Constructing Control Lyapunov-Value Functions Using Hamilton-Jacobi Reachability Analysis

In this letter, we seek to build connections between control Lyapunov functions (CLFs) and Hamilton-Jacobi (HJ) reachability analysis. CLFs have been used extensively in the control community for synthesizing stabilizing feedback controllers. However, there is no systematic way to construct CLFs for general nonlinear systems and the problem can become more complex with input constraints. HJ reachability is a formal method that can be used to guarantee safety or reachability for general nonlinear systems with input constraints. The main drawback is the well-known “curse of dimensionality.” In this letter we modify HJ reachability to construct what we call a control Lyapunov-Value Function (CLVF) which can be used to find and stabilize to the smallest control invariant set <inline-formula> <tex-math notation="LaTeX">$(\mathcal I_{m}^{\infty })$ </tex-math></inline-formula> around a point of interest. We prove that the CLVF is the viscosity solution to a modified HJ variational inequality (VI), and can be computed numerically, during which the input constraints and exponential decay rate <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula> are incorporated. This process identifies the region of exponential stability to <inline-formula> <tex-math notation="LaTeX">$\mathcal I_{m}^{\infty }$ </tex-math></inline-formula> given the desired input bounds and <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>. Finally, a feasibility-guaranteed quadratic program (QP) is proposed for online implementation.