1,008 research outputs found
Final-State Constrained Optimal Control via a Projection Operator Approach
In this paper we develop a numerical method to solve nonlinear optimal
control problems with final-state constraints. Specifically, we extend the
PRojection Operator based Netwon's method for Trajectory Optimization (PRONTO),
which was proposed by Hauser for unconstrained optimal control problems. While
in the standard method final-state constraints can be only approximately
handled by means of a terminal penalty, in this work we propose a methodology
to meet the constraints exactly. Moreover, our method guarantees recursive
feasibility of the final-state constraint. This is an appealing property
especially in realtime applications in which one would like to be able to stop
the computation even if the desired tolerance has not been reached, but still
satisfy the constraints. Following the same conceptual idea of PRONTO, the
proposed strategy is based on two main steps which (differently from the
standard scheme) preserve the feasibility of the final-state constraints: (i)
solve a quadratic approximation of the nonlinear problem to find a descent
direction, and (ii) get a (feasible) trajectory by means of a feedback law
(which turns out to be a nonlinear projection operator). To find the (feasible)
descent direction we take advantage of final-state constrained Linear Quadratic
optimal control methods, while the second step is performed by suitably
designing a constrained version of the trajectory tracking projection operator.
The effectiveness of the proposed strategy is tested on the optimal state
transfer of an inverted pendulum
A feedback approach to bifurcation analysis in biochemical networks with many parameters
Feedback circuits in biochemical networks which underly cellular signaling
pathways are important elements in creating complex behavior. A specific aspect
thereof is how stability of equilibrium points depends on model parameters. For
biochemical networks, which are modelled using many parameters, it is typically
very difficult to estimate the influence of parameters on stability. Finding
parameters which result in a change in stability is a key step for a meaningful
bifurcation analysis. We describe a method based on well known approaches from
control theory, which can locate parameters leading to a change in stability.
The method considers a feedback circuit in the biochemical network and relates
stability properties to the control system obtained by loop--breaking. The
method is applied to a model of a MAPK cascade as an illustrative example
Continuation-minimization methods for stability problems
AbstractWe study the solution branches of stable and unstable bifurcations in certain semilinear elliptic eigenvalue problems with Dirichlet boundary conditions. A secant predictor-line search backtrack corrector continuation method is described to trace the solution curves numerically. Sample numerical results with computer graphic output are reported
Applying numerical continuation to the parameter dependence of solutions of the Schr\"odinger equation
In molecular reactions at the microscopic level the appearance of resonances
has an important influence on the reactivity. It is important to predict when a
bound state transitions into a resonance and how these transitions depend on
various system parameters such as internuclear distances. The dynamics of such
systems are described by the time-independent Schr\"odinger equation and the
resonances are modeled by poles of the S-matrix. Using numerical continuation
methods and bifurcation theory, techniques which find their roots in the study
of dynamical systems, we are able to develop efficient and robust methods to
study the transitions of bound states into resonances. By applying Keller's
Pseudo-Arclength continuation, we can minimize the numerical complexity of our
algorithm. As continuation methods generally assume smooth and well-behaving
functions and the S-matrix is neither, special care has been taken to ensure
accurate results. We have successfully applied our approach in a number of
model problems involving the radial Schr\"odinger equation
On the Complexity of Exclusion Algorithms for Optimization
AbstractExclusion algorithms are a well-known tool in the area of interval analysis for finding all solutions of a system of nonlinear equations or for finding the global minimum of a function over a compact domain. The present paper discusses a new class of tests for such algorithms in the context of global optimization and presents complexity results concerning the resulting algorithms
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