Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T07:21:55.159Z Has data issue: false hasContentIssue false
  • English
  • Français

A deterministic affine-quadratic optimal control problem∗

Published online by Cambridge University Press:  21 May 2014

Yuanchang Wang  and
Jiongmin Yong
Yuanchang Wang
Affiliation:
School of Mathematics, Yunnan Normal University, Kunming, 650500, P.R. China Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA. [email protected]
Jiongmin Yong
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA. [email protected]
Get access

Abstract

A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional conditions, it is shown that the value function is actually twice differentiable and the so-called quasi-Riccati equation is derived, whose solution can be used to construct the state feedback representation for the optimal control.

Keywords

Affine quadratic optimal controldynamic programmingHamilton–Jacobi–Bellman equationquasi-Riccati equationstate feedback representation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 633 - 661
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banks, S.P. and Cimen, T., Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria. System Control Lett. 53 (2004) 327–346. Google Scholar
S.P. Banks and T. Cimen, Optimal control of nonlinear systems, Optimization and Control with Applications. In vol. 96 of Appl. Optim. Springer, New York (2005) 353–367.
Banks, H.T., Lewis, B.M. and Tran, H.T., Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach. Comput. Optim. Appl. 37 (2007) 177–218. Google Scholar
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997).
Bardi, M. and DaLio, F., On the Bellman equation for some unbounded control problems. Nonlinear Differ. Eqs. Appl. 4 (1997) 491–510. Google Scholar
Benveniste, L.M. and Scheinkman, J.A., On the differentiability of the value function in dynamic models of economics. Econometrica 47 (1979) 727–732. Google Scholar
L.D. Berkovitz, Optimal Control Theory. Springer-Verlag, New York (1974).
J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000).
Cannarsa, P. and Frankowska, H., Some characterizatins of optimal trajecotries in control theory. SIAM J. Control Optim. 29 (1991) 1322–1347. Google Scholar
T. Cimen, State-dependent Riccati equation (SDRE) control: a survey. Proc. 17th World Congress IFAC (2008) 3761–3775.
H. Frankowska, Value Function in Optimal Control, Mathematical Control Theory, Part 1, 2 (2001) 516–653.
Hildebrandt, T. and Graves, L., Implicit functions and their differentials in general analysis. Trans. Amer. Math. Soc. 29 (1927) 127–153. Google Scholar
Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations. Probab. Theory Rel. Fields 103 (1995) 273–283. Google Scholar
Kalman, R.E., Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5 (1960) 102–119. Google Scholar
J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications. Vol. 1702 of Lect. Notes Math. Springer-Verlag (1999).
Qiu, H. and Yong, J., Hamilton-Jacobi equations and two-person zero-sum differential games with unbounded controls. ESAIM: COCV 19 (2013) 404–437. Google Scholar
Rincón-Zapatero, J.P. and Santos, M.S., Differentiability of the value function in continuous-time economic models. J. Math. Anal. Appl. 394 (2012) 305–323. Google Scholar
Yong, J., Finding adapted solutions of forward-backward stochastic differential equations – method of continuation, Probab. Theory Rel. Fields 107 (1997) 537–572. Google Scholar
Yong, J., Stochastic optimal control and forward-backward stochastic differential equations. Comput. Appl. Math. 21 (2002) 369–403. Google Scholar
Yong, J., Forward backward stochastic differential equations with mixed initial and terminal conditions. Trans. AMS 362 (2010) 1047–1096. Google Scholar
J. Yong and X.Y. Zhou, Stochastic Control: Hamiltonian Systems and HJB Equations. Springer-Verlag (1999).
You, Y., A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems. SIAM J. Control Optim. 25 (1987) 905–920. Google Scholar
You, Y., Synthesis of time-variant optimal control with nonquadratic criteria. J. Math. Anal. Appl. 209 (1997) 662–682. Google Scholar
E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems. Springer-Verlag, New York (1986) 150–151.
E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators. Springer-Verlag, New York (1990).