Modern research in control is undergoing a paradigm shift from finite-dimensional first-principles models to data-driven non-parametric representations, driven by the need for scalability and adaptability in complex, real-world systems. This transition poses both opportunities and challenges for optimal control theory, algorithm, and software development, particularly in balancing model fidelity with computational tractability. In this talk, I will present a critical assessment of this shift, arguing that operator-theoretic frameworks---rooted in reproducing kernel Hilbert spaces (RKHS), partial differential equation systems, and infinite-dimensional linear systems theory---offer a promising path forward for addressing data-driven optimal control problems. By leveraging linear algebraic structures and convex optimization, these models enable the treatment of nonlinear dynamics through the lens of stochastic diffusion processes, learned directly from data.

In detail, I will discuss recent advances in non-parametric learning of controlled Markov transitions via RKHS embeddings, capturing probabilistic uncertainties in system behavior without explicit parametric assumptions. Moreover, I will discuss recent advances in the field of operator-theoretic dynamic programming for infinite-dimensional systems, which enables the efficient solution of optimal control problems through low-rank regression based operator approximations while preserving theoretical guarantees. The talk will conclude with open challenges in probabilistic forecasting as well as robust and data-driven control.


REFERENCES:
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[1] B. Houska. Convex operator-theoretic methods in stochastic control. Automatica, 2025.
[2] P. Bevanda, et.al. Kernel-based optimal control: an infinitesimal generator approach. In Proceedings of the 7th Annual Learning for Dynamics and Control Conference (L4DC), 2025.