Abstract. In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.

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Nowadays, rehabilitation training for stroke survivors is mainly completed under the guidance of the physician. There are various treatment ways, however, most of them are affected by various factors such as experience of physician and training intensity. The treatment effect cannot be fed back in time, and objective evaluation data is lacking. In addition, the treatment method is complicated, costly, and highly dependent on physicians. Moreover, stroke survivors' compliance is poor, which leads to various limitations. This paper combines the Internet-of-Things, machine learning, and intelligence system technologies to design a smartphone-based intelligence system to help stroke survivors to improve upper limb rehabilitation. With the built-in multi-modal sensors of the smart phone, training action data of users can be obtained, and then transfer to the server through the Internet. This research presents a DTW-KNN joint algorithm to recognize accuracy of rehabilitation actions and classify to multiple training completion levels. The experimental results show that the DTW-KNN algorithm can evaluate the rehabilitation actions, the accuracy rates of the classification in excellent, good, and normal are 85.7%, 66.7%, and 80% respectively. The intelligence system presented in this paper can help stroke survivors to proceed rehabilitation training independently and remotely, which reduces medical costs and psychological burden.

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