Second order conditions for periodic optimal control problems

• Autorzy: Allwright J. , Vinter R.
• Języki publikacji: EN

Abstrakty

EN

This paper concerns second order sufficient conditions of optimality, involving the Riccati equation, for optimal control problems with periodic boundary conditions. The problems considered involve no pathwise constraints and are 'regular', in the sense that the strengthened Legendre-Clebsch condition is assumed to be satisfied. A well-known sufficient, condition, which we refer to as the Riccati sufficient condition, requires the existence of a global solution to the Riccati equation whose endpoint values satisfy a certain inequality. A sharper condition, named the extended sufficient, condition, takes the form of an inequality involving the solutions of a Riccati equation and two additional linear matrix equations. We highlight the superiority of the extended Riccati sufficient condition and develop a number of equivalent formulations of this condition. Not only does the extended Riccati sufficient, condition supply more information about, minimizers, but it is the basis of simpler numerical tests for assessing whether an extremal is a minimizer, at least in a local sense. The Riccati and also the extended Riccati sufficient conditions are applied to a variant of Speyer's 'sailboat' problem, involving parameters. It is found that the extended Riccati sufficient condition identifies a much larger set of points on parameter space for which a nominal control is optimal, in comparison to the Riccati sufficient condition.

Słowa kluczowe

EN

second order conditions; periodic optimal control; Riccati equations; dynamic programming

PL

równanie Riccatiego; programowanie dynamiczne

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 3
• Strony: 617--643
• Opis fizyczny: Bibliogr. 7 poz., wykr.

Twórcy

autor

Allwright J.

• Department of Electrical and Electronic Engineering, Imperial College London SW7 2BT, United Kingdom

autor

Vinter R.

• Department of Electrical and Electronic Engineering, Imperial College London SW7 2BT, United Kingdom

Bibliografia

• Hestenes, M. R. (1951) Applications of the Theory of Quadratic Forms in Hilbert Space to Calculus of Variations. Pacific J. Math 1, 525-581.
• Maurer, H. and Pickenhain, S. (1995) Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theory and Applic. 86, 649-667.
• Speyer, J.L. (1996) Periodic Optimal Flight. J. Guidance, Control and Dynamics 61,745-754.
• Stefani, G. and Zezza, P. (1997) Constrained Regular LQ-Control Problems. SIAM J. Control and Optim. 35, 876-900.
• Vinter, R.B. (2000) Optimal Control. Birkh¨auser, Boston.
• Wang, Q. and Speyer, J.L. (1990) Necessary and Sufficient Conditions for Local Optimality of a Periodic Process. SIAM J. Control and Optim. 28, 482-497.
• Zeidan, V. (2001) New Second-Order Optimality Conditions for Variational Problems with C2-Hamiltonians. SIAM J. Control and Optim. 40, 577-609.