Abstract
In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.
Similar content being viewed by others
References
Bryson A.E. Jr, Ho Y.C.: Applied Optimal Control. Blaisdell Publishing Company, Massachusetts (1969)
Tsuchiya T., Suzuki S.: Computational method of optimal control problem using mathematical programming method (1st report) introduction of sensitivity differential equations. J. Jpn. Soc. Aeronaut. Space Sci. 45, 231–237 (1997) (in Japanese)
Hull D.G.: Conversion of optimal control problems into parameter optimization problems. J. Guid. Control Dyn. 20, 57–60 (1997)
Hargraves C.R., Paris S.W.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dyn. 10, 338–342 (1987)
Elnagar G., Kazemi M.A., Razzaghi M.: The pseudospectral legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40, 1793–1796 (1995)
Ali M.M., Storey C., Törn A.: Application of stochastic global optimization algorithms to practical problems. J. Optim. Theory Appl. 95, 545–563 (1997)
Esposito W.R., Floudas C.A.: Deterministic global optimization in nonlinear optimal control problems. J. Glob. Optim. 17, 97–126 (2000)
Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)
Parrilo P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. Ser. B 96, 293–320 (2003)
Pardalos P.M., Wolkowicz H.: Topics in Semidefinite and Interior-Point Methods. American Mathematical Society, Providence (1998)
Lasserre J.B., Henrion D., Prieur C., Trélat E.: Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM J. Control Optim. 47, 1643–1666 (2008)
Meziat R., Patiño D., Pedregal P.: An alternative approach for non-linear optimal control problems based on the method of moments. Comput. Optim. Appl. 38, 147–171 (2007)
Lasserre J.B.: A sum of squares approximation of nonnegative polynomials. SIAM J. Optim. 16, 751–765 (2006)
Waki H., Kim S., Kojima M., Muramatsu M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)
Ohtsuka T.: Model structure simplification of nonlinear systems via immersion. IEEE Trans. Autom. Control 50, 607–618 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tsuchiya, T. Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation. J Glob Optim 54, 831–854 (2012). https://doi.org/10.1007/s10898-011-9797-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9797-8