Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit‐time optimal control Part I: Theory

Abstract

This work studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton-Jacobi-Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite-dimensional nonlinear programming problems. We propose to use exit-time optimal control for that purpose. This article is the first part of a two-part work. First, we state an exit-time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null-controllability. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well.Our developments serve as a theoretical basis for a curse-of-dimensionality-free approach to feedback stabilization, that is presented in the second part of this work together with supporting numerical simulation results.

Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit‐time optimal control Part II: Numerical approach

“…are closed ellipsoidal domains in R n . SinceV(⋅) is a local CLF for (1), there exists a level c ′ > 0 satisfying…”

“…The aim of the first example in this section is to numerically test the theoretical results of the previous study 1 with regard to a simple nonlinear control system that has two state coordinates and can be handled analytically to some extent. The second example exploits the developments of Sections 2 and 3 to design, train, and simulate a stabilizing MPC controller for a nonlinear system with six-dimensional state space.…”

“…The previous study 1 has developed theoretical results regarding the use of exit-time optimal control to characterize control Lyapunov functions (CLFs) for deterministic nonlinear systems described by ordinary differential equations. As noted in conclusion of Reference 1, such characterizations can help to attenuate the curse of dimensionality, but the curse of complexity remains a significant issue for obtaining global or semi-global approximations of CLFs and associated feedback strategies in high-dimensional regions of the state space.…”

“…Its purpose is to practically test the theoretical results of the first part 1 as applied to a simple nonlinear control system with two-dimensional state space. This system can be treated analytically to some extent, so that the exact and numerical solutions can be easily compared with each other.…”

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

“…Such problems are called optimal impulse control problems. In previous works (Bressan et al, 1991;1994;Dykhta, 1995;Yegorov, 2004), the optimal impulse control problems for lumped parameter systems have been considered. These problems have been considered for distributed parameter systems, when the impulse effect occurs only in one point (Yegorov, 1978;Mamedov, 2006).…”