Adaptive time--mesh refinement in optimal control problems with state constraints

Luís Tiago Paiva; Fernando A. C. C. Fontes

doi:10.3934/dcds.2015.35.4553

Article Contents

2015, Volume 35, Issue 9: 4553-4572. Doi: 10.3934/dcds.2015.35.4553

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Adaptive time--mesh refinement in optimal control problems with state constraints

Luís Tiago Paiva^1, and
Fernando A. C. C. Fontes^2,

1.
SYSTEC-ISR, Faculdade de Engenharia, Universidade do Porto, 4200-465, Porto
2.
ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto

Received: April 30, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

When using direct methods to solve continuous-time nonlinear optimal control problems, regular time meshes having equidistant spacing are most frequently used. However, in some cases, these meshes cannot cope accurately with nonlinear behaviour and increasing uniformly the number of mesh nodes may lead to a more complex problem. We propose an adaptive time--mesh refinement algorithm, considering different levels of refinement and several mesh refinement criteria. Namely, we use information of the adjoint multipliers to decide where to refine further. This technique is here tested to solve two optimal control problems. One involving nonholonomic vehicles with state constraints which is characterized by having strong nonlinearities and by discontinuous controls; the other is also a nonlinear problem of a compartmental SEIR system. The proposed strategy leads to results with higher accuracy and yet with lower overall computational time, when compared to results obtained by meshes having equidistant spacing. We also apply the necessary condition of optimality in the form of the Maximum Principle of Pontryagin to characterize the solution and to validate the numerical results.

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Mathematics Subject Classification: Primary: 49J15, 65L50.

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