Geodesic
A deterministic affine-quadratic optimal control problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 633-661

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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.

DOI : 10.1051/cocv/2013078
Classification : 49J15, 49K15, 49L20, 49N10
Keywords: affine quadratic optimal control, dynamic programming, Hamilton-Jacobi-Bellman equation, quasi-Riccati equation, state feedback representation 
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     author = {Wang, Yuanchang and Yong, Jiongmin},
     title = {A deterministic affine-quadratic optimal control problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {633--661},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {3},
     year = {2014},
     doi = {10.1051/cocv/2013078},
     mrnumber = {3270127},
     zbl = {1293.49004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013078/}
}
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Wang, Yuanchang; Yong, Jiongmin. A deterministic affine-quadratic optimal control problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 633-661. doi: 10.1051/cocv/2013078

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