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

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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

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

References:

[1]	J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming,, SIAM, (2001).
[2]	J. T. Betts, N. Biehn, S. L. Campbell and W. P. Huffman, Compensating for order variation in mesh refinement for direct transcription methods,, Journal of Computational and Applied Mathematics, 125 (2000), 147. doi: 10.1016/S0377-0427(00)00465-9.
[3]	J. T. Betts and W. P. Huffman, Mesh refinement in direct transcription methods for optimal control,, Optimal Control Applications and Methods, 19 (1998), 1. doi: 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q.
[4]	M. H. A. Biswas, L. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints,, Mathematical Biosciences and Engineering, 11 (2014), 761. doi: 10.3934/mbe.2014.11.761.
[5]	P. Falugi, E. Kerrigan and E. Van Wyk, Imperial College London Optimal Control Software. User Guide (ICLOCS),, Department of Electrical Engineering, (2010).
[6]	F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems and Control Letters, 42 (2001), 127. doi: 10.1016/S0167-6911(00)00084-0.
[7]	F. A. C. C. Fontes and H. Frankowska, Normality and nondegeneracy for optimal control problems with state constraints,, Journal of Optimization Theory and Applications, 22 (2015). doi: 10.1007/s10957-015-0704-1.
[8]	F. A. C. C. Fontes and S. O. Lopes, Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints,, Journal of Mathematical Analysis and Applications, 399 (2013), 27. doi: 10.1016/j.jmaa.2012.09.049.
[9]	I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems,, IEEE Control Systems, 15 (1995), 20. doi: 10.1109/37.476384.
[10]	I. Kornienko, L. T. Paiva and M. d. R. d. Pinho, Introducing state constraints in optimal control for health problems,, Procedia Technology, 17 (2014), 415. doi: 10.1016/j.protcy.2014.10.249.
[11]	S. O. Lopes, F. A. Fontes and M. d. R. de Pinho, On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems,, Discrete and Continuous Dynamical Systems (DCDS-A), 29 (2011), 559. doi: 10.3934/dcds.2011.29.559.
[12]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling,, Modeling paradigms and analysis of disease transmission models, 75 (2010), 67.
[13]	L. T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces,, Technical report, (2013).
[14]	L. T. Paiva and F. A. C. C. Fontes, Mesh refinement strategy for optimal control problems,, AIP Conference Proceedings, 1558 (2013), 590. doi: 10.1063/1.4825560.
[15]	L. T. Paiva and F. A. C. C. Fontes, Time-mesh refinement in optimal control problems for nonholonomic vehicles,, Procedia Technology, 17 (2014), 178. doi: 10.1016/j.protcy.2014.10.226.
[16]	M. A. Patterson, W. W. Hager and A. V. Rao, A ph mesh refinement method for optimal control,, Optimal Control Applications and Methods, ().
[17]	R. B. Vinter, Optimal Control,, Springer, (2000).
[18]	A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25. doi: 10.1007/s10107-004-0559-y.
[19]	Y. Zhao and P. Tsiotras, Density functions for mesh refinement in numerical optimal control,, Journal of Guidance, 34 (2011), 271. doi: 10.2514/1.45852.

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References:

[1]	J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming,, SIAM, (2001).
[2]	J. T. Betts, N. Biehn, S. L. Campbell and W. P. Huffman, Compensating for order variation in mesh refinement for direct transcription methods,, Journal of Computational and Applied Mathematics, 125 (2000), 147. doi: 10.1016/S0377-0427(00)00465-9.
[3]	J. T. Betts and W. P. Huffman, Mesh refinement in direct transcription methods for optimal control,, Optimal Control Applications and Methods, 19 (1998), 1. doi: 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q.
[4]	M. H. A. Biswas, L. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints,, Mathematical Biosciences and Engineering, 11 (2014), 761. doi: 10.3934/mbe.2014.11.761.
[5]	P. Falugi, E. Kerrigan and E. Van Wyk, Imperial College London Optimal Control Software. User Guide (ICLOCS),, Department of Electrical Engineering, (2010).
[6]	F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems and Control Letters, 42 (2001), 127. doi: 10.1016/S0167-6911(00)00084-0.
[7]	F. A. C. C. Fontes and H. Frankowska, Normality and nondegeneracy for optimal control problems with state constraints,, Journal of Optimization Theory and Applications, 22 (2015). doi: 10.1007/s10957-015-0704-1.
[8]	F. A. C. C. Fontes and S. O. Lopes, Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints,, Journal of Mathematical Analysis and Applications, 399 (2013), 27. doi: 10.1016/j.jmaa.2012.09.049.
[9]	I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems,, IEEE Control Systems, 15 (1995), 20. doi: 10.1109/37.476384.
[10]	I. Kornienko, L. T. Paiva and M. d. R. d. Pinho, Introducing state constraints in optimal control for health problems,, Procedia Technology, 17 (2014), 415. doi: 10.1016/j.protcy.2014.10.249.
[11]	S. O. Lopes, F. A. Fontes and M. d. R. de Pinho, On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems,, Discrete and Continuous Dynamical Systems (DCDS-A), 29 (2011), 559. doi: 10.3934/dcds.2011.29.559.
[12]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling,, Modeling paradigms and analysis of disease transmission models, 75 (2010), 67.
[13]	L. T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces,, Technical report, (2013).
[14]	L. T. Paiva and F. A. C. C. Fontes, Mesh refinement strategy for optimal control problems,, AIP Conference Proceedings, 1558 (2013), 590. doi: 10.1063/1.4825560.
[15]	L. T. Paiva and F. A. C. C. Fontes, Time-mesh refinement in optimal control problems for nonholonomic vehicles,, Procedia Technology, 17 (2014), 178. doi: 10.1016/j.protcy.2014.10.226.
[16]	M. A. Patterson, W. W. Hager and A. V. Rao, A ph mesh refinement method for optimal control,, Optimal Control Applications and Methods, ().
[17]	R. B. Vinter, Optimal Control,, Springer, (2000).
[18]	A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25. doi: 10.1007/s10107-004-0559-y.
[19]	Y. Zhao and P. Tsiotras, Density functions for mesh refinement in numerical optimal control,, Journal of Guidance, 34 (2011), 271. doi: 10.2514/1.45852.

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