11,850 research outputs found
Steering the distribution of agents in mean-field and cooperative games
The purpose of this work is to pose and solve the problem to guide a
collection of weakly interacting dynamical systems (agents, particles, etc.) to
a specified terminal distribution. The framework is that of mean-field and of
cooperative games. A terminal cost is used to accomplish the task; we establish
that the map between terminal costs and terminal probability distributions is
onto. Our approach relies on and extends the theory of optimal mass transport
and its generalizations.Comment: 20 pages, 8 figure
Optimal control of the state statistics for a linear stochastic system
We consider a variant of the classical linear quadratic Gaussian regulator
(LQG) in which penalties on the endpoint state are replaced by the
specification of the terminal state distribution. The resulting theory
considerably differs from LQG as well as from formulations that bound the
probability of violating state constraints. We develop results for optimal
state-feedback control in the two cases where i) steering of the state
distribution is to take place over a finite window of time with minimum energy,
and ii) the goal is to maintain the state at a stationary distribution over an
infinite horizon with minimum power. For both problems the distribution of
noise and state are Gaussian. In the first case, we show that provided the
system is controllable, the state can be steered to any terminal Gaussian
distribution over any specified finite time-interval. In the second case, we
characterize explicitly the covariance of admissible stationary state
distributions that can be maintained with constant state-feedback control. The
conditions for optimality are expressed in terms of a system of dynamically
coupled Riccati equations in the finite horizon case and in terms of algebraic
conditions for the stationary case. In the case where the noise and control
share identical input channels, the Riccati equations for finite-horizon
steering become homogeneous and can be solved in closed form. The present paper
is largely based on our recent work in arxiv.org/abs/1408.2222,
arxiv.org/abs/1410.3447 and presents an overview of certain key results.Comment: 7 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1410.344
Optimal transport over a linear dynamical system
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 () 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 Schroedinger 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
Steering state statistics with output feedback
Consider a linear stochastic system whose initial state is a random vector
with a specified Gaussian distribution. Such a distribution may represent a
collection of particles abiding by the specified system dynamics. In recent
publications, we have shown that, provided the system is controllable, it is
always possible to steer the state covariance to any specified terminal
Gaussian distribution using state feedback. The purpose of the present work is
to show that, in the case where only partial state observation is available, a
necessary and sufficient condition for being able to steer the system to a
specified terminal Gaussian distribution for the state vector is that the
terminal state covariance be greater (in the positive-definite sense) than the
error covariance of a corresponding Kalman filter.Comment: 10 pages, 2 figure
Entropic and displacement interpolation: a computational approach using the Hilbert metric
Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for
geometries in the space of positive densities -- it quantifies the cost of
transporting a mass distribution into another. In particular, it provides
natural options for interpolation of distributions (displacement interpolation)
and for modeling flows. As such it has been the cornerstone of recent
developments in physics, probability theory, image processing, time-series
analysis, and several other fields. In spite of extensive work and theoretical
developments, the computation of OMT for large scale problems has remained a
challenging task. An alternative framework for interpolating distributions,
rooted in statistical mechanics and large deviations, is that of Schroedinger
bridges (entropic interpolation). This may be seen as a stochastic
regularization of OMT and can be cast as the stochastic control problem of
steering the probability density of the state-vector of a dynamical system
between two marginals. In this approach, however, the actual computation of
flows had hardly received any attention. In recent work on Schroedinger bridges
for Markov chains and quantum evolutions, we noted that the solution can be
efficiently obtained from the fixed-point of a map which is contractive in the
Hilbert metric. Thus, the purpose of this paper is to show that a similar
approach can be taken in the context of diffusion processes which i) leads to a
new proof of a classical result on Schroedinger bridges and ii) provides an
efficient computational scheme for both, Schroedinger bridges and OMT. We
illustrate this new computational approach by obtaining interpolation of
densities in representative examples such as interpolation of images.Comment: 20 pages, 7 figure
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