A Matlab Package (OCMat) for Analyzing Optimal Control Problems

01.11.2010 - 28.02.2014
Research funding project
In economic and management applications optimal control models with an infinite time horizon play an important role. Thus Pontryagin's Maximumprinciple yields an boundary value problem (BVP) on an infinite time interval. Therefore the in a numerical calculation the transversality condition has to be replaced by an asymptotic transversality condition. Or in a different approach the time interval is transformed yielding a singular BVP. In this proposal we develop a Matlab package allowing us to solve such problems. We therefore implement a BVP solver under Matlab allowing continuation. Using this Matlab package the (un)stable paths and center paths of steady states and limit cycles can be computed. Subsequently a bifurcation analysis of the canonical system can be carried out, including global bifurcations, such as heteroclinic bifurcations. Moreover we use the object oriented programming environment of Matlab to create an object representing the optimal control model for specific parameter values. After initialization of the model the first order necessary optimality conditions are calculated by using the symbolic toolbox. Thereafter the Matlab files for the analysis are automatically generated. The results can then be stored within the object and saved as a datafile. Implemented plot functions allow an easy graphical representation of the calculated results and paths. Heteroclinic bifurcations of the canonical system often correspond with the occurrence of DNSS or Skiba points. These are points of the state space exhibiting multiple optimal solutions. In the optimal vector field the occurrence of DNSS points give rise to a bifurcation, e.g., an ¿indifference-attractor-bifurcation¿ as is described by Florian Wagener. Therefore our software package is well suited to the numerical bifurcation analysis of the optimal vector field. For models with two or more states DNSS curves or surfaces can exist. These manifolds may also be computed using our continuation algorithm, starting at a single DNSS point. For an efficient calculation of solution paths it is necessary to use a BVP solver, which is well adapted to the specific problems occurring in the context of optimal control problems. Some of these are singularities, as already mentioned at the beginning but also the treatment of algebraic differential equations and specifically the inclusion of a continuation algorithm. Therefore we decided to take advantage of the experience of the research group of Ewa Weinmüller to adapt and further develop their implemented BVP solver.

People

Project leader

Institute

Grant funds

  • FWF - Österr. Wissenschaftsfonds (National) Austrian Science Fund (FWF)

Research focus

  • Modeling and Simulation: 100%

Keywords

GermanEnglish
optimale KontrolltheorieOptimal Control Theory
mehrfache optimale LösungenMultiple Optimal Solutions
BifurkationstheorieBifurcation Theory
Randwertproblem auf unendlichem ZeithorizontInfinite Time BVP

Publications