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Control of continuous time dynamics with multiplicative noise is a classic topic in stochastic optimal control. This work addresses the problem of designing infinite horizon optimal controls with stability guarantees for \textit{a single agent or large population systems} of identical, non-cooperative and non-networked agents, with multi-dimensional and nonlinear stochastic dynamics excited by multiplicative noise. For agent dynamics belonging to the class of reversible diffusion processes, we provide constraints on the state and control cost functions which guarantee stability of the closed-loop system under the action of the individual optimal controls. A condition relating the state-dependent control cost and volatility is introduced to prove the stability of the equilibrium density. This condition is a special case of the constraint required to use the path integral Feynman-Kac formula for computing the control. We investigate the connection between the stabilizing optimal control and the path integral formalism, leading us to a control law formulation expressed exclusively in terms of the desired equilibrium density., 6 pages, 0 figures, IEEE Conference on Decision and Control 2020 (accepted)

In this work we present the first systematic framework to sculpt active nematics systems, using optimal control theory and a hydrodynamic model of active nematics. We demonstrate the use of two different control fields, (1) applied vorticity and (2) activity strength, to shape the dynamics of an extensile active nematic that is confined to a disk. In the absence of control inputs, the system exhibits two attractors, clockwise and counterclockwise circulating states characterized by two co-rotating topological $+\frac{1}{2}$ defects. We specifically seek spatiotemporal inputs that switch the system from one attractor to the other; we also examine phase-shifting perturbations. We identify control inputs by optimizing a penalty functional with three contributions: total control effort, spatial gradients in the control, and deviations from the desired trajectory. This work demonstrates that optimal control theory can be used to calculate non-trivial inputs capable of restructuring active nematics in a manner that is economical, smooth, and rapid, and therefore will serve as a guide to experimental efforts to control active matter.

We consider the problem of incentivization and optimal control of autonomous vehicles for improving traffic congestion. In our scenario, autonomous vehicles must be incentivized in order to participate in traffic improvement. Using the theory and methods of optimal transport, we propose a constrained optimization framework over dynamics governed by partial differential equations, so that we can optimally select a portion of vehicles to be incentivized and controlled. The goal of the optimization is to obtain a uniform distribution of vehicles over the spatial domain. To achieve this, we consider two types of penalties on vehicle density, one is the $L^2$ cost and the other is a multiscale-norm cost, commonly used in fluid-mixing problems. To solve this non-convex optimization problem, we introduce a novel algorithm, which iterates between solving a convex optimization problem and propagating the flow of uncontrolled vehicles according to the Lighthill-Whitham-Richards model. We perform numerical simulations, which suggest that the optimization of the $L^2$ cost is ineffective while optimization of the multiscale norm is effective. The results also suggest the use of a dedicated lane for this type of control in practice., 6 pages, 10 figures, submitted to the 2020 IEEE Intelligent Vehicles Symposium

We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge-Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems., Comment: With additional numerics. 23 pages, 17 figures. To appear in CNSNS

We consider the problem of optimally controlling turbulent buoyancy-driven flows in the built environment, focusing on a model test case of displacement ventilation with a time-varying heat source. The flow is modeled using the unsteady Reynolds-averaged equations (URANS). A direct-adjoint-looping implementation of the nonlinear optimal control problem yields time-varying values of temperature and velocity of the inlet flow that lead to ‘optimal’ time-averaged temperature relative to appropriate objective functionals in a region of interest. The resulting dynamics of both ‘filling’ and ‘intruding’ added layers due to a time-varying source and inlet flow are discussed. The robustness of the optimal solution is demonstrated. It is found that for large enough values of time horizon the optimal steady solution is recovered, while for intermediate values a non-trivial deviation from this optimal steady state design is achieved. The computational framework is flexible, and can be applied to several problems of interest in optimal design and control of indoor airflow.

Empirically derived continuum models of collective behavior among large populations of dynamic agents are a subject of intense study in several fields, including biology, engineering and finance. We formulate and study a mean-field game model whose behavior mimics an empirically derived non-local homogeneous flocking model for agents with gradient self-propulsion dynamics. The mean-field game framework provides a non-cooperative optimal control description of the behavior of a population of agents in a distributed setting. In this description, each agent's state is driven by optimally controlled dynamics that result in a Nash equilibrium between itself and the population. The optimal control is computed by minimizing a cost that depends only on its own state, and a mean-field term. The agent distribution in phase space evolves under the optimal feedback control policy. We exploit the low-rank perturbative nature of the non-local term in the forward-backward system of equations governing the state and control distributions, and provide a linear stability analysis demonstrating that our model exhibits bifurcations similar to those found in the empirical model. The present work is a step towards developing a set of tools for systematic analysis, and eventually design, of collective behavior of non-cooperative dynamic agents via an inverse modeling approach., Comment: Revised to incorporate reviewers' suggestions. Accepted to Chaos journal

