2,883 research outputs found
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
Consistent Approximations for the Optimal Control of Constrained Switched Systems
Though switched dynamical systems have shown great utility in modeling a
variety of physical phenomena, the construction of an optimal control of such
systems has proven difficult since it demands some type of optimal mode
scheduling. In this paper, we devise an algorithm for the computation of an
optimal control of constrained nonlinear switched dynamical systems. The
control parameter for such systems include a continuous-valued input and
discrete-valued input, where the latter corresponds to the mode of the switched
system that is active at a particular instance in time. Our approach, which we
prove converges to local minimizers of the constrained optimal control problem,
first relaxes the discrete-valued input, then performs traditional optimal
control, and then projects the constructed relaxed discrete-valued input back
to a pure discrete-valued input by employing an extension to the classical
Chattering Lemma that we prove. We extend this algorithm by formulating a
computationally implementable algorithm which works by discretizing the time
interval over which the switched dynamical system is defined. Importantly, we
prove that this implementable algorithm constructs a sequence of points by
recursive application that converge to the local minimizers of the original
constrained optimal control problem. Four simulation experiments are included
to validate the theoretical developments
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Partitions of Minimal Length on Manifolds
We study partitions on three dimensional manifolds which minimize the total
geodesic perimeter. We propose a relaxed framework based on a
-convergence result and we show some numerical results. We compare our
results to those already present in the literature in the case of the sphere.
For general surfaces we provide an optimization algorithm on meshes which can
give a good approximation of the optimal cost, starting from the results
obtained using the relaxed formulation
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