4 research outputs found
Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations
We consider integer-restricted optimal control of systems governed by
abstract semilinear evolution equations. This includes the problem of optimal
control design for certain distributed parameter systems endowed with multiple
actuators, where the task is to minimize costs associated with the dynamics of
the system by choosing, for each instant in time, one of the actuators together
with ordinary controls. We consider relaxation techniques that are already used
successfully for mixed-integer optimal control of ordinary differential
equations. Our analysis yields sufficient conditions such that the optimal
value and the optimal state of the relaxed problem can be approximated with
arbitrary precision by a control satisfying the integer restrictions. The
results are obtained by semigroup theory methods. The approach is constructive
and gives rise to a numerical method. We supplement the analysis with numerical
experiments
k-Extreme Points in Symmetric Spaces of Measurable Operators
Let be a semifinite von Neumann algebra with a faithful,
normal, semifinite trace and be a strongly symmetric Banach function
space on . We show that an operator in the unit sphere of
is -extreme, , whenever its
singular value function is -extreme and one of the following
conditions hold (i) or (ii)
and , where and
are null and support projections of , respectively. The converse is
true whenever is non-atomic. The global -rotundity property
follows, that is if is non-atomic then is -rotund if and
only if is -rotund. As a consequence of the
noncommutive results we obtain that is a -extreme point of the unit ball
of the strongly symmetric function space if and only if its decreasing
rearrangement is -extreme and . We conclude
with the corollary on orbits and . We get that is a
-extreme point of the orbit , , or
, , , if and only if
and . From this we obtain a
characterization of -extreme points in Marcinkiewicz spaces.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s00020-014-2206-