American Institute of Mathematical Sciences

Advanced
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 - 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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Falugi, E. Kerrigan and E. Van Wyk, Imperial College London Optimal Control Software. User Guide (ICLOCS),, Department of Electrical Engineering, (2010).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems,, IEEE Control Systems, 15 (1995), 20.  doi: 10.1109/37.476384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces,, Technical report, (2013).   Google Scholar

[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.  Google Scholar

[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.  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]

R. B. Vinter, Optimal Control,, Springer, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming,, SIAM, (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Falugi, E. Kerrigan and E. Van Wyk, Imperial College London Optimal Control Software. User Guide (ICLOCS),, Department of Electrical Engineering, (2010).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems,, IEEE Control Systems, 15 (1995), 20.  doi: 10.1109/37.476384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces,, Technical report, (2013).   Google Scholar

[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.  Google Scholar

[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.  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]

R. B. Vinter, Optimal Control,, Springer, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220
[2]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521
[3]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092
[4]

Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009
[5]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
[6]

M. H. A. Biswas, L. T. Paiva, MdR de Pinho. A SEIR model for control of infectious diseases with constraints. Mathematical Biosciences & Engineering, 2014, 11 (4) : 761-784. doi: 10.3934/mbe.2014.11.761
[7]

Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456
[8]

Zhong Wan, Jingjing Liu, Jing Zhang. Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-23. doi: 10.3934/jimo.2019046
[9]

Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences & Engineering, 2017, 14 (1) : 111-126. doi: 10.3934/mbe.2017008
[10]

Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
[11]

Markus Thäter, Kurt Chudej, Hans Josef Pesch. Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth. Mathematical Biosciences & Engineering, 2018, 15 (2) : 485-505. doi: 10.3934/mbe.2018022
[12]

Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211
[13]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
[14]

Jie Sun. On methods for solving nonlinear semidefinite optimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 1-14. doi: 10.3934/naco.2011.1.1
[15]

Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112
[16]

Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515
[17]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15
[18]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
[19]

Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078
[20]

Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences