Formally Verified Neural Control Lyapunov Functions for Stabilizing Nonlinear Systems
Core Concepts
A principled approach to learning and formally verifying neural networkbased control Lyapunov functions that can significantly outperform traditional methods in estimating the nullcontrollability set for nonlinear systems.
Abstract
The paper investigates the problem of constructing control Lyapunov functions (CLFs) for stabilizing nonlinear dynamical systems. The key insights are:

The authors propose a physicsinformed learning approach to solve the transformed HamiltonJacobiBellman (ZubovHJB) equation using neural networks. This allows them to compute nearmaximal estimates of the nullcontrollability set, which characterizes the region of initial conditions that can be asymptotically stabilized to the origin.

The learned neural network CLFs are then formally verified using satisfiability modulo theories (SMT) solvers. This provides rigorous guarantees on the stability and nullcontrollability properties of the closedloop system.

Numerical examples demonstrate that the neural network CLFs computed using the proposed approach significantly outperform traditional methods, such as sumofsquares and rational CLFs, in estimating the nullcontrollability set.

The authors also show that the neural network CLFs can be used to derive nearoptimal stabilizing controllers with formal guarantees of stability.
The paper presents a principled framework that combines physicsinformed learning and formal verification to tackle the challenging problem of constructing CLFs for nonlinear systems. The results highlight the benefits of this approach in terms of obtaining more accurate estimates of the nullcontrollability set and deriving nearoptimal stabilizing controllers.
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Formally Verified PhysicsInformed Neural Control Lyapunov Functions
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The paper does not provide any specific numerical data or metrics to support the key claims. However, it presents several numerical examples that illustrate the effectiveness of the proposed approach.
Quotes
"By approximately solving the optimal value functions, our approach also allows for the derivation of nearoptimal controllers with formal guarantees of stability."
"Clearly, the PDE loss significantly enhances extrapolation on larger domains beyond where the data were taken."
Deeper Inquiries
How can the proposed framework be extended to handle more complex nonlinear systems, such as those with bounded control inputs or state constraints?
The proposed framework for computing and verifying neural networkbased Control Lyapunov Functions (CLFs) can be extended to handle more complex nonlinear systems by incorporating techniques that address bounded control inputs and state constraints. One approach is to modify the Pontryagin’s Maximum Principle (PMP) to account for constraints on the control inputs. This can be achieved by reformulating the optimization problem to include constraints directly in the Hamiltonian, ensuring that the optimal control law respects the bounds on the control inputs.
Additionally, state constraints can be integrated into the framework by employing barrier functions or penalty methods that penalize states that approach the boundaries of the feasible region. This can be done by augmenting the loss function used in the physicsinformed neural network (PINN) training process to include terms that enforce these constraints. For instance, one could introduce a term that penalizes the neural network output when the state approaches the constraint boundaries, thereby guiding the learning process to respect these constraints.
Moreover, compositional verification techniques can be utilized to break down complex systems into simpler subsystems, each of which can be analyzed and verified separately. This modular approach allows for the handling of more intricate dynamics while maintaining the formal verification guarantees provided by the existing framework. By leveraging these strategies, the framework can be adapted to effectively manage a broader class of nonlinear systems with various constraints.
What are the theoretical guarantees on the convergence and optimality of the neural network solutions to the ZubovHJB equation?
The theoretical guarantees on the convergence and optimality of neural network solutions to the ZubovHamiltonJacobiBellman (HJB) equation are an area of ongoing research. While the paper presents a heuristic approach to solving the ZubovHJB equation using physicsinformed neural networks (PINNs), establishing rigorous theoretical guarantees remains a challenge.
In general, the convergence of neural network solutions to PDEs, including the ZubovHJB equation, can be supported by results from approximation theory, which suggest that sufficiently expressive neural networks can approximate any continuous function to arbitrary precision. This implies that, under appropriate conditions on the architecture and training process, the neural network can converge to the true solution of the ZubovHJB equation.
However, the optimality of the neural network solutions is more complex. The paper indicates that the neural network solution ( W_N ) can approximate the true solution ( W ) of the ZubovHJB equation, which characterizes the nullcontrollability set. If ( W ) is positive definite and satisfies the conditions derived from the ZubovHJB equation, then the closedloop system can be shown to be asymptotically stable. Thus, while the neural network can provide nearoptimal solutions, formal guarantees on optimality would require further theoretical development, including the exploration of the conditions under which the neural network approximations maintain the properties of the true solutions.
Can the formal verification techniques be further improved or combined with other methods to handle a wider range of nonlinear dynamics and control problems?
Yes, the formal verification techniques presented in the paper can be further improved and combined with other methods to handle a wider range of nonlinear dynamics and control problems. One potential improvement is the integration of more advanced satisfiability modulo theories (SMT) solvers that can handle a broader class of nonlinear functions and constraints. By leveraging solvers that support nonpolynomial dynamics, the verification process can be extended to more complex systems.
Additionally, combining formal verification with machine learning techniques, such as reinforcement learning or model predictive control (MPC), could enhance the robustness and adaptability of the control strategies. For instance, reinforcement learning can be used to explore the state space and identify regions where the neural CLFs are valid, while formal verification can ensure that the learned policies adhere to safety and stability constraints.
Another avenue for improvement is the use of compositional verification techniques, which allow for the modular analysis of complex systems. By breaking down a system into smaller, manageable components, each can be verified independently, and the results can be combined to ensure the overall system's safety and performance.
Finally, incorporating uncertainty quantification methods into the verification process can help address the challenges posed by external disturbances and model inaccuracies. By analyzing the robustness of the neural CLFs under various perturbations, the framework can be made more resilient to realworld conditions, thereby expanding its applicability to a wider range of nonlinear dynamics and control problems.