A MATLAB Package for Analyzing Optimal Control Problem

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.

The main topic of this project was the development of a MATLAB based software package(OCMat), that allows the analysis of optimal control problems over an infinite time horizon. Since most of the models are not analytically tractable the analysis relies on numerical methods. One of the difficulties for problems over an infinite time horizon is the provision of conditions that the solution has to satisfy at the end. In a first step the models limiting behaviour is analyzed.This includes the calculation of steady states, periodic solutions and possible divergent behaviour as well. When all possible long-run solutions are determined this information is used to calculate the solutions for arbitrary initial points. Therefore, the numerical algorithm bases on a boundary value solver combined with a continuation step. This ansatz allows to start from the long-run solution to continue and find the solution for arbitrary initial states or parameter values.OCMat is implemented within the MATLAB environment. Using the symbolic computation capabilities of MATLAB many of the necessary recurrent steps were automatized. Specifically this includes the derivation of the first order necessary optimality conditions and the generation of the MATLAB files that are needed for the numerical computations. Therefore, the user writes an initiation function and usually the automatization procedure allows to start a few minutes later with the numerical analysis. Even though this task is not of specific mathematical interest it eases the handling tremendously and changes in the model formulation can be adapted very fast and flexible.A further task, beside the development of the numerical algorithms, was the provision of a programming environment that facilitates the storing and reloading of (numerical) results. But also the graphical depiction and printing used for presentations and publications. Therefore, the object oriented programming environment of MATLAB was used. This allows a consistent representation of the different models and the provision of specific functions for loading, storing, plotting, etc.Beyond the scientific work another important aspect is the dissemination of the results within the non-scientific community. A promising approach is the graphical presentation as short animations, where the entire information about the models behaviour can be maintained but the underlying mathematics is removed. For that purpose an interface to the opens source software Synfig Studio and Blender (for 2D/3D animations) were programmed. This opens the possibility for a discussion between the scientist and public on equal terms. Examples from a talk held at the workshop The Economics of Complex Systems workshop, April 28-29 at the Beijer Institute of Ecological Economics, can be downloaded from http://orcos.tuwien.ac.at/research/ocmat_software/#c50711.