Abstract. The aim of this paper is to investigate the large deviations for a class of slow-fast mean-field diffusions, where the slow process depends on not only the fast process but also their distributions. Due to the perturbations of the fast process and its time marginal law, one cannot prove the large deviations based on verifying the powerful weak convergence criterion directly. To overcome this problem, we employ the functional occupation measure, which, combined with the notion of viable pair and the controls of the feedback form, characterize the limits of controlled sequences and justify the upper and lower bounds of the Laplace principle. As a consequence, the explicit representation formula of the rate function for large deviations is also presented.
Abstract. Controlling a damped oscillator is crucial in various technological and scientific fields, such as structural engineering, aerospace, and noise reduction device design. This article considers a classical underdamped harmonic oscillator, focusing on its minimal-time control by modulating its time-dependent frequency. The goal is to connect in minimal time two states with zero kinetic energy but different displacements. We provide a comprehensive theoretical analysis of this problem, characterizing the set of reachable states and detailing the structure of the optimal trajectories, which we precisely describe in phase space. These trajectories correspond to bang-bang controls, and the number of commutations depends on the ratio between the final and initial states. Our approach is based on combining the Pontryagin principle with a suitable choice of “energy-angle” variables and a Bellman-like optimality principle.
Abstract. This paper studies singular perturbations of discrete-time Markov chains in general state spaces, which may include the transient set. By leveraging the idea of state aggregation, we derive a Taylor series expansion for the invariant probability measure of the singularly perturbed Markov chain. We then apply this expansion to analyze the expected long-run average cost (EAC) for singularly perturbed Markov decision processes (MDPs). For this EAC problem, we establish the existence of an optimal pure stationary policy and construct an explicit asymptotically optimal policy. A contribution of our work lies in the application of strong stability conditions (see [N. V. Kartashov, J. Soviet Math., 34 (1986), pp. 1493–1498]), allowing us to relax the widely used Doeblin conditions (see [S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd ed., Cambridge University Press, Cambridge, UK, 2009]) and thereby extend the theory’s applicability to a broader class of practical systems. The efficacy of our results is demonstrated through a maintenance system as an example.
Abstract. With Ekeland’s variational principle, we prove a general stochastic maximum principle (SMP) for square-integrable optimal control of linearly growing stochastic differential systems subject to a quadratically growing cost functional. The diffusion coefficient is allowed to depend on the control variable, and the admissible control range is allowed to be nonconvex. We relax the existing assumptions of finite moments of arbitrary order on optimal control (see [S. G. Peng, SIAM J. Control Optim., 28 (1990), pp. 966–979]), and of control-bounded coefficients (see [J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Appl. Math. (N. Y.) 43, Springer, 1999, pp. 114 and 118]), so that the typical linear quadratic optimal stochastic control problem is included to satisfy all the assumptions of our SMP.
Abstract. We investigate the homogenization of boundary optimal control problems governed by an elliptic partial differential equation (PDE) in pillar-type domains with highly oscillating boundaries. The pillars are composed of high-conductivity and low-conductivity materials, leading to a PDE with high-contrast diffusive coefficients. Two boundary control problems are considered. More precisely, in the first problem, control is applied to the boundary of the high-conductivity region whereas in the 2nd one, it is on the boundary of the low-conductivity region. Due to rapid oscillations in the boundary, the control is appropriately scaled. Depending on the choice of scaling and the controlled boundary, three different limit problems arise: a combination of interior, boundary, and interface control problems; a boundary and interface control problem; and an interior and interface control problem.
