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September  2015, 35(9): 4553-4572. doi: 10.3934/dcds.2015.35.4553

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, Portugal
2. 

ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal

Received  May 2014 Revised  October 2014 Published  April 2015

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.
Keywords: nonlinear optimization, numerical methods, grid refinement, Optimal control, nonholonomic vehicles, SEIR model..
Mathematics Subject Classification: Primary: 49J15, 65L5.
Citation: Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
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show all references

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SIAM, 2001.  Google Scholar

[2]

Journal of Computational and Applied Mathematics, 125 (2000), 147-158. doi: 10.1016/S0377-0427(00)00465-9.  Google Scholar

[3]

Optimal Control Applications and Methods, 19 (1998), 1-21. doi: 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q.  Google Scholar

[4]

Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761.  Google Scholar

[5]

Department of Electrical Engineering, Imperial College London, London, UK, 2010. Google Scholar

[6]

Systems and Control Letters, 42 (2001), 127-143. doi: 10.1016/S0167-6911(00)00084-0.  Google Scholar

[7]

Journal of Optimization Theory and Applications, 22 (2015), p30. doi: 10.1007/s10957-015-0704-1.  Google Scholar

[8]

Journal of Mathematical Analysis and Applications, 399 (2013), 27-37. doi: 10.1016/j.jmaa.2012.09.049.  Google Scholar

[9]

IEEE Control Systems, 15 (1995), 20-36. doi: 10.1109/37.476384.  Google Scholar

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Procedia Technology, 17 (2014), 415-422. doi: 10.1016/j.protcy.2014.10.249.  Google Scholar

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Discrete and Continuous Dynamical Systems (DCDS-A), 29 (2011), 559-575. doi: 10.3934/dcds.2011.29.559.  Google Scholar

[12]

Modeling paradigms and analysis of disease transmission models, 75 (2010), 67-81. Google Scholar

[13]

Technical report, Faculdade de Engenharia, Universidade do Porto, 2013. Google Scholar

[14]

AIP Conference Proceedings, 1558 (2013), 590-593, Proceding of the ICNAAM 2013 - 11th International Conference on Numerical Analysis and Applied Mathematics. doi: 10.1063/1.4825560.  Google Scholar

[15]

Procedia Technology, 17 (2014), 178-185. doi: 10.1016/j.protcy.2014.10.226.  Google Scholar

[16]

M. A. Patterson, W. W. Hager and A. V. Rao, A ph mesh refinement method for optimal control,, Optimal Control Applications and Methods, ().   Google Scholar

[17]

Springer, 2000. Google Scholar

[18]

Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.  Google Scholar

[19]

Journal of Guidance, Control, and Dynamics, 34 (2011), 271-277. doi: 10.2514/1.45852.  Google Scholar

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