We consider the problem of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy control. The ...system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise" forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a pair of differential Lyapunov equations that are nonlinearly coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to quality control, control of industrial processes, as well as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schrödinger bridges) and the present case of steering a linear stochastic system constitutes such a Schrödinger bridge for possibly degenerate diffusions. Our results provide the first implementable form of the optimal control for a general Gauss-Markov process. Illustrative examples are provided for steering inertial particles and for "cooling" a stochastic oscillator. A final result establishes directly the property of Schrödinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems. A second part to this work, that is to appear as part II, addresses the general situation where the stochastic excitation enters through channels that may differ from those used to control.
We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the ...two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a
Fisher information functional
. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of
optimal transport with prior
. This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.
We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics ...correspond to an integrator (ẋ(t) = u(t)) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrödinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.
In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one ...and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.
We consider damped stochastic systems in a controlled (time varying) potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of ...thermodynamics, the minimum amount of work needed for the transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates nonequilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of the classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.
We introduce an optimal mass transport framework on the space of Gaussian mixture models. These models are widely used in statistical inference. Specifically, we treat the Gaussian mixture models as ...a submanifold of probability densities equipped with the Wasserstein metric. The topology induced by optimal transport is highly desirable and natural because, in contrast to total variation and other metrics, the Wasserstein metric is weakly continuous (i.e., convergence is equivalent to the convergence of moments). Thus, our approach provides natural ways to compare, interpolate, and average Gaussian mixture models. Moreover, the approach has low computational complexity. Different aspects of the framework are discussed, and examples are presented for illustration purposes.
The isolation of various two-dimensional (2D) materials, and the possibility to combine them in vertical stacks, has created a new paradigm in materials science: heterostructures based on 2D ...crystals. Such a concept has already proven fruitful for a number of electronic applications in the area of ultrathin and flexible devices. Here, we expand the range of such structures to photoactive ones by using semiconducting transition metal dichalcogenides (TMDCs)/graphene stacks. Van Hove singularities in the electronic density of states of TMDC guarantees enhanced light-matter interactions, leading to enhanced photon absorption and electron-hole creation (which are collected in transparent graphene electrodes). This allows development of extremely efficient flexible photovoltaic devices with photoresponsivity above 0.1 ampere per watt (corresponding to an external quantum efficiency of above 30%).
Although brain functionality is often remarkably robust to lesions and other insults, it may be fragile when these take place in specific locations. Previous attempts to quantify ...robustness and fragility sought to understand how the functional connectivity of brain networks is affected by structural changes, using either model-based predictions or empirical studies of the effects of lesions. We advance a geometric viewpoint relying on a notion of network curvature, the so-called Ollivier-Ricci curvature. This approach has been proposed to assess financial market robustness and to differentiate biological networks of cancer cells from healthy ones. Here, we apply curvature-based measures to brain structural networks to identify robust and fragile brain regions in healthy subjects. We show that curvature can also be used to track changes in brain connectivity related to age and autism spectrum disorder (ASD), and we obtain results that are in agreement with previous MRI studies.
An obstacle to the use of graphene as an alternative to silicon electronics has been the absence of an energy gap between its conduction and valence bands, which makes it difficult to achieve low ...power dissipation in the OFF state. We report a bipolar field-effect transistor that exploits the low density of states in graphene and its one-atomic-layer thickness. Our prototype devices are graphene heterostructures with atomically thin boron nitride or molybdenum disulfide acting as a vertical transport barrier. They exhibit room-temperature switching ratios of ≈50 and ≈10,000, respectively. Such devices have potential for high-frequency operation and large-scale integration.
Multiplex networks have been intensively studied during the last few years as they offer a more realistic representation of many interdependent and multilevel complex networked systems. However, even ...if most real networks have some degree of directionality, the vast majority of the existent literature deals with multiplex networks where all layers are undirected. Here, we study the dynamics of diffusion processes acting on coupled multilayer networks where at least one layer consists of a directed graph; we call these directed multiplex networks. We reveal a new and unexpected signature of diffusion dynamics on directed multiplex networks, namely, that different from their undirected counterparts, they can exhibit a nonmonotonic rate of convergence to steady state as a function of the degree of coupling, resulting in a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. We use synthetic multiplex examples and real-world topologies to illustrate the characteristics of the underlying dynamics that give rise to a regime in which an optimal coupling exists. We further provide analytical and numerical evidence that this new phenomenon is solely a property of directed multiplex, where at least one of the layers exhibits sufficient directionality quantified by a normalized metric of asymmetry in directional path lengths. Given the ubiquity of both directed and multilayer networks in nature, our results have important implications for studying the dynamics of multilevel complex systems.
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