Abstract
In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.
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Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)
Teo, K.L., Jennings, L.S.: Nonlinear optimal control problems with continuous state inequality constraints. J. Optim. Theory Appl. 63(1), 1–22 (1989)
Loxton, R.C., Teo, K.L., Rehbock, V., Yiu, K.F.C.: Optimal control problems with a continuous inequality constraint on the state and control. Automatica 45, 2250–2257 (2009)
Buskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real time control. J. Comput. Appl. Math. 120(1–2), 85–108 (2000)
Chen, T.W.C., Vassiliadis, V.S.: Inequality path constraints in optimal control: a finite iteration ε-convergent scheme based on pointwise discretization. J. Process Control 15(3), 353–362 (2005)
Huyer, W., Neumaier, A.: A new exact penalty function. SIAM J. Optim. 13(4), 1141–1158 (2003)
Gerdts, M.: Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems. SIAM J. Control Optim. 19(1), 326–350 (2008)
Gerdts, M.: A nonsmooth Newton’s method for control-state constrained optimal control problems. Math. Comput. Simul. 79(4), 925–936 (2008)
Gerdts, M., Kunkel, M.: A nonsmooth Newton’s method for discretized optimal control problems with state and control constraints. J. Ind. Manage. Optim. 4(2), 247–270 (2008)
Teo, K.L., Jennings, L.S., Lee, H.W.J., Rehbock, V.L.: The control parameterization enhancing transformation for constrained optimal control problems. J. Aust. Math. Soc. Ser. B, Appl. Math 40, 314–335 (1997)
Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach for Optimal Control Problems. Longman, New York (1991)
Yu, C., Teo, K.L., Zhang, L., Bai, Y.: A new exact penalty function method for continuous inequality constrained optimization problems. J. Ind. Manag. Optim. 6(4), 895–910 (2010)
Mehra, R.K., Davis, R.E.: A generalized gradient method for optimal control problems with inequality constraints and singular Arcs. IEEE Trans. Autom. Control AC-17, 69–78 (1972)
Sakawa, Y., Shindo, Y.: Optimal control of container cranes. Automatica 18, 257–266 (1982)
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Li, B., Yu, C.J., Teo, K.L. et al. An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem. J Optim Theory Appl 151, 260–291 (2011). https://doi.org/10.1007/s10957-011-9904-5
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DOI: https://doi.org/10.1007/s10957-011-9904-5