@article{63557,
  abstract     = {{We discretise a recently proposed new Lagrangian approach to optimal control problems with dynamics described by force-controlled Euler-Lagrange equations (Konopik et al., in Nonlinearity 38:11, 2025). The resulting discretisations are in the form of discrete Lagrangians. We show that the discrete necessary conditions for optimality obtained provide variational integrators for the continuous problem, akin to Karush-Kuhn-Tucker (KKT) conditions for standard direct approaches. This approach paves the way for the use of variational error analysis to derive the order of convergence of the resulting numerical schemes for both state and costate variables and to apply discrete Noether’s theorem to compute conserved quantities, distinguishing itself from existing geometric approaches. We show for a family of low-order discretisations that the resulting numerical schemes are ‘doubly-symplectic’, meaning they yield forced symplectic integrators for the underlying controlled mechanical system and overall symplectic integrators in the state-adjoint space. Multi-body dynamics examples are solved numerically using the new approach. In addition, the new approach is compared to standard direct approaches in terms of computational performance and error convergence. The results highlight the advantages of the new approach, namely, better performance and convergence behaviour of state and costate variables consistent with variational error analysis and automatic preservation of certain first integrals.}},
  author       = {{Konopik, Michael and Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Sato Martín de Almagro, Rodrigo T.}},
  issn         = {{1384-5640}},
  journal      = {{Multibody System Dynamics}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{On the variational discretisation of optimal control problems for unconstrained Lagrangian dynamics}}},
  doi          = {{10.1007/s11044-025-10138-1}},
  year         = {{2026}},
}

@article{59792,
  abstract     = {{<jats:title>Abstract</jats:title>
          <jats:p>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 <jats:italic>static turnpike to manifold turnpike</jats:italic> we extend the <jats:italic>exponential turnpike property</jats:italic> to the <jats:italic>exponential trim turnpike</jats:italic> 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 <jats:italic>trim turnpike theorem</jats:italic> for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem.
</jats:p>}},
  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{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{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{59797,
  author       = {{Konopik, Michael and T. Sato Martín de Almagro, Rodrigo and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Leyendecker, Sigrid}},
  journal      = {{Journal of Nonlinear Science}},
  number       = {{11}},
  title        = {{{Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term}}},
  doi          = {{10.1007/s00332-025-10229-5}},
  volume       = {{36}},
  year         = {{2025}},
}

@article{59799,
  author       = {{Konopik, Michael and Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and T. Sato Martín de Almagro, Rodrigo}},
  journal      = {{Nonlinearity}},
  number       = {{11}},
  title        = {{{A new Lagrangian approach to optimal control of second-order systems}}},
  doi          = {{10.1088/1361-6544/ae1d08}},
  volume       = {{38}},
  year         = {{2025}},
}

@article{53101,
  abstract     = {{In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.}},
  author       = {{Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Almagro, Rodrigo T. Sato Martín de and Szemenyei, Flóra Orsolya}},
  issn         = {{2158-2491}},
  journal      = {{Journal of Computational Dynamics}},
  keywords     = {{Optimal control problem, Lagrangian system, Hamiltonian system, Variations, Pontryagin's maximum principle.}},
  pages        = {{0--0}},
  publisher    = {{American Institute of Mathematical Sciences (AIMS)}},
  title        = {{{A new Lagrangian approach to control affine systems with a quadratic Lagrange term}}},
  doi          = {{10.3934/jcd.2024017}},
  year         = {{2024}},
}

@inproceedings{59791,
  author       = {{Maslovskaya, Sofya and Ober-Blöbaum, Sina}},
  booktitle    = {{IFAC-PapersOnLine}},
  issn         = {{2405-8963}},
  number       = {{17}},
  pages        = {{85--90}},
  publisher    = {{Elsevier BV}},
  title        = {{{Symplectic Methods in Deep Learning}}},
  doi          = {{10.1016/j.ifacol.2024.10.118}},
  volume       = {{58}},
  year         = {{2024}},
}

@unpublished{59801,
  author       = {{Jean, Frédéric and Maslovskaya, Sofya}},
  title        = {{{Inverse optimal control problem in the non autonomous linear-quadratic case}}},
  year         = {{2024}},
}

