Jorge Cortés

Understanding the relationship between the structure and function of the brain is one of the key questions in neuroscience. An approach to this question that has gained popularity over the last twenty years is the use of control theoretic techniques, motivated by the ability to model the brain as an interconnected dynamical system. In this work we consider a firing-rate model of the brain, where the activity of populations of neurons is averaged over time and their impact on each other is considered. We focus on the linear-threshold rate model, which exhibits varying behaviors including multistability, limit cycles, and chaos, opening the door to use the dynamics to model multiple complex brain processes. In particular, we provide computational results for three brain processes: goal-driven selective attention, declarative memory, and seizure dynamics in epilepsy. These processes relate to the dynamical properties of stabilization, multistability, and bifurcations and oscillations, respectively.

This paper presents a method for incorporating control analysis into design optimization for highly-maneuverable aircraft. By studying reachable sets for aircraft dynamics, we ensure that the optimizer will take the aircraft's controlled capabilities into account. We compute reachable sets of linear dynamics for computational efficiency, and account for aircraft trim points to factor in asymmetric magnitude bounds on the input signals. We demonstrate the proposed method in design optimization of a blended-wing-body aircraft. Considering its wing half-span and center half-span as design variables, we optimize the aircraft based on its longitudinal dynamics' reachable sets to yield improvements in its controlled performance. When designing a reference tracking controller, we find up to 30\% less tracking error for angle of attack of the optimized model's nonlinear dynamics.

This paper presents a method for incorporating control analysis into design optimization for highly-maneuverable aircraft. By studying reachable sets for aircraft dynamics, we ensure that the optimizer will take the aircraft's controlled capabilities into account. We compute reachable sets of linear dynamics for computational efficiency, and account for aircraft trim points to factor in asymmetric magnitude bounds on the input signals. We demonstrate the proposed method in design optimization of a blended-wing-body aircraft. Considering its wing half-span and center half-span as design variables, we optimize the aircraft based on its longitudinal dynamics' reachable sets to yield improvements in its controlled performance. When designing a reference tracking controller, we find up to 30\% less tracking error for angle of attack of the optimized model's nonlinear dynamics.

Koopman operator theory offers a rigorous treatment of dynamics, emerging as a robust alternative for learning-based control in robotics. By representing nonlinear dynamics as a linear, higher-dimensional operator, it provides a fresh lens for modeling complex systems. Its ability to support incremental updates and low computational cost makes it particularly appealing for real-time applications and online learning. This review delves deeply into the foundations, systematically bridging theoretical principles to practical robotic applications. We explain mathematical underpinnings, approximation approaches for inputs, data collection strategies, and lifting function design. We explore how Koopman models unify tasks like model-based control, state estimation, and motion planning. The review surveys cutting-edge research across domains ranging from aerial and legged platforms to manipulators, soft robots, and multi-agent networks. We also present advanced theoretical topics and reflect on open challenges and future research directions. To support adoption, we provide a hands-on tutorial with code at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/sunnyshi0310/KoopmanRobo/tree/main</uri>.

Energy-based models have become a central paradigm for understanding computation and stability in both theoretical neuroscience and machine learning. However, the energetic framework typically relies on symmetry in synaptic or weight matrices - a constraint that excludes biologically realistic systems such as excitatory-inhibitory (E-I) networks. When symmetry is relaxed, the classical notion of a global energy landscape fails, leaving the dynamics of asymmetric neural systems conceptually unanchored. In this work, we extend the energetic framework to asymmetric firing rate networks, revealing an underlying game-theoretic structure for the neural dynamics in which each neuron is an agent that seeks to minimize its own energy. In addition, we exploit rigorous stability principles from network theory to study regulation and balancing of neural activity in E-I networks. We combine the novel game-energetic interpretation and the stability results to revisit standard frameworks in theoretical neuroscience, such as the Wilson-Cowan and lateral inhibition models. These insights allow us to study cortical columns of lateral inhibition microcircuits as contrast enhancer - with the ability to selectively sharpen subtle differences in the environment through hierarchical excitation-inhibition interplay. Our results bridge energetic and game-theoretic views of neural computation, offering a pathway toward the systematic engineering of biologically grounded, dynamically stable neural architectures.

