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