86,452 research outputs found
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras
Let be an irreducible smooth complex representation of a general
linear -adic group and let be an irreducible complex supercuspidal
representation of a classical -adic group of a given type, so that
is a representation of a standard Levi subgroup of a
-adic classical group of higher rank. We show that the reducibility of the
representation of the appropriate -adic classical group obtained by
(normalized) parabolic induction from does not depend on
, if is "separated" from the supercuspidal support of . (Here, "separated" means that, for each factor of a representation
in the supercuspidal support of , the representation parabolically
induced from is irreducible.) This was conjectured by E.
Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen,
that this induced representation is always reducible if the supercuspidal
support is not separated.)
More generally, we study, for a given set of inertial orbits of
supercuspidal representations of -adic general linear groups, the category
\CC _{I,\sigma} of smooth complex finitely generated representations of
classical -adic groups of fixed type, but arbitrary rank, and supercuspidal
support given by and , show that this category is equivalent to a
category of finitely generated right modules over a direct sum of tensor
products of extended affine Hecke algebras of type , and and
establish functoriality properties, relating categories with disjoint 's. In
this way, we extend results of C. Jantzen who proved a bijection between
irreducible representations corresponding to these categories. The proof of the
above reducibility result is then based on Hecke algebra arguments, using
Kato's exotic geometry.Comment: 21 pages, the results of the paper have been improved thanks to the
remarks and encouragements of the anonymous refere
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