The Koopman operator, originally defined for dynamical systems without input, has inspired many applications in control. Yet, the theoretical foundations underpinning this progress in control remain underdeveloped. This paper investigates the theoretical structure and connections between two extensions of Koopman theory to control: (i) Koopman operator via infinite input sequences and (ii) the Koopman control family. Although these frameworks encode system information in fundamentally different ways, we show that under certain conditions on the function spaces they operate on, they are equivalent. The equivalence is both in terms of the actions of the Koopman-based formulations in each framework as well as the function values on the system trajectories. Our analysis provides constructive tools to translate between the frameworks, offering a unified perspective for Koopman methods in control.

This paper considers the problem of solving constrained reinforcement learning (RL) problems with anytime guarantees, meaning that the algorithmic solution must yield a constraint-satisfying policy at every iteration of its evolution. Our design is based on a discretization of the Robust Safe Gradient Flow (RSGF), a continuous-time dynamics for anytime constrained optimization whose forward invariance and stability properties we formally characterize. The proposed strategy, termed RSGF-RL, is an off-policy algorithm which uses episodic data to estimate the value functions and their gradients and updates the policy parameters by solving a convex quadratically constrained quadratic program. Our technical analysis combines statistical analysis, the theory of stochastic approximation, and convex analysis to determine the number of episodes sufficient to ensure that safe policies are updated to safe policies and to recover from an unsafe policy, both with an arbitrary user-specified probability, and to establish the asymptotic convergence to the set of KKT points of the RL problem almost surely. Simulations on a navigation example and the cart-pole system illustrate the superior performance of RSGF-RL with respect to the state of the art.

We consider the problem of designing learning-based reactive power controllers that perform voltage regulation in distribution grids while ensuring closed-loop system stability. In contrast to existing methods, where the provably stable controllers are restricted to be decentralized, we propose a unified design framework that enables the controllers to take advantage of an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arbitrary</i> communication infrastructure on top of the physical power network. This allows the controllers to incorporate information beyond their local bus, covering existing methods as a special case and leading to less conservative constraints on the controller design. We then provide a design procedure to construct input convex neural network (ICNN) based controllers that satisfy the identified stability constraints by design under arbitrary communication scenarios, and train these controllers using supervised learning. Simulation results on the University of California, San Diego (UCSD) microgrid testbed illustrate the effectiveness of the framework and highlight the role of communication in improving control performance.

We propose a control barrier function (CBF) formulation for enforcing equality and inequality constraints in variational inference. The key idea is to define a barrier functional on the space of probability density functions that encode the desired constraints imposed on the variational density. By leveraging the Liouville equation, we establish a connection between the time derivative of the variational density and the particle drift, which enables the systematic construction of corresponding CBFs associated to the particle drift. Enforcing these CBFs gives rise to the safe particle flow and ensures that the variational density satisfies the original constraints imposed by the barrier functional. This formulation provides a principled and computationally tractable solution to constrained variational inference, with theoretical guarantees of constraint satisfaction. The effectiveness of the method is demonstrated through numerical simulations.

We consider the problem of designing learning-based reactive power controllers that perform voltage regulation in distribution grids while ensuring closed-loop system stability. In contrast to existing methods, where the provably stable controllers are restricted to be decentralized, we propose a unified design framework that enables the controllers to take advantage of an arbitrary communication infrastructure on top of the physical power network. This allows the controllers to incorporate information beyond their local bus, covering existing methods as a special case and leading to less conservative constraints on the controller design. We then provide a design procedure to construct input convex neural network (ICNN) based controllers that satisfy the identified stability constraints by design under arbitrary communication scenarios, and train these controllers using supervised learning. Simulation results on the the University of California, San Diego (UCSD) microgrid testbed illustrate the effectiveness of the framework and highlight the role of communication in improving control performance.

