A central theme of our research is the control of probability distributions rather than individual trajectories. We develop exact and approximate methods for steering the state density of dynamical systems, including mechanical systems on Riemannian manifolds, between prescribed endpoint distributions. Equivalently, we treat ensembles of systems as the fundamental object of control.
This thrust also investigates the geometric structure of optimal mass transport itself, including the group-theoretic properties of transport maps, holonomy phenomena that arise when transport plans are composed, and kinetic formulations of transport and Schrödinger bridge problems on Lie groups and compact manifolds. These structural results inform, and are informed by, the design of steering controllers.
References
2026
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The holonomy of optimal mass transport: The Gaussian-linear case
M. Abdelgalil and T. T. Georgiou
IEEE Transactions on Automatic Control, 2026
The theory of Monge-Kantorovich Optimal Mass Transport (OMT) has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in engineering and science. We herein introduce a new type of transportation problems. The salient feature of these problems is that particles/agents in the ensemble are labeled and their relative position along their journey is of interest. Of particular importance in our program are control laws that steer ensembles along cycles ensuring that individual particles return to their original position. This feature is in contrast with the classical theory of optimal transport where the primary object of study is the path of probability densities, without any concern about particle labels. In the theory that we present, we focus on the case Gaussian distributions and linear dynamics, and explore a hitherto unstudied sub-Riemannian structure of Monge-Kantorovich transport where the relative position of particles along their journey is modeled by the holonomy of the transportation schedule. From this vantage point, we discuss several other problems of independent interest.
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Control of the state transition matrix via factorization into positive definite factors
M. Abdelgalil and T. T. Georgiou
IEEE Control Systems Letters, 2026
The factorization of an arbitrary matrix Φ with positive determinant into a product of positive-definite ones provides a piece-wise smooth time-varying control protocol to realize Φ as the state transition matrix of a linear system via state feedback. The factors, being symmetric and positive-definite, realize a succession of gradient flows ensuring the avoidance of shocks. We relate such a factorization to a composition of optimal transport maps between suitably constructed Gaussian distributions. Our analysis provides a principled construction of such factorizations that, in addition, allows a handle on the condition number of the factors which relates to control effort.
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On the isospectral nature of minimum-shear covariance control
R. Sabbagh, A. Eldesoukey, M. Abdelgalil, and 1 more author
IEEE Control Systems Letters, 2026
We revisit Brockett’s attention in the context of bilinear gradient flow of an ensemble, and explore an alternative formalism that aims to reduce shear by minimizing the conditioning number of the dynamics; equivalently, we minimize the range of the eigenvalues of the dynamics. Remarkably, the evolution is isospectral, and this property is inherited by the coupled nonlinear dynamics of the control problem from a Lax isospectral flow.
2025
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Collective steering in finite time: Controllability on GL⁺(n, ℝ)
M. Abdelgalil and T. T. Georgiou
IEEE Transactions on Automatic Control, 2025
We consider the problem of steering a collection ofnparticles that obey identicaln-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.
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Control of the state transition via state feedback
M. Abdelgalil and T. T. Georgiou
In Proc. IEEE Conference on Decision and Control, 2025
We study a nonlinear control problem in which linear dynamics are regulated by a time-varying state-feedback law. The nonlinearity arises from the bilinear interaction between the feedback gain and the state. We show that the classical Kalman controllability condition, easily seen to be necessary, is also sufficient in this setting. More strongly, we establish that one can not only prescribe the terminal state vector over any finite horizon, but also steer the entire state-transition matrix to an arbitrary target with positive determinant.Our interest in controlling the state-transition matrix is motivated by the problem of regulating ensembles of agents that evolve according to the given dynamics, where a common time-varying feedback gain is broadcast and applied by all agents. We show that, under a suitable protocol, a collection of particles can be repositioned arbitrarily fast into any admissible terminal configuration.While this work was initially intended as an extended abstract for the journal article [1], which develops broader results on controllability over diffeomorphism groups, the present article now provides an independent constructive derivation of main result and of the feedback protocols that achieve prescribed terminal state-transition matrices.
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Collective steering: Tracer-informed dynamics
A. Eldesoukey, M. Abdelgalil, and T. T. Georgiou
In Proc. IEEE Conference on Decision and Control, 2025
We consider control and inference problems where control protocols and internal dynamics are informed by two types of constraints. Our data consist of i) statistics on the ensemble and ii) trajectories or final disposition of selected tracer particles embedded in the flow. Our aim is i’) to specify a control protocol to realize a flow that meets such constraints or ii’) to recover the internal dynamics that are consistent with such a data set. We analyze these problems in the setting of linear flows and Gaussian distributions. The control cost is taken to be a suitable action integral constrained by either the trajectories of tracer particles or their terminal placements.