Kernel-Based Optimal Control: An Infinitesimal Generator Approach (2412.01591v3)

Abstract: This paper presents a novel operator-theoretic approach for optimal control of nonlinear stochastic systems within reproducing kernel Hilbert spaces. Our learning framework leverages data samples of system dynamics and stage cost functions, with only control penalties and constraints provided. The proposed method directly learns the infinitesimal generator of a controlled stochastic diffusion in an infinite-dimensional hypothesis space. We demonstrate that our approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-BeLLMan recursions, enabling a data-driven solution to the optimal control problems. Furthermore, our learning framework includes nonparametric estimators for uncontrolled infinitesimal generators as a special case. Numerical experiments, ranging from synthetic differential equations to simulated robotic systems, showcase the advantages of our approach compared to both modern data-driven and classical nonlinear programming methods for optimal control.

• The paper introduces a novel data-driven approach integrating RKHS theory with diffusion operators to solve stochastic control problems.
• It develops a continuous-time kernel HJB recursion for deriving globally optimal solutions to nonlinear stochastic systems.
• Numerical experiments on systems like the Van der Pol oscillator and inverted pendulum demonstrate improved robustness and efficiency.

An Expert Analysis of "Kernel-Based Optimal Control: An Infinitesimal Generator Approach"

The paper introduces a significant advancement in the domain of optimal control for nonlinear stochastic systems, deploying a novel method grounded in the principles of infinitesimal generator learning within infinite-dimensional reproducing kernel Hilbert spaces (RKHS). This methodology is particularly relevant for systems where traditional control approaches, which typically rely on first-principle models, encounter substantial difficulties. The primary contribution of this work lies in its integration of the diffusion operator for the controlled Fokker-Planck-Kolmogorov (FPK) equation within an RKHS framework, offering a data-driven solution to optimal control problems.

Key Contributions and Theoretical Insights

The paper provides a comprehensive paper of two main areas: system identification and stochastic optimal control. The authors present a method to derive non-parametric estimators for infinitesimal generators related to optimally controlled diffusion processes. This method utilizes the properties of RKHS to enable learning operators that govern the strongly parabolic FPK equation—a significant theoretical innovation enabling seamless integration with Hamilton-Jacobi-BeLLMan (HJB) frameworks.

A pivotal proposition of the paper is the formulation of a continuous-time Kernel HJB recursion for managing data-driven estimates from FPK operators, thereby facilitating the calculation of globally optimal solutions to stochastic control challenges. The analysis extends the state of the art by capturing the intricacies of stochastic differential equations through this new lens, particularly in how these can be leveraged for more efficient control policy derivations.

Numerical Experiments and Comparative Analysis

The authors conduct numerical experiments demonstrating the utility of their approach over both synthetic and realistic robotic systems. Key test cases include the Van der Pol oscillator and robotic control systems such as the inverted pendulum and cartpole—scenarios that are illustrative of the method’s practical superiority. The numerical results underscore the robustness and efficiency of the approach, highlighting its advantages in comparison to established nonlinear programming ap

Implications and Future Directions

This research holds substantial implications for both theoretical exploration and practical application in AI-driven control systems. Practically, the proposed framework introduces new possibilities for designing controllers that are less reliant on accurate system models, thus extending their applicability to complex and uncertain environments. Theoretically, the paper paves the path for further studies into the use of infinite-dimensional representations for control tasks, suggesting a fertile ground for future research, particularly in enhancing model scalability and integrating with other machine learning techniques.

The transition from parametric to nonparametric models in control systems, as enacted in this paper, exemplifies a critical shift that may redefine system identification and control algorithms. Future investigations may focus on optimizing computational efficiency and extending these methodologies to broader classes of dynamic systems. Further refinement in kernel selection and adaptation mechanisms can also contribute to improved performance metrics and extend the adaptability of these control systems in increasingly diverse operational contexts.

In conclusion, this paper sets a foundational milestone in optimal control research, challenging traditional paradigms and offering a promising direction towards more adaptive and resilient control processes in dynamic and uncertain environments.