physical Review Research)
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.

Commutator-free Cayley methods - Wembe Moafo, Boris Edgar; Offen, Cristian; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Singh, Pranav (J. Comput. Appl. Math)
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.

Multiphoton, multimode state classification for nonlinear optical circuits - Kopylov, Denis; Offen, Christian; Ares, Laura; Wembe Moafo, Boris Edgar; Ober-Blöbaum, Sina; Meier, Torsten; Sharapova, Polina; Sperling, Jan
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.

Accessibility properties of abnormal geodesics in optimal control illustrated by two case studies - Bonnard, Bernard; Rouot, Jérémy; Wembe Moafo, Boris Edgar (Mathematical Control and Related Fields)
Keywords: Applied Mathematics; Control and Optimization; General Medicine

Abnormal geodesics in 2D-Zermelo navigation problems in the case of revolution and the fan shape of the small time balls - Bonnard, B.; Cots, O.; Gergaud, J.; Wembe Moafo, Boris Edgar (Systems & Control Letters)
Keywords: Electrical and Electronic Engineering; Mechanical Engineering; General Computer Science; Control and Systems Engineering

Homotopic approach for turnpike and singularly perturbed optimal control problems - Cots, Olivier; Gergaud, Joseph; Wembe Moafo, Boris Edgar (ESAIM: Proceedings and Surveys)
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.
Keywords: Immunology

Geometric and Numerical Methods in Optimal Control and Zermelo Problems on Revolution Surfaces - Applications - Wembe Moafo, Boris Edgar

Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow - Balsa, Carlos; Cots, Olivier; Gergaud, Joseph; Wembe Moafo, Boris Edgar (Lecture Notes in Electrical Engineering)

A Zermelo navigation problem with a vortex singularity - Bonnard, Bernard; Cots, Olivier; Wembe Moafo, Boris Edgar (ESAIM: Control, Optimisation and Calculus of Variations)
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.
Keywords: Computational Mathematics; Control and Optimization; Control and Systems Engineering