"Backward reachable tubes (BRTs), computed via viscous Hamilton-Jacobi (HJ) partial differential equations, provide principled safety certificates for learned controllers and planning algorithms in trustworthy machine learning. However, classical grid-based HJ solvers require $O(M^n)$ memory footprint for $M$ grid points per $n$ state dimension. This renders them impractical for high-dimensional systems. We address this bottleneck with a local PDE linearization that enables a frozen-coefficient sampling scheme for the viscous HJ PDE: a generalized Cole-Hopf-type transformation reduces the nonlinear HJ equation to a sequence of linear heat equations whose solutions admit Gaussian heat-kernel representations. The value function and its spatial gradient are then recovered via roll-outs of Monte Carlo expectations on Gaussian densities, yielding a storage and grid-free algorithm that scales as $N\cdot n$ for $N$ samples. This decoupling of memory from dimensionality enables reachability analysis on problems where grid-based methods are simply impossible. We prove a finite-sample concentration bound $O(N^{-1/2}$ error and conditional linear convergence for the introduced Monte-Carlo Picard iterative scheme. Numerical validation on pursuit-evasion games demonstrates relative $L^2_{\text{rel}}$ errors of $0.03 - 0.20$, with $14-26$ second wall-clock times per 2D slice on a CPU. Crucially, the method scales with validation on up to (but not limited to) $n=45$-dimensional multi-agent games."

Finite-dimensional (FD) diffusion policies are well-known to degrade in performance in long-horizon control prediction settings owing to temporal drift of the backward action trajectory and the spectral artifacts associated with the finite grid discretization from which actions are sampled. This hinders their reliable deployment when policies must satisfy all allocated and functional baselines with certifiable robustness guarantees. To address this deficiency, we lift diffusion to an infinite-dimensional subset of a Hilbert Space and transform the Monte-Carlo action sampling generation into a deterministic PDE regression. This reparameterization (i) attenuates the instability associated with Monte Carlo rollouts in FD settings; whilst (ii) enabling reconfigurability of the action space in spectral form to support the entire data manifold; and (iii) provides dimension-independent convergence guarantees. Specifically, we lift the backward Kolmogorov score function to a Cameron-Martin space, introducing a physics-grounded Kolmogorov residual as a diagnostic during learning. Our schemes are validated on contact-rich manipulation (\texttt{PushT}) and manufacturing constant work-in-processes (\texttt{CONWIP}) flow forecasting. Across action (reverse) denoising steps on \texttt{PushT}, we notice an inter-step drift convergence accuracy of 98% on our infinite-dimensional formulation versus 85% on the FD visuomotor policy tasks. In addition, we achieved a 28.4% improvement in forecasting accuracy over LSTM baselines and near-perfect recall in bottleneck event detection for manufacturing \texttt{CONWIP}. To further imbue robustness into diffusion workflows, we integrate our approach with Hamilton–Jacobi reachability safety analysis, so that our approach yields certified safe policies that reduce deadlocks by 96% in stochastic factory automation discrete event systems.

This paper revisits the classic generalized Nash-equilibrium-seeking games over networks and casts the local objectives of players as a set-valued, rather than point-valued, objective to be optimized. We concretely postulate the learning algorithm for real-time decision-making in environments where decisions are not clear-cut. Our formulation is backed by empirical evidence that demonstrate the feasibility of our ideas on real-world behavior systems. These algorithms may become consequential as a means for rationalizing agentic behavior in the future.

Diffusion policies are very consequential in learning the visuomotor policies of many autonomous systems. Despite the underlying diffusion process' existence in an infinite-dimensional (ID) function space, practical autonomous systems implementations discretize diffused control trajectories within finite-dimensional (FD) vector spaces. We set diffusion visuomotor policies in the natural ID fabric of the problems they attempt to solve, approximating controls ideally with all potential solutions within the function space, including their relative probabilities, while precisely deferring discretization as long as possible. We show that the Backward Kolmogorov Equation (BKE) in a Hilbert space turns stochastic diffusion score regression into deterministic PDE regression with a Cameron-Martin (CM) loss that achieves a dimension-independent total-variation bound on the underlying probability distribution.

We develop a theoretical framework for multi-agent proximal policy optimization that simultaneously provides convergence guarantees, robustness certificates, constraint satisfaction for safety, and convergence to generalized Nash equilibria in mixed cooperative-competitive settings.

We present a universal collision-free planner for open-embodiment robots that combines compact implicit scene representations with memory-efficient incremental occupancy mapping. The method generalizes across robot morphologies without retraining and achieves real-time planning in cluttered, partially observed environments.

We investigate whether value function approximation via Bellman residual minimization is sufficient for stable and optimal control. Examining failure modes in continuous-time systems, we identify conditions under which the Bellman equation is necessary but not sufficient, and propose an augmented PSD-constrained network architecture that recovers stability guarantees.

We formulate trauma resuscitation as a multi-agent control problem and propose a generalized Nash equilibrium-seeking policy that coordinates multiple clinical agents under resource constraints, improving patient outcomes while satisfying safety and fairness requirements.

Supplementary material accompanying LevelSetPy, providing extended benchmark evaluations, numerical examples, and performance comparisons against MATLAB toolbox baselines on Hamilton-Jacobi reachability problems in up to six dimensions.

We present a singularly-perturbed backstepping controller for whole-body strain regulation in multi-section continuum robots modeled by discrete Cosserat rods. The composite fast-slow design decouples the bending and shear strain dynamics, achieving convergence guarantees and superior transient performance compared to monolithic control laws.

LevelSetPy is a Python package for numerically solving hyperbolic Hamilton-Jacobi PDEs that arise in reachability analysis and level set methods. Leveraging JAX-based GPU acceleration, it achieves order-of-magnitude speedups over the standard MATLAB toolbox while providing a modern, differentiable programming interface for robotics and control researchers.

This paper introduces LevelSetPy, a Python-native reimplementation of the Hamilton-Jacobi reachability toolbox with GPU support via JAX. We demonstrate its correctness and efficiency on canonical pursuit-evasion games and high-dimensional reachability problems relevant to autonomous systems.

We analyze the structural (controllability, observability, and passivity) properties of multi-section soft robots modeled by discrete Cosserat rod theory. Exploiting these properties, we design an energy-based controller with provable stability guarantees and validate it in simulation on a three-section planar manipulator.

We propose a counterexample-guided training procedure for neural network barrier functions that incorporates a formal verifier in the training loop. The algorithm terminates with a certificate of correctness and significantly reduces the barrier function training time compared to purely adversarial approaches.

PCLAST learns a continuous latent state space from high-dimensional observations in which planning with simple interpolation is provably correct. By optimizing for topological planability rather than reconstruction fidelity, the learned representations support reliable goal-conditioned planning in complex visual environments without environment-specific reward engineering.

We formulate and solve a continuous-time mixed H2/H-infinity reinforcement learning problem for output-feedback control synthesis without a known system model. The algorithm iterates policy evaluation and improvement steps on data collected from the system, converging to an optimal robust policy under disturbance attenuation constraints.

We study robust policy optimization in continuous-time stochastic systems under the mixed H2/H-infinity performance criterion. Combining stochastic differential equation theory with modern policy gradient methods, we derive convergence rates and demonstrate improved robustness to model uncertainty in uncertain dynamical systems arising in robotics and medical automation.

We prove that multi-step inverse dynamics models recover controllable latent state representations with provable guarantees under a rich-observation MDP assumption. The theoretical analysis resolves an open question on when self-supervised representation learning methods succeed for downstream control tasks.

We extend interaction-grounded learning (IGL) to settings where actions affect the feedback signal itself. By incorporating action-inclusive feedback into the IGL framework, we derive provably efficient algorithms for learning reward-free representations and policies in partially observable environments, with applications to healthcare decision support.

We present the mechanical design, kinematic modeling, and experimental validation of a soft pneumatic robot that immobilizes a patient's head during frameless MRI-guided radiation therapy. The mechanism compensates for involuntary motion in real time using a vision-based feedback loop, reducing positioning error to sub-millimeter accuracy.

We present a convolutional neural network that predicts near-optimal beam orientations for IMRT prostate cancer treatment plans in under one second, matching the quality of column-generation-based optimization while reducing planning time by two orders of magnitude.

This dissertation develops a multi-degree-of-freedom soft robotic system for real-time patient positioning and beam orientation selection in intensity-modulated radiation therapy (IMRT). Contributions include continuum robot modeling via Cosserat rod theory, neuro-adaptive control for 6-DOF pose correction, deep reinforcement learning for beam angle optimization, and clinical validation on a CyberKnife platform.

We frame beam orientation optimization (BOO) in radiation therapy as a sequential decision problem and train a deep neural network policy using imitation learning from a column-generation solver. The learned policy selects coplanar and non-coplanar beam configurations that rival expert-optimized plans while running in real time.

We present a minimax iterative dynamic game algorithm for robust nonlinear robot control under adversarial perturbations. The method frames the control problem as a two-player zero-sum game and computes feedback policies via saddle-point iterations on the Hamilton-Jacobi-Isaacs equation, achieving superior robustness on a 6-DOF arm tracking task compared to standard optimal control baselines.

Soft-NeuroAdapt is a three-degree-of-freedom soft actuator system coupled with a neuro-adaptive controller that corrects patient head pose in real time for maskless frameless cancer radiotherapy. The neural adaptive law compensates for actuator and tissue nonlinearities, achieving sub-millimeter positioning accuracy without requiring a rigid fixation mask.

We present a vision-based closed-loop control architecture for a pneumatic soft robot that positions a patient's head during maskless cancer radiotherapy. A model-free adaptive controller uses real-time pose feedback from a stereo camera to drive positioning error below the clinical tolerance of 1 mm.

We describe the first real-time soft-robotic patient positioning system for maskless head-and-neck cancer radiotherapy, demonstrating a proof-of-concept platform that uses pneumatic actuators and image-based feedback to maintain sub-millimeter positioning accuracy during treatment fractions.

We develop a deep reinforcement learning algorithm for zero-sum two-player games that is robust to function approximation errors and sparse rewards. The approach uses adversarial training and minimax Q-learning, achieving stronger worst-case guarantees than standard DRL baselines on robotic manipulation and locomotion tasks.

We identify nonlinear dynamical systems using deep recurrent neural networks trained end-to-end on input-output data. Applied to soft-robot dynamics identification, the approach outperforms classical system identification methods and provides a differentiable model suitable for model-based control.