arXiv subjects

Sean Kragelund

Publications and source records attributed to Sean Kragelund.

5 recordsLinked to original sources

arXiv2026

Mixed Bernstein-Fourier Approximants for Optimal Trajectory Generation with Periodic Behavior

Efficient trajectory generation is crucial for autonomous systems; however, current numerical methods often struggle to handle periodic behaviors effectively, particularly when the onboard sensors require equidistant temporal sampling. This paper introduces a novel mixed Bernstein-Fourier approximation framework tailored explicitly for optimal motion planning. Our proposed methodology leverages the uniform convergence properties of Bernstein polynomials for nonperiodic behaviors while effectively capturing periodic dynamics through the Fourier series. Theoretical results are established, including uniform convergence proofs for approximations of functions, derivatives, and integrals, as well as detailed error bound analyses. We further introduce a regulated least squares approach for determining approximation coefficients, enhancing numerical stability and practical applicability. Within an optimal control context, we establish the feasibility and consistency of approximated solutions to their continuous counterparts. We also extend the covector mapping theorem, providing theoretical guarantees for approximating dual variables crucial in verifying the necessary optimality conditions from Pontryagin's Maximum Principle. Numerical examples illustrate the method's superior performance, demonstrating substantial improvements in computational efficiency and precision in scenarios with complex periodic constraints and dynamics. Our mixed Bernstein-Fourier methodology thus presents a robust, theoretically grounded, and computationally efficient approach for advanced optimal trajectory planning in autonomous systems.

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arXiv2026

Saddle Point Evasion via Curvature-Regularized Gradient Dynamics

Nonconvex optimization underlies many modern machine learning and control tasks, where saddle points pose the dominant obstacle to reliable convergence in high-dimensional settings. Escaping these saddle points deterministically using continuous-time optimization remains an open challenge: gradient descent is blind to curvature, stochastic perturbation methods lack deterministic guarantees, and Newton-type approaches suffer from Hessian singularity. Adopting the perspective of viewing optimization algorithms as dynamical systems, we present Curvature-Regularized Gradient Dynamics (CRGD), which augments the objective with a smooth penalty on the negative Hessian eigenvalues, yielding an augmented cost that serves as an optimization Lyapunov function with user-selectable convergence rates to second-order stationary points. Numerical experiments confirm that CRGD converges to second-order stationary points, even in regimes where gradient descent fails.

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arXiv2026

Prescribed-Time Distributed Generalized Nash Equilibrium Seeking

Safety-critical multi-agent systems, from cooperative guidance to collision avoidance, must often reach a coordinated decision by a hard deadline rather than merely converge to one eventually. This paper proposes the first fully distributed algorithm that solves the generalized Nash equilibrium (GNE) problem, a non-cooperative game with shared coupling constraints and general cost coupling, at a user-prescribed time $T$ independent of initial conditions. The foundation is a centralized, prescribed-time result built on the optimization Lyapunov function framework and implemented via unnormalized Hessian-gradient feedback, chosen because, unlike the Newton and normalized Hessian-gradient realizations, it naturally splits into per-agent computations. Distributing this feedback requires each agent to run three coupled processes simultaneously: a prescribed-time observer of the global state, a local optimization law, and a dual-consensus mechanism that enforces the shared multipliers of the variational GNE. Their simultaneous operation is the core difficulty, as the optimization continually displaces the states the observers track, while estimation errors corrupt the gradients that drive the optimization. We resolve this coupling with a multi-rate gain schedule whose observer and dual-consensus layers contract strictly faster than the optimization layer, so that every error component vanishes exactly at $T$. A Fischer-Burmeister reformulation keeps the design projection-free while enforcing the constraints at the deadline. Numerical results for a Cournot game and a time-critical sensor-coverage problem validate the approach and demonstrate its use as a solver-in-the-loop for time-critical autonomy.

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arXiv2026

Optimization via a Control-Centric Framework

Optimization plays a central role in intelligent systems and cyber-physical technologies, where speed and reliability of convergence directly impact performance. In control theory, optimization is the foundation of widely-used design methodologies such as linear quadratic regulation, $H_\infty$ control, and model predictive control. In contrast, this paper develops a control-centric framework for optimization itself, where algorithms are constructed directly from Lyapunov stability principles rather than being proposed first and analyzed afterward. A key element is the stationarity vector, which encodes first-order optimality conditions and enables Lyapunov-based convergence analysis. By pairing a Lyapunov function with a selectable decay law, we obtain continuous-time dynamics with guaranteed exponential, finite-time, fixed-time, or prescribed-time convergence. Within this framework, we introduce three feedback realizations of increasing restrictiveness: the Hessian-gradient, Newton, and gradient dynamics. Each realization shapes the decay of the stationarity vector to achieve the desired rate. These constructions unify unconstrained optimization, extend to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability. In total, the framework spans six problem classes and four convergence regimes, yielding a unified design recipe across twenty-four combinations, nine of which, to the best of our knowledge, have no direct continuous-time counterpart in the prior literature. The framework provides systematic design tools for optimization algorithms in control and game-theoretic problems.

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arXiv2026

Multi-UAV Trajectory Optimization for Bearing-Only Localization in GPS Denied Environments

Accurate localization of maritime targets by unmanned aerial vehicles (UAVs) remains challenging in GPS-denied environments. UAVs equipped with gimballed electro-optical sensors are typically used to localize targets, however, reliance on these sensors increases mechanical complexity, cost, and susceptibility to single-point failures, limiting scalability and robustness in multi-UAV operations. This work presents a new trajectory optimization framework that enables cooperative target localization using UAVs with fixed, non-gimballed cameras operating in coordination with a surface vessel. This estimation-aware optimization generates dynamically feasible trajectories that explicitly account for mission constraints, platform dynamics, and out-of-frame events. Estimation-aware trajectories outperform heuristic paths by reducing localization error by more than a factor of two, motivating their use in cooperative operations. Results further demonstrate that coordinated UAVs with fixed, non-gimballed cameras achieve localization accuracy that meets or exceeds that of single gimballed systems, while substantially lowering system complexity and cost, enabling scalability, and enhancing mission resilience.

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