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We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

This paper focuses on stability of stochastic hybrid systems with random delay driven by a singularly perturbed Markov chain. By weak convergence and the martingale method, the limit system is obtained. Using the limit system as a bridge, this paper establishes the moment exponential stability of the original system by Razumikhin-type techniques. Finally, an example is given to illustrate this result. [ABSTRACT FROM AUTHOR]