We introduce a novel method for mobile robot navigation in dynamic, unknown environments, leveraging onboard sensing and distributionally robust optimization to impose probabilistic safety constraints. Our method introduces a distributionally robust control barrier function (DR-CBF) that directly integrates noisy sensor measurements and state estimates to define safety constraints. This approach is applicable to a wide range of control-affine dynamics, generalizable to robots with complex geometries, and capable of operating at real-time control frequencies. Coupled with a control Lyapunov function (CLF) for path following, the proposed CLF-DR-CBF control synthesis method achieves safe, robust, and efficient navigation in challenging environments. We demonstrate the effectiveness and robustness of our approach for safe autonomous navigation under uncertainty and dynamic obstacles in simulations and real-world experiments with differential-drive robots.

Motivated by the key role of control barrier functions (CBFs) in assessing safety and enabling the synthesis of safe controllers in nonlinear control systems, this paper presents a suite of converse results on CBFs. Given any safe set, we first identify a set of general sufficient conditions which guarantee the existence of a CBF. Our technical analysis also enables us to define an extended notion of CBF which is always guaranteed to exist if the set is safe. We next turn our attention to the problem of joint safety and stability, and give conditions under which the notions of control Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF) and CBF pair are guaranteed to exist. We also show via a counterexample that the existence of a CLF and a CBF does not imply the existence of a strictly compatible CLF-CBF pair, but provide conditions under which this holds. Throughout the paper, we intersperse different examples and counterexamples to motivate our results and position them within the state of the art.

This paper considers a class of bilinear systems with a neural network in the loop. These arise naturally when employing machine learning techniques to approximate general, non-affine in the input, control systems. We propose a controller design framework that combines linear fractional representations and tools from linear parameter varying control to guarantee local exponential stability of a desired equilibrium. The controller is obtained from the solution of linear matrix inequalities, which can be solved offline, making the approach suitable for online applications. The proposed methodology offers tools for stability and robustness analysis of deep neural networks interconnected with dynamical systems.

This paper focuses on an AC optimal power flow (OPF) problem for distribution feeders equipped with controllable distributed energy resources (DERs). We consider a solution method that is based on a continuous approximation of the projected gradient flow - referred to as the safe gradient flow - that incorporates voltage and current information obtained either through real-time measurements or power flow computations. These two setups enable both online and offline implementations. The safe gradient flow involves the solution of convex quadratic programs (QPs). To enhance computational efficiency, we propose a novel framework that employs a neural network approximation of the optimal solution map of the QP. The resulting method has two key features: (a) it ensures that the DERs' setpoints are practically feasible, even for an online implementation or when an offline algorithm has an early termination; (b) it ensures convergence to a neighborhood of a strict local optimizer of the AC OPF. The proposed method is tested on a 93-node distribution system with realistic loads and renewable generation. The test shows that our method successfully regulates voltages within limits during periods with high renewable generation.

The purpose of the germination test is to determine the viability of a batch of seeds, which is determined through the percentage of seeds that have the capacity to generate normal seedlings under optimal conditions of light, water, air and temperature, which establishes the germination capacity.It should be noted that physiological quality refers to intrinsic mechanisms of the seed that determine its germination capacity, the emergence and development of those structures essential to produce a normal seedling under favorable conditions.The work consisted in making a comparison on germination quality between creole corn from the states of Guanajuato and Michoacan, in the Chemistry laboratory of the Agronomist Engineer Degree in Production at the University Center UAEM Zumpango, it is concluded that they had a germination behavior between 93 to 99% both, so it can be said that it is a seed that meets the physiological quality and ensures a good population in the production unit.

This paper provides a formulation of the log-homotopy particle flow from the perspective of variational inference. We show that the transient density used to derive the particle flow follows a time-scaled trajectory of the Fisher-Rao gradient flow in the space of probability densities. The Fisher-Rao gradient flow is obtained as a continuous-time algorithm for variational inference, minimizing the Kullback-Leibler divergence between a variational density and the true posterior density. When considering a parametric family of variational densities, the function space Fisher-Rao gradient flow simplifies to the natural gradient flow of the variational density parameters. By adopting a Gaussian variational density, we derive a Gaussian approximated Fisher-Rao particle flow and show that, under linear Gaussian assumptions, it reduces to the Exact Daum and Huang particle flow. Additionally, we introduce a Gaussian mixture approximated Fisher-Rao particle flow to enhance the expressive power of our model through a multi-modal variational density. Simulations on low- and high-dimensional estimation problems illustrate our results.