@article{30861,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>We consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). According to Pontryagin’s maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or <jats:italic>Fuller</jats:italic> phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the <jats:italic>turnpike</jats:italic> property. This property is further used in order to design simple and realistic suboptimal control strategies.</jats:p>}},
  author       = {{Caillau, Jean-Baptiste and Djema, Walid and Gouzé, Jean-Luc and Maslovskaya, Sofya and Pomet, Jean-Baptiste}},
  issn         = {{0022-3239}},
  journal      = {{Journal of Optimization Theory and Applications}},
  keywords     = {{Applied Mathematics, Management Science and Operations Research, Control and Optimization}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Turnpike Property in Optimal Microbial Metabolite Production}}},
  doi          = {{10.1007/s10957-022-02023-0}},
  year         = {{2022}},
}

@article{29543,
  author       = {{Djema, Walid and Giraldi, Laetitia and Maslovskaya, Sofya and Bernard, Olivier}},
  issn         = {{0005-1098}},
  journal      = {{Automatica}},
  keywords     = {{Electrical and Electronic Engineering, Control and Systems Engineering}},
  publisher    = {{Elsevier BV}},
  title        = {{{Turnpike features in optimal selection of species represented by quota models}}},
  doi          = {{10.1016/j.automatica.2021.109804}},
  volume       = {{132}},
  year         = {{2021}},
}

@inproceedings{20812,
  author       = {{Jean, Frederic and Maslovskaya, Sofya}},
  booktitle    = {{2019 IEEE 58th Conference on Decision and Control (CDC)}},
  isbn         = {{9781728113982}},
  title        = {{{Injectivity of the inverse optimal control problem for control-affine systems}}},
  doi          = {{10.1109/cdc40024.2019.9028877}},
  year         = {{2020}},
}

@article{29545,
  author       = {{Jean, Frédéric and Maslovskaya, Sofya and Zelenko, Igor}},
  issn         = {{0046-5755}},
  journal      = {{Geometriae Dedicata}},
  keywords     = {{Geometry and Topology}},
  number       = {{1}},
  pages        = {{295--314}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry}}},
  doi          = {{10.1007/s10711-020-00581-z}},
  volume       = {{213}},
  year         = {{2020}},
}

@inproceedings{29546,
  author       = {{Maslovskaya, Sofya and Caillau, Jean-Baptiste and Djema, Walid and Giraldi, Laetitia and Jean-Luc, Jean-Luc and Pomet, Jean-Baptiste}},
  title        = {{{The turnpike property in maximization of microbial metabolite production}}},
  year         = {{2020}},
}

@inproceedings{20813,
  author       = {{Caillau, Jean-Baptiste and Maslovskaya, Sofya and Mensch, Thomas and Moulinier, Timothee and Pomet, Jean-Baptiste}},
  booktitle    = {{2019 IEEE 58th Conference on Decision and Control (CDC)}},
  isbn         = {{9781728113982}},
  title        = {{{Zermelo-Markov-Dubins problem and extensions in marine navigation}}},
  doi          = {{10.1109/cdc40024.2019.9029293}},
  year         = {{2020}},
}

@inproceedings{20810,
  author       = {{Jean, Frederic and Maslovskaya, Sofya}},
  booktitle    = {{2018 IEEE Conference on Decision and Control (CDC)}},
  isbn         = {{9781538613955}},
  title        = {{{Inverse optimal control problem: the linear-quadratic case}}},
  doi          = {{10.1109/cdc.2018.8619204}},
  year         = {{2019}},
}

@article{20811,
  author       = {{Jean, Frédéric and Maslovskaya, Sofya and Zelenko, Igor}},
  issn         = {{0046-5755}},
  journal      = {{Geometriae Dedicata}},
  pages        = {{279--319}},
  title        = {{{On projective and affine equivalence of sub-Riemannian metrics}}},
  doi          = {{10.1007/s10711-019-00437-1}},
  year         = {{2019}},
}

@phdthesis{20815,
  author       = {{Maslovskaya, Sofya}},
  title        = {{{Inverse Optimal Control : theoretical study}}},
  year         = {{2018}},
}

@article{20809,
  author       = {{Jean, Frédéric and Maslovskaya, Sofya and Zelenko, Igor}},
  issn         = {{2405-8963}},
  journal      = {{IFAC-PapersOnLine}},
  pages        = {{500--505}},
  title        = {{{Inverse Optimal Control Problem: the Sub-Riemannian Case }}},
  doi          = {{10.1016/j.ifacol.2017.08.105}},
  year         = {{2017}},
}