# A deterministic affine-quadratic optimal control problem

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 3, page 633-661
- ISSN: 1292-8119

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topWang, Yuanchang, and Yong, Jiongmin. "A deterministic affine-quadratic optimal control problem." ESAIM: Control, Optimisation and Calculus of Variations 20.3 (2014): 633-661. <http://eudml.org/doc/272934>.

@article{Wang2014,

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.},

author = {Wang, Yuanchang, Yong, Jiongmin},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {affine quadratic optimal control; dynamic programming; Hamilton–Jacobi–Bellman equation; quasi-Riccati equation; state feedback representation; affine-quadratic optimal control problem; Hamilton-Jacobi-Bellman equation},

language = {eng},

number = {3},

pages = {633-661},

publisher = {EDP-Sciences},

title = {A deterministic affine-quadratic optimal control problem},

url = {http://eudml.org/doc/272934},

volume = {20},

year = {2014},

}

TY - JOUR

AU - Wang, Yuanchang

AU - Yong, Jiongmin

TI - A deterministic affine-quadratic optimal control problem

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2014

PB - EDP-Sciences

VL - 20

IS - 3

SP - 633

EP - 661

AB - 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.

LA - eng

KW - affine quadratic optimal control; dynamic programming; Hamilton–Jacobi–Bellman equation; quasi-Riccati equation; state feedback representation; affine-quadratic optimal control problem; Hamilton-Jacobi-Bellman equation

UR - http://eudml.org/doc/272934

ER -

## References

top- [1] S.P. Banks and T. Cimen, Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria. System Control Lett.53 (2004) 327–346. Zbl1157.49313MR2097793
- [2] 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. Zbl1089.49034MR2144384
- [3] H.T. Banks, B.M. Lewis and H.T. Tran, Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach. Comput. Optim. Appl.37 (2007) 177–218. Zbl1117.49032MR2325656
- [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997). Zbl0890.49011MR1484411
- [5] M. Bardi and F. DaLio, On the Bellman equation for some unbounded control problems. Nonlinear Differ. Eqs. Appl.4 (1997) 491–510. Zbl0894.49017MR1485734
- [6] L.M. Benveniste and J.A. Scheinkman, On the differentiability of the value function in dynamic models of economics. Econometrica47 (1979) 727–732. Zbl0435.90031MR533081
- [7] L.D. Berkovitz, Optimal Control Theory. Springer-Verlag, New York (1974). Zbl0295.49001MR372707
- [8] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000). Zbl0966.49001MR1756264
- [9] P. Cannarsa and H. Frankowska, Some characterizatins of optimal trajecotries in control theory. SIAM J. Control Optim.29 (1991) 1322–1347. Zbl0744.49011MR1132185
- [10] T. Cimen, State-dependent Riccati equation (SDRE) control: a survey. Proc. 17th World Congress IFAC (2008) 3761–3775.
- [11] H. Frankowska, Value Function in Optimal Control, Mathematical Control Theory, Part 1, 2 (2001) 516–653. Zbl1098.49501MR1972793
- [12] T. Hildebrandt and L. Graves, Implicit functions and their differentials in general analysis. Trans. Amer. Math. Soc.29 (1927) 127–153. Zbl53.0234.02MR1501380JFM53.0234.02
- [13] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations. Probab. Theory Rel. Fields103 (1995) 273–283. Zbl0831.60065MR1355060
- [14] R.E. Kalman, Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana5 (1960) 102–119. Zbl0112.06303MR127472
- [15] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications. Vol. 1702 of Lect. Notes Math. Springer-Verlag (1999). Zbl0927.60004MR1704232
- [16] H. Qiu and J. Yong, Hamilton-Jacobi equations and two-person zero-sum differential games with unbounded controls. ESAIM: COCV 19 (2013) 404–437. Zbl1263.49024MR3049717
- [17] J.P. Rincón-Zapatero and M.S. Santos, Differentiability of the value function in continuous-time economic models. J. Math. Anal. Appl.394 (2012) 305–323. Zbl1251.91046MR2926223
- [18] J. Yong, Finding adapted solutions of forward-backward stochastic differential equations – method of continuation, Probab. Theory Rel. Fields107 (1997) 537–572. Zbl0883.60053MR1440146
- [19] J. Yong, Stochastic optimal control and forward-backward stochastic differential equations. Comput. Appl. Math.21 (2002) 369–403. Zbl1123.60313MR2009959
- [20] J. Yong, Forward backward stochastic differential equations with mixed initial and terminal conditions. Trans. AMS362 (2010) 1047–1096. Zbl1185.60067MR2551515
- [21] J. Yong and X.Y. Zhou, Stochastic Control: Hamiltonian Systems and HJB Equations. Springer-Verlag (1999). Zbl0943.93002MR1696772
- [22] Y. You, A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems. SIAM J. Control Optim.25 (1987) 905–920. Zbl0632.49004MR893989
- [23] Y. You, Synthesis of time-variant optimal control with nonquadratic criteria. J. Math. Anal. Appl.209 (1997) 662–682. Zbl0872.49015MR1474631
- [24] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems. Springer-Verlag, New York (1986) 150–151. Zbl0583.47050MR816732
- [25] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators. Springer-Verlag, New York (1990). Zbl0684.47029MR1033498

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