63 research outputs found
Infinite horizon sparse optimal control
A class of infinite horizon optimal control problems involving -type
cost functionals with is discussed. The existence of optimal
controls is studied for both the convex case with and the nonconvex case
with , and the sparsity structure of the optimal controls promoted by
the -type penalties is analyzed. A dynamic programming approach is
proposed to numerically approximate the corresponding sparse optimal
controllers
An Efficient Policy Iteration Algorithm for Dynamic Programming Equations
We present an accelerated algorithm for the solution of static
Hamilton-Jacobi-Bellman equations related to optimal control problems. Our
scheme is based on a classic policy iteration procedure, which is known to have
superlinear convergence in many relevant cases provided the initial guess is
sufficiently close to the solution. In many cases, this limitation degenerates
into a behavior similar to a value iteration method, with an increased
computation time. The new scheme circumvents this problem by combining the
advantages of both algorithms with an efficient coupling. The method starts
with a value iteration phase and then switches to a policy iteration procedure
when a certain error threshold is reached. A delicate point is to determine
this threshold in order to avoid cumbersome computation with the value
iteration and, at the same time, to be reasonably sure that the policy
iteration method will finally converge to the optimal solution. We analyze the
methods and efficient coupling in a number of examples in dimension two, three
and four illustrating its properties
Optimal actuator design based on shape calculus
An approach to optimal actuator design based on shape and topology
optimisation techniques is presented. For linear diffusion equations, two
scenarios are considered. For the first one, best actuators are determined
depending on a given initial condition. In the second scenario, optimal
actuators are determined based on all initial conditions not exceeding a chosen
norm. Shape and topological sensitivities of these cost functionals are
determined. A numerical algorithm for optimal actuator design based on the
sensitivities and a level-set method is presented. Numerical results support
the proposed methodology.Comment: 41 pages, several figure
Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES
© 2018 Society for Industrial and Applied Mathematics. A procedure for the numerical approximation of high-dimensional Hamilton–Jacobi–Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen
Invisible control of self-organizing agents leaving unknown environments
In this paper we are concerned with multiscale modeling, control, and
simulation of self-organizing agents leaving an unknown area under limited
visibility, with special emphasis on crowds. We first introduce a new
microscopic model characterized by an exploration phase and an evacuation
phase. The main ingredients of the model are an alignment term, accounting for
the herding effect typical of uncertain behavior, and a random walk, accounting
for the need to explore the environment under limited visibility. We consider
both metrical and topological interactions. Moreover, a few special agents, the
leaders, not recognized as such by the crowd, are "hidden" in the crowd with a
special controlled dynamics. Next, relying on a Boltzmann approach, we derive a
mesoscopic model for a continuum density of followers, coupled with a
microscopic description for the leaders' dynamics. Finally, optimal control of
the crowd is studied. It is assumed that leaders exploit the herding effect in
order to steer the crowd towards the exits and reduce clogging. Locally-optimal
behavior of leaders is computed. Numerical simulations show the efficiency of
the optimization methods in both microscopic and mesoscopic settings. We also
perform a real experiment with people to study the feasibility of the proposed
bottom-up crowd control technique.Comment: in SIAM J. Appl. Math, 201
Supervised learning for kinetic consensus control
In this paper, how to successfully and efficiently condition a target
population of agents towards consensus is discussed. To overcome the curse of
dimensionality, the mean field formulation of the consensus control problem is
considered. Although such formulation is designed to be independent of the
number of agents, it is feasible to solve only for moderate intrinsic
dimensions of the agents space. For this reason, the solution is approached by
means of a Boltzmann procedure, i.e. quasi-invariant limit of controlled binary
interactions as approximation of the mean field PDE. The need for an efficient
solver for the binary interaction control problem motivates the use of a
supervised learning approach to encode a binary feedback map to be sampled at a
very high rate. A gradient augmented feedforward neural network for the Value
function of the binary control problem is considered and compared with direct
approximation of the feedback law.Comment: 6 pages, 3 figure
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