The ARC pilot plant conceptual design study has been extended beyond its initial scope [B. N. Sorbom et al., FED 100 (2015) 378] to explore options for managing ∼525 MW of fusion power generated in a compact, high field (B0 = 9.2 T) tokamak that is approximately the size of JET (R0 = 3.3 m). Taking advantage of ARC's novel design – demountable high temperature superconductor toroidal field (TF) magnets, poloidal magnetic field coils located inside the TF, and vacuum vessel (VV) immersed in molten salt FLiBe blanket – this follow-on study has identified innovative and potentially robust power exhaust management solutions. The superconducting poloidal field coil set has been reconfigured to produce double-null plasma equilibria with a long-leg X-point target divertor geometry. This design choice is motivated by recent modeling which indicates that such configurations enhance power handling and may attain a passively-stable detachment front that stays in the divertor leg over a wide power exhaust window. A modified VV accommodates the divertor legs while retaining the original core plasma volume and TF magnet size. The molten salt FLiBe blanket adequately shields all superconductors, functions as an efficient tritium breeder, and, with augmented forced flow loops, serves as an effective single-phase, low-pressure coolant for the divertor, VV, and breeding blanket. Advanced neutron transport calculations (MCNP) indicate a tritium breeding ratio of ∼1.08. The neutron damage rate (DPA/year) of the remote divertor targets is ∼3–30 times lower than that of the first wall. The entire VV (including divertor and first wall) can tolerate high damage rates since the demountable TF magnets allow the VV to be replaced every 1–2 years as a single unit, employing a vertical maintenance scheme. A tungsten swirl tube FLiBe coolant channel design, similar in geometry to that used by ITER, is considered for the divertor heat removal and shown capable of exhausting divertor heat flux levels of up to 12 MW/m2. Several novel, neutron tolerant diagnostics are explored for sensing power exhaust and for providing feedback control of divertor conditions over long time scales. These include measurement of Cherenkov radiation emitted in FLiBe to infer DT fusion reaction rate, measurement of divertor detachment front locations in the divertor legs with microwave interferometry, and monitoring “hotspots” on the divertor chamber walls via IR imaging through the FLiBe blanket. ©2018, DOE NNSA Stewardship Science Graduate Fellowship (No. DE-NA0002135), National Science Foundation Graduate Research Fellowship (Grant No. DGE1122374)

We demonstrate the use of the ‘Direct-Adjoint-Looping method’ for the identification of optimal buoyancy-driven ventilation flows governed by Boussinesq equations. We use the incompressible Reynolds-averaged Navier-Stokes (RANS) equations, derive the corresponding adjoint equations and solve the resulting sensitivity equations with respect to inlet conditions. We focus on a displacement ventilation scenario with a steady plume due to a line source. Subject to an enthalpy flux constraint on the incoming flow, we identify boundary conditions leading to ‘optimal’ temperature distributions in the occupied zone. Our results show that depending on the scaled volume and momentum flux of the inlet flow, qualitatively different flow regimes may be obtained. The numerical optimal results agree with analytically obtained optimal inlet conditions available from classical plume theory in an ‘intermediate’ regime of strong stratification and two-layer flow.

We present results on stabilization for reduced order models (ROM) of partial differential equations using learning. Stabilization is achieved via closure models for ROMs, where we use a model-free extremum seeking (ES) dither-based algorithm to learn the best closure models' parameters, for optimal ROM stabilization. We first propose to auto-tune linear closure models using ES, and then extend the results to a closure model combining linear and nonlinear terms, for better stabilization performance. The coupled Burgers' equation is employed as a test-bed for the proposed tuning method.

We study the problem of optimizing the performance of a nonlinear spring-mass-damper attached to a class of multiple-degree-of-freedom systems. We aim to maximize the rate of one-way energy transfer from primary system to the attachment, and focus on impulsive excitation of a two-degree-of-freedom primary system with an essentially nonlinear attachment. The nonlinear attachment is shown to be able to perform as a `nonlinear energy sink' (NES) by taking away energy from the primary system irreversibly for some types of impulsive excitations. Using perturbation analysis and exploiting separation of time scales, we perform dimensionality reduction of this strongly nonlinear system. Our analysis shows that efficient energy transfer to nonlinear attachment in this system occurs for initial conditions close to homoclinic orbit of the slow time-scale undamped system, a phenomenon that has been previously observed for the case of single-degree-of-freedom primary systems. Analytical formulae for optimal parameters for given impulsive excitation input are derived. Generalization of this framework to systems with arbitrary number of degrees-of-freedom of the primary system is also discussed. The performance of both linear and nonlinear optimally tuned attachments is compared. While NES performance is sensitive to magnitude of the initial impulse, our results show that NES performance is more robust than linear tuned-mass-damper to several parametric perturbations. Hence, our work provides evidence that homoclinic orbits of the underlying Hamiltonian system play a crucial role in efficient nonlinear energy transfers, even in high dimensional systems, and gives new insight into robustness of systems with essential nonlinearity., Comment: Updated results. 34 pages, 29 figures

We present a sparse sensing framework based on Dynamic Mode Decomposition (DMD) to identify flow regimes and bifurcations in large-scale thermo-fluid systems. Motivated by real-time sensing and control of thermal-fluid flows in buildings and equipment, we apply this method to a Direct Numerical Simulation (DNS) data set of a 2D laterally heated cavity. The resulting flow solutions can be divided into several regimes, ranging from steady to chaotic flow. The DMD modes and eigenvalues capture the main temporal and spatial scales in the dynamics belonging to different regimes. Our proposed classification method is data-driven, robust w.r.t measurement noise, and exploits the dynamics extracted from the DMD method. Namely, we construct an augmented DMD basis, with "built-in" dynamics, given by the DMD eigenvalues. This allows us to employ a short time-series of data from sensors, to more robustly classify flow regimes, particularly in the presence of measurement noise. We also exploit the incoherence exhibited among the data generated by different regimes, which persists even if the number of measurements is small compared to the dimension of the DNS data. The data-driven regime identification algorithm can enable robust low-order modeling of flows for state estimation and control., Expanded discussion. Fixed some typos and figures

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler, Ross, Grover, Kumar, Topological chaos and periodic braiding of almost-cyclic sets. Physical Review Letters 106 (2011), 114101]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or `ghost rods' around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems., Comment: Submitted to Chaos