This paper considers the problem of solving constrained reinforcement learning problems with anytime guarantees, meaning that the algorithmic solution returns a safe policy regardless of when it is terminated. Drawing inspiration from anytime constrained optimization, we introduce Reinforcement Learning-based Safe Gradient Flow (RL-SGF), an on-policy algorithm which employs estimates of the value functions and their respective gradients associated with the objective and safety constraints for the current policy, and updates the policy parameters by solving a convex quadratically constrained quadratic program. We show that if the estimates are computed with a sufficiently large number of episodes (for which we provide an explicit bound), safe policies are updated to safe policies with a probability higher than a prescribed tolerance. We also show that iterates asymptotically converge to a neighborhood of a KKT point, whose size can be arbitrarily reduced by refining the estimates of the value function and their gradients. We illustrate the performance of RL-SGF in a navigation example.

This paper considers non-smooth optimization problems where we seek to minimize the pointwise maximum of a continuously parameterized family of functions. Since the objective function is given as the solution to a maximization problem, neither its values nor its gradients are available in closed form, which calls for approximation. Our approach hinges upon extending the so-called gradient sampling algorithm, which approximates the Clarke generalized gradient of the objective function at a point by sampling its derivative at nearby locations. This allows us to select descent directions around points where the function may fail to be differentiable and establish algorithm convergence to a stationary point from any initial condition. Our key contribution is to prove this convergence by alleviating the requirement on continuous differentiability of the objective function on an open set of full measure. We further provide assumptions under which a desired convex subset of the decision space is rendered attractive for the iterates of the algorithm.

This paper considers the problem of controlling distributed energy resources (DERs) in a distribution network (DN); the paper focuses on the voltage regulation task and on the concept ofvirtual power plant(VPP). For the latter, we envision an aggregation of DERs as a VPP that tracks power setpoints at the point of common coupling to provide ancillary services to the bulk power system. We propose a feedback-based controller that pursues solutions to a time-varying AC optimal power flow problem. This controller ensures voltage constraints are met in real time, assuming no voltage measurement errors. The feedback from the system mitigates the controller sensitivity to model uncertainties and eliminates the need for measurements of uncontrollable loads at every node. The feedback-based controller is tested on the IEEE37-node system with delta and wye-connected devices in a strongly unbalanced configuration.

The existence of a control barrier function (CBF) for a control-affine system provides a powerful design tool to ensure safety. Any controller that satisfies the CBF condition and ensures that the trajectories of the closed-loop system are well defined makes the zero superlevel set forward invariant. Such a controller is referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safe</i>. This article studies the regularity properties of the minimum-norm safe controller as a stepping stone toward the design of general continuous safe feedback controllers. We characterize the set of points where the minimum-norm safe controller is discontinuous and show that it depends solely on the safe set and not on the particular CBF that describes it. Our analysis of the controller behavior as we approach a point of discontinuity allows us to identify sufficient conditions determining whether the controller grows unbounded or remains bounded. Examples illustrate our results, providing insight into the conditions that lead to (un)bounded discontinuous minimum-norm controllers.

Designing controllers that accomplish tasks while guaranteeing safety constraints remains a significant challenge. We often want an agent to perform well in a nominal task, such as environment exploration, while ensuring it can avoid unsafe states and return to a desired target by a specific time. In particular we are motivated by the setting of safe, efficient, hands-off training for reinforcement learning in the real world. By enabling a robot to safely and autonomously reset to a desired region (e.g., charging stations) without human intervention, we can enhance efficiency and facilitate training. Safety filters, such as those based on control barrier functions, decouple safety from nominal control objectives and rigorously guarantee safety. Despite their success, constructing these functions for general nonlinear systems with control constraints and system uncertainties remains an open problem. This paper introduces a safety filter obtained from the value function associated with the reach-avoid problem. The proposed safety filter minimally modifies the nominal controller while avoiding unsafe regions and guiding the system back to the desired target set. By preserving policy performance while allowing safe resetting, we enable efficient hands-off reinforcement learning and advance the feasibility of safe training for real world robots. We demonstrate our approach using a modified version of soft actor-critic to safely train a swing-up task on a modified cartpole stabilization problem.

This paper studies regularity properties of optimization-based controllers, which are obtained by solving optimization problems where the parameter is the system state and the optimization variable is the input to the system. Under a wide range of assumptions on the optimization problem data, we provide an exhaustive collection of results about their regularity, and examine their implications on the existence and uniqueness of solutions and the forward invariance guarantees for the resulting closed-loop systems. We discuss the broad relevance of the results in different areas of systems and controls.

The increasing integration of renewable energy resources into power grids has led to time-varying system inertia and consequent degradation in frequency dynamics. A promising solution to alleviate performance degradation is using power electronics interfaced energy resources, such as renewable generators and battery energy storage for primary frequency control, by adjusting their power output set-points in response to frequency deviations. However, designing a frequency controller under <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">time-varying inertia</i> is challenging. Specifically, the stability or optimality of controllers designed for time-invariant systems can be compromised once applied to a time-varying system. We model the frequency dynamics under time-varying inertia as a nonlinear switching system, where the frequency dynamics under each mode are described by the nonlinear swing equations and different modes represent different inertia levels. We identify a key controller structure, named Neural Proportional-Integral (Neural-PI) controller, that guarantees exponential input-to-state stability for each mode. To further improve performance, we present an online event-triggered switching algorithm to select the most suitable controller from a set of Neural-PI controllers, each optimized for specific inertia levels. Simulations on the IEEE 39-bus system validate the effectiveness of the proposed online switching control method with stability guarantees and optimized performance for frequency control under time-varying inertia.

The implementation of the Koopman operator on digital computers often relies on the approximation of its action on finite-dimensional function spaces. This approximation is generally done by orthogonally projecting on the subspace. Extended Dynamic Mode Decomposition (EDMD) is a popular, special case of this projection procedure in a data-driven setting. Importantly, the accuracy of the model obtained by EDMD depends on the quality of the finite-dimensional space, specifically on how close it is to being invariant under the Koopman operator. This paper presents a data-driven algebraic search algorithm, termed Recursive Forward-Backward EDMD, for subspaces close to being invariant under the Koopman operator. Relying on the concept of temporal consistency, which measures the quality of the subspace, our algorithm recursively decomposes the search space into two subspaces with different prediction accuracy levels. The subspace with lower level of accuracy is removed if it does not reach a satisfactory threshold. The algorithm allows for tuning the level of accuracy depending on the underlying application and is endowed with convergence and accuracy guarantees.

Control barrier functions (CBFs) offer a powerful tool for enforcing safety specifications in control synthesis. This paper deals with the problem of constructing valid CBFs. Given a second-order system and any desired safety set with linear boundaries in the position space, we construct a provably control-invariant subset of this desired safety set. The constructed subset does not sacrifice any positions allowed by the desired safety set, which can be nonconvex. We show how our construction can also meet safety specification on the velocity. We then demonstrate that if the system satisfies standard Euler-Lagrange systems properties then our construction can also handle constraints on the allowable control inputs. We finally show the efficacy of the proposed method in a numerical example of keeping a 2D robot arm safe from collision.

This paper considers a class of bilinear systems with a neural network in the loop. These arise naturally when employing machine learning techniques to approximate general, non-affine in the input, control systems. We propose a controller design framework that combines linear fractional representations and tools from linear parameter varying control to guarantee local exponential stability of a desired equilibrium. The controller is obtained from the solution of linear matrix inequalities, which can be solved offline, making the approach suitable for online applications. The proposed methodology offers tools for stability and robustness analysis of deep neural networks interconnected with dynamical systems.

Motivated by multi-agent Q-learning scenarios, this paper introduces a distributed action selection algorithm that relies on individual agents interacting with local neighbors to learn a joint action. The algorithm, termed Best Choice Dynamics, has each agent communicate its current planned action to its neighbors, who in turn utilize this information to update their own actions. We characterize the convergence and robustness of the algorithm against message losses, showing that it converges to locally optimal joint actions in finite time. We also discuss its relative advantages with respect to message-passing algorithms and best response dynamics regarding convergence guarantees, lack of oscillations, and communication complexity. We illustrate the algorithm performance in various simulation scenarios, including both on-line training and offline training with distributed on-line roll-out.

We propose a control barrier function (CBF) formulation for enforcing equality and inequality constraints in variational inference. The key idea is to define a barrier functional on the space of probability density functions that encode the desired constraints imposed on the variational density. By leveraging the Liouville equation, we establish a connection between the time derivative of the variational density and the particle drift, which enables the systematic construction of corresponding CBFs associated to the particle drift. Enforcing these CBFs gives rise to the safe particle flow and ensures that the variational density satisfies the original constraints imposed by the barrier functional. This formulation provides a principled and computationally tractable solution to constrained variational inference, with theoretical guarantees of constraint satisfaction. The effectiveness of the method is demonstrated through numerical simulations.

This article considers the stabilization of unknown switched linear systems using data. Instead of a full system model, we have access to a finite number of trajectories of each of the different modes prior to the online operation of the system. On the basis of informative enough measurements, we design an online switched controller that alternates between a mode detection phase and a stabilization phase. Since the currently active mode is unknown, the controller employs online measurements to determine it by implementing computationally efficient tests that check compatibility with the set of systems consistent with the precollected measurements. The stabilization phase applies a stabilizing feedback gain corresponding to the identified active mode and monitors the evolution of the associated Lyapunov function to detect switches. When a switch is detected, the controller returns to the mode detection phase. Under average dwell- and activation-time assumptions on the switching signal, we show that the proposed controller guarantees a practical stability property of the closed-loop switched system. Various simulations illustrate our results.

This paper studies the dynamical properties of closed-loop systems obtained from control barrier function-based safety filters. We provide a sufficient and necessary condition for the existence of undesirable equilibria and show that the Jacobian matrix of the closed-loop system evaluated at an undesirable equilibrium always has a nonpositive eigenvalue. In the special case of linear planar systems and ellipsoidal obstacles, we give a complete characterization of the dynamical properties of the corresponding closed-loop system. We show that for underactuated systems, the safety filter always introduces a single undesirable equilibrium, which is a saddle-point. We prove that all trajectories outside the global stable manifold of such equilibrium converge to the origin. In the fully actuated case, we discuss how the choice of nominal controller affects the stability properties of the closed-loop system. Various simulations illustrate our results.

This paper proposes a control design approach for stabilizing nonlinear control systems. Our key observation is that the set of points where the decrease condition of a control Lyapunov function (CLF) is feasible can be regarded as a safe set. By leveraging a nonsmooth version of control barrier functions (CBFs) and a weaker notion of CLF, we develop a control design that forces the system to converge to and remain in the region where the CLF decrease condition is feasible. We characterize the conditions under which our controller asymptotically stabilizes the origin or a small neighborhood around it, even in the cases where it is discontinuous. We illustrate our design in various examples.

This article considers the problem of controlling inverter-interfaced distributed energy resources (DERs) in a distribution grid to solve an ac optimal power flow (OPF) problem in real time. The ac OPF includes voltage constraints and seeks to minimize costs associated with the economic operation, power losses, or the power curtailment from renewables. We develop an online feedback optimization method to drive the DERs’ power setpoints to solutions of an ac OPF problem based only on voltage measurements (and without requiring measurements of the power consumption of noncontrollable assets). The proposed method—grounded on the theory of control barrier functions (CBFs)—is based on a continuous approximation of the projected gradient flow, appropriately modified to accommodate measurements from the power network. We provide results in terms of local exponential stability and assess the robustness to errors in the measurements and in the system Jacobian matrix. We show that the proposed method ensures anytime satisfaction of the voltage constraints when no model and measurement errors are present; if these errors are present and are small, the voltage violation is practically negligible. We also discuss extensions of the framework to virtual power plant (VPP) setups and cases where constraints on power flows and currents must be enforced. Numerical experiments on a 93-bus distribution system with realistic load and production profiles show superior performance in terms of voltage regulation relative to existing methods.

