The optimal control problem for stability analysis of general slopes

In general, a slope is a statically indeterminate system of infinite order. In the framework of limit equilibrium methods, where the slip body is treated as a rigid body, non-physical assumptions about the internal forces must be introduced to make the system statically determinate. Different assumptions lead to distinct limit equilibrium methods, all of which can bring the slope into a limit equilibrium state. Although so-called rigorous methods (satisfying all equilibrium conditions) produce relatively small variations in the factor of safety, none of these methods guarantees a statically admissible force system a priori, which leads to inefficiencies in slope stabilization design. By defining the normal stress on the slip surface at the limit equilibrium state and the critical sliding direction vector as control variables, with the factor of safety as the state variable, this study demonstrates that the slope stability analysis can be reduced to an optimal control problem of integral equations. The state equations represent the limit equilibrium equations of the slip body, while the cost functional depends on the specific problem. In this study, the cost functional is defined as the factor of safety of slip body, derived from Pan's maximum principle applied to the stability analysis of the given slip body. By solving this optimal control problem, the critical sliding direction and safety factor for a three-dimensional asymmetric slip body are obtained simultaneously, along with a statically admissible force system. The analysis of several classical examples and a well-known real case demonstrates the accuracy and robustness of the optimal control model proposed in this paper.