Formulate and solve continuous-time and discrete-time optimal control problems for high-DoF robotic manipulators, bipedal/humanoid locomotion, and autonomous guidance systems.
Synthesize formal stability and safety certificates using Control Lyapunov Functions (CLF), Control Barrier Functions (CBF), and Robust H∞ / μ-synthesis for safety-critical dynamical systems under bounded disturbances.
Design multi-step Bellman backup verifiers, formal process reward models (PRM), and contraction-mapping policy gradient pipelines for frontier reasoning models and sequential decision agents.
Build massively parallel differentiable rigid-body and contact-rich simulation kernels enabling 100,000+ simultaneous closed-loop control rollouts per second on GPU clusters.
Develop stochastic control architectures, Itô-diffusion HJB-Fokker-Planck solvers, Risk-Sensitive LQG controllers, and decentralized multi-agent cybernetic game equilibria.
Dedicated remote research track for top MSc/PhD candidates, recent graduates, and postdoctoral researchers from 139 indexed universities worldwide specializing in control theory, cybernetics, dynamical systems, and applied mathematics.
Exact and approximate dynamic programming, value/policy iteration, contraction mappings in Banach spaces, neuro-dynamic programming, and multi-horizon sequential decision processes.
Viscosity solutions to nonlinear first- and second-order HJB PDEs, covector adjoint equations, minimum-time/minimum-fuel optimal control, and symplectic boundary-value solvers.
Real-time receding-horizon nonlinear MPC, Differential Dynamic Programming (DDP), iterative LQR (iLQR), sequential quadratic programming (SQP), and contact-implicit trajectory optimization.
Closed-loop feedback regulation, Wiener-Bellman cybernetic feedback architectures, Control Lyapunov Functions (CLF), Control Barrier Functions (CBF-QP), and robust H∞ disturbance rejection.
Bridging classical control theory with modern foundation model post-training: verifiable multi-step reasoning rewards, differentiable physics engines (CUDA/Triton/MJX), and sim-to-real policy distillation.
| Formulation | Canonical Equation / Operator | Physical & Algorithmic Domain |
|---|---|---|
| Discrete Bellman Optimality Bellman (1957) |
V*(s) = max_{a ∈ A(s)} [ R(s,a) + γ ∑_{s′} P(s′|s,a) V*(s′) ] | Markov Decision Processes (MDPs), exact dynamic programming, contraction mapping fixed-point iteration. |
| Hamilton-Jacobi-Bellman PDE Continuous-Time Optimal Control |
&partial;V(x,t)/&partial;t + min_{u ∈ U} { ⟨∇_x V(x,t), f(x,u,t)⟩ + L(x,u,t) } = 0 | Continuous-time nonlinear trajectory optimization, aerospace guidance, viscosity solutions on manifolds. |
| Pontryagin Maximum Principle Adjoint Covector Dynamics |
ẋ = ∇_p H(x,p,u*), ṗ = −∇_x H(x,p,u*), u* = argmin_u H | Two-point boundary value problems (TPBVP), indirect trajectory optimization, minimum-time/fuel control. |
| Algebraic Riccati Equation (CARE) Kalman-Bellman LQR Dual |
A^⊤ P + P A − P B R^{-1} B^⊤ P + Q = 0, K = R^{-1} B^⊤ P | Infinite-horizon Linear Quadratic Regulation (LQR), iLQR backward Riccati recursion, Kalman filtering dual. |
| CLF-CBF Safety-Critical QP Nonlinear Cybernetic Invariance |
L_f V(x) + L_g V(x)u ≤ −c V(x), L_f h(x) + L_g h(x)u + α(h(x)) ≥ 0 | Real-time quadratic programming (QP) safety filters for bipedal humanoids, surgical robotics, and autonomous vehicles. |