Project: Theory, Applications and Numerics of Optimal Control

Researchers from University of Porto (Faculdade de Engenharia e Faculdade de Economia), University of Minho and University of Aveiro (Departamento de Matemática) and a senior foreigner researcher from Germany get together as a team in this project.

The following research Units are also involved: SYSTEC, CMUP, CIDMA.

Team Member

Maria do Rosário de Pinho (PI),

Amélia C. D. Caldeira,

António Pedro Aguiar,

Cristiana J. Silva,

Dalila Fontes,

Delfim F. M. Torres,

Fernando A. C. C. Fontes,

Fernando Lobo Pereira

Georgi Smirnov,

Helmut Maurer,

Luís Tiago Paiva ,

Manuel Guerra,

M. Fernanda P. Costa,

M. M. A. Ferreira,

Paula Rocha

Sílvio Gama,

Sofia Oliveira Lopes.

Grantees:

Sasa Rakovi (From November 2016 to April 2017)

Diyako Ghaderyan (Research Grantee since March 2017 to January 2018)

Pos- Doc Researchers:

Filipa N. Nogueira (from January 2018 to January 2019),

Zahra Foroozandeh (since February 19, 2018)

Jorge Becerril(since September 3, 2018).

The participation of Karla Lorena Cortez (prodomus) is also gratefully acknowledged.

Short Description of the Project

This project aims to contribute to the development of Optimal Control methods and applications. It focuses on building bridges between theory, applications and numerical treatment of optimal control problems by solving a selected number of problems from different areas.

This project is launched as a continuation of the work initialized with the FCT project PTDC/EEI-AUT/1450/2012|FCOMP-01-0124-FEDER-028894.

It brings together a renewed researcher team with a multidisciplinary background including Optimal Control Theory, Control Engineering, Optimization and Numerical Analysis to work on theoretical developments in conjunction with interesting and relevant applications of optimal control.

We seek to bring together major Portuguese research centers in Optimal Control to promote the exchange of experience and knowledge based on on-going research activities, the training of young researchers and to deliver value to the society. The proposed Research Team will collaborate closely with EU and American research groups to explore new research methods and to promote the introduction of optimal control methods on different areas. We aim to contribute to the creation of a vibrant, productive and efficient optimal control research community, to attract young researchers, to increase awareness of the benefits of optimal control areas as different as biomedicine and economics and to deliver value to the society.

While the driving force behind the choice of applications to be studied will be the need to test and illustrate recent theoretical advances, we expect the applications themselves to trigger new theoretical development. Many results leading to a characterization of solution to optimal control problems may be used to validate numerical solutions (i.e., to verify that a certain solution is possibly an optimal solution). However, the numerical verification of optimality is in general a hard task. We hope to remedy some of these situations by proposing numerically verifiable optimality conditions and such conditions on regularity and normality. Special attention will be paid to problems with state constraints, mixed constraints, bang-bang controls, singular controls, bang-singular junctions. We shall also be concerned with the synthesis of controlled trajectories, sensitivity analysis of the solutions to parameters and Model Predictive Control techniques.

Selected problems from biomedicine, such as epidemiology, power systems; economics and also more classical areas such as the planning of trajectories of unmanned vehicles will be considered. The challenges related to applied problems that will be addressed can be divided into four classes

a) formulation of the problems for some applications;

b) solving the problems numerically via optimization methods;

c) establishing the optimality of the numerical solution;

d) connections between the types of solutions and concepts of the application under study.

While b) and c) fall within the expertise of the research team, their collaborators and the project advisory board, items a) and d) above will be conducted in close consultation with experts on the specific area of the applied problems.

The choice of problems is based primarily on their societal interest but it is done to ascertain that their analytical treatment will trigger the need for improvements of theoretical results. We do not intend to address modeling issues. Rather we shall work with known models. However, we will seek to break new ground and to obtain new insight on the problems with innovative choices of objective functions, reformulations, introduction of non-standard constraints as well as a thorough sensitivity analysis of the solutions.

Although the problems under consideration will be solved via direct and indirect methods using well known optimization software (for example WORHP, IPOPTS, KNITRO, GPOS, to name but a few) and software based on Hamilton Jacobi equations (for exemple, RJO, PRONTO), we hope to contribute on those numerical schemes with routines for automatic refinement of the mesh and to allow the numerical verification of optimality and regularity conditions.