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On Weak Topology for Optimal Control of Switched Nonlinear Systems
Optimal control of switched systems is challenging due to the discrete nature
of the switching control input. The embedding-based approach addresses this
challenge by solving a corresponding relaxed optimal control problem with only
continuous inputs, and then projecting the relaxed solution back to obtain the
optimal switching solution of the original problem. This paper presents a novel
idea that views the embedding-based approach as a change of topology over the
optimization space, resulting in a general procedure to construct a switched
optimal control algorithm with guaranteed convergence to a local optimizer. Our
result provides a unified topology based framework for the analysis and design
of various embedding-based algorithms in solving the switched optimal control
problem and includes many existing methods as special cases
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T-optimal designs formulti-factor polynomial regressionmodelsvia a semidefinite relaxation method
We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models wherethe design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets.Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When theregression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtainan equivalent semidefinite program.When there are two or more factors in the models, we apply a moment relaxation techniqueand approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When therelaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and itsoptimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples
Robust Transmission in Downlink Multiuser MISO Systems: A Rate-Splitting Approach
We consider a downlink multiuser MISO system with bounded errors in the
Channel State Information at the Transmitter (CSIT). We first look at the
robust design problem of achieving max-min fairness amongst users (in the
worst-case sense). Contrary to the conventional approach adopted in literature,
we propose a rather unorthodox design based on a Rate-Splitting (RS) strategy.
Each user's message is split into two parts, a common part and a private part.
All common parts are packed into one super common message encoded using a
public codebook, while private parts are independently encoded. The resulting
symbol streams are linearly precoded and simultaneously transmitted, and each
receiver retrieves its intended message by decoding both the common stream and
its corresponding private stream. For CSIT uncertainty regions that scale with
SNR (e.g. by scaling the number of feedback bits), we prove that a RS-based
design achieves higher max-min (symmetric) Degrees of Freedom (DoF) compared to
conventional designs (NoRS). For the special case of non-scaling CSIT (e.g.
fixed number of feedback bits), and contrary to NoRS, RS can achieve a
non-saturating max-min rate. We propose a robust algorithm based on the
cutting-set method coupled with the Weighted Minimum Mean Square Error (WMMSE)
approach, and we demonstrate its performance gains over state-of-the art
designs. Finally, we extend the RS strategy to address the Quality of Service
(QoS) constrained power minimization problem, and we demonstrate significant
gains over NoRS-based designs.Comment: Accepted for publication in IEEE Transactions on Signal Processin
The value function of an asymptotic exit-time optimal control problem
We consider a class of exit--time control problems for nonlinear systems with
a nonnegative vanishing Lagrangian. In general, the associated PDE may have
multiple solutions, and known regularity and stability properties do not hold.
In this paper we obtain such properties and a uniqueness result under some
explicit sufficient conditions. We briefly investigate also the infinite
horizon problem
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