A unified optimal control framework: time-optimal control and stochastic optimal control
Shuzhen Yang
TL;DR
The paper develops a unified stochastic optimal control framework that couples time-optimal control with classical stochastic control by enforcing a minimum-time constraint $\tau^{u}=\inf\{t:\mathbb{E}[Y^{u}(t)]\le 0\}\wedge T$ on a two-state system $(X^{u},Y^{u})$, rendering the terminal time endogenous. An extended stochastic maximum principle is derived, introducing adjoint processes and a Hamiltonian that balances time-to-target with running costs, and a bang-bang structure is shown for the linear-time case. The framework is demonstrated through a portfolio optimization example that jointly minimizes a running-cost integral and achieves a statistically defined target, illustrating practical integration of time and cost objectives. Overall, the approach provides a versatile tool for decision-making in finance, autonomous systems, and supply chains, enabling strategies that reach targets as early as possible while controlling costs.
Abstract
In this paper, we propose a unified stochastic optimal control framework that integrates time-optimal control problems with classical stochastic optimal control formulations. Unlike conventional deterministic time-optimal control models, our approach incorporates a generalized stochastic control structure subject to minimum-time constraints. In this setting, the minimum-time condition is defined as the earliest achievable moment-in expectation-for reaching a target state, making the terminal time an endogenous and control-dependent variable. The main contributions of this work are twofold: first, we derive an extended stochastic maximum principle for the proposed model; second, we establish a bang-bang type optimal control for the linear time-optimal control problem. This unified stochastic optimal control framework facilitates the design of optimal strategies across diverse fields-such as finance, autonomous systems, and supply chain management-by enabling simultaneous minimization of operational costs and achievement of statistically-defined targets at the earliest feasible time. As an application, we solve a financial portfolio optimization problem within the proposed framework.