Abstract This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G -invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Over the past two decades, an increasing array of control-theoretic methods have been used to study the brain as a complex dynamical system and better understand its structure-function relationship. This article provides an overview on one such family of methods, based on the linear-threshold rate (LTR) dynamics, which arises when modeling the spiking activity of neuronal populations and their impact on each other. LTR dynamics exhibit a wide range of behaviors based on network topologies and inputs, including mono- and multi-stability, limit cycles, and chaos, allowing it to be used to model many complex brain processes involving fast and slow inhibition, multiple time and spatial scales, different types of neural behavior, and higher-order interactions. Here we investigate how the versatility of LTR dynamics paired with concepts and tools from systems and control can provide a computational theory for explaining the dynamical mechanisms enabling different brain processes. Specifically, we illustrate stability and stabilization properties of LTR dynamics and how they are related to goal-driven selective attention, multistability and its relationship with declarative memory, and bifurcations and oscillations and their role in modeling seizure dynamics in epilepsy. We conclude with a discussion on additional properties of LTR dynamics and an outlook on other brain processess that for which they might be play a similar role.

This paper considers the problem of designing motion planning algorithms for control-affine systems that generate collision-free paths from an initial to a final destination and can be executed using safe and dynamically-feasible controllers. We introduce the C-CLF-CBF-RRT algorithm, which produces paths with such properties and leverages rapidly exploring random trees (RRTs), control Lyapunov functions (CLFs) and control barrier functions (CBFs). We show that C-CLF-CBF-RRT is computationally efficient for linear systems with polytopic and ellipsoidal constraints, and establish its probabilistic completeness. We showcase the performance of C-CLF-CBF-RRT in different simulation and hardware experiments.

The increasing integration of renewable energy resources into power grids has led to time-varying system inertia and consequent degradation in frequency dynamics. A promising solution to alleviate performance degradation is using power electronics interfaced energy resources, such as renewable generators and battery energy storage for primary frequency control, by adjusting their power output set-points in response to frequency deviations. However, designing a frequency controller under time-varying inertia is challenging. Specifically, the stability or optimality of controllers designed for time-invariant systems can be compromised once applied to a time-varying system. We model the frequency dynamics under time-varying inertia as a nonlinear switching system, where the frequency dynamics under each mode are described by the nonlinear swing equations and different modes represent different inertia levels. We identify a key controller structure, named Neural Proportional-Integral (Neural-PI) controller, that guarantees exponential input-to-state stability for each mode. To further improve performance, we present an online event-triggered switching algorithm to select the most suitable controller from a set of Neural-PI controllers, each optimized for specific inertia levels. Simulations on the IEEE 39-bus system validate the effectiveness of the proposed online switching control method with stability guarantees and optimized performance for frequency control under time-varying inertia.

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Projected dynamical systems</i> (PDSs) form a class of discontinuous constrained dynamical systems, and have been used widely to solve optimization problems and variational inequalities. Recently, they have also gained significant attention for control purposes, such as high-performance integrators, saturated control, and feedback optimization. In this work, we establish that locally Lipschitz continuous dynamics, involving <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Control Barrier Functions</i> (CBFs), namely, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CBF-based dynamics</i>, approximate PDSs. Specifically, we prove that trajectories of CBF-based dynamics <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniformly converge</i> to trajectories of PDSs, as a CBF-parameter approaches infinity. Toward this, we also prove that CBF-based dynamics are perturbations of PDSs, with quantitative bounds on the perturbation. Our results pave the way to implement discontinuous PDS-based controllers in a continuous fashion, employing CBFs. We demonstrate this on an example on synchronverter control. Moreover, our results can be employed to numerically simulate PDSs, overcoming disadvantages of existing discretization schemes, such as computing projections to possibly nonconvex sets. Finally, this bridge between CBFs and PDSs may yield other potential benefits, including novel insights on stability.