Geodesic
Comparison principle for equations of the Hamilton--Jacobi type in control theory
Trudy Instituta matematiki i mehaniki, Dynamical systems: modeling, optimization, and control, Tome 12 (2006) no. 1, pp. 173-183

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This paper deals with the comparison principle for the first-order ODEs of the Hamilton–Jacobi–Bellman and Hamilton–Jacobi–Bellman–Isaacs type which describe solutions to the problems of reachability and control synthesis under complete as well as under limited information on the system disturbances. Since the exact solutions require fairly complicated calculation, this paper presents the upper and lower bounds to these solutions, which in some cases may suffice for solving such problems as the investigation of safety zones in motion planning, verification of control strategies or of conditions for the nonintersection of reachability tubes, etc. For systems with original linear structure it is indicated that present among the suggested estimates are those of ellipsoidal type, which ensure tight approximations of the convex reachability sets as well as of the solvability sets for the problem of control synthesis. 
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     title = {Comparison principle for equations of the {Hamilton--Jacobi} type in control theory},
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A. B. Kurzhanskii. Comparison principle for equations of the Hamilton--Jacobi type in control theory. Trudy Instituta matematiki i mehaniki, Dynamical systems: modeling, optimization, and control, Tome 12 (2006) no. 1, pp. 173-183. http://geodesic.mathdoc.fr/item/TIMM_2006_12_1_a12/