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Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. (English) Zbl 0796.49024

The value function of Mayer’s problem arising in optimal control is investigated. Lower semicontinuous solutions of the associated Hamilton- Jacobi-Bellman equation (HJB)

$-\frac{\partial V}{\partial t}\left(t,x\right)+H\left(t,x,-\frac{\partial V}{\partial t}\left(t,x\right)\right)=0,\phantom{\rule{1.em}{0ex}}V\left(T,·\right)=g\left(·\right)\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\text{Dom}\left(V\right)·$

are defined in three equivalent ways including the one given by Barron and Jensen. The value function is proved to be the unique solution under quite weak assumptions about the control system. Such a solution is stable with respect to the perturbations of the control system and the cost, and it coincides with the viscosity solution whenever it is continuous.

The outline is as follows: In §2 the main results is stated, including five equivalent statements about the value function and the condition under which $V$ is a viscosity solution of the HJB equation. §3 is devoted to the monotone behavior of contingent solutions and §4 to solutions formulated in terms of subdifferentials. §5 is dedicated to the discussion of the definition of solution proposed by Barron and Jensen, using alternative boundary conditions. A comparison with continuous viscosity solutions is provided in §6. In §7 the author associate with a HJB equation an optimal control problem whose value function is the only solution of the HJB equation. This leads to both existence and uniqueness of the results. Finally, in §8 the stability of the value function is investigated and this is to help to the stability study of solution of the HJB equation under perturbation of $H$ and $g$.

##### MSC:
 49L20 Dynamic programming method (infinite-dimensional problems) 49L25 Viscosity solutions (infinite-dimensional problems) 35B37 PDE in connection with control problems (MSC2000) 49J15 Optimal control problems with ODE (existence) 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 49K40 Sensitivity, stability, well-posedness of optimal solutions