Liraz Mudrik

Liraz Mudrik, Yaakov Oshman

We address the problem of optimal evasion in a planar endgame engagement, where a target with bounded lateral acceleration seeks to avoid interception by a missile guided by a linear feedback law. Contrary to existing approaches, that assume perfect information or use heuristic maneuver models in stochastic settings, we formulate the problem in an inherently stochastic framework involving imperfect information and bounded controls. Complying with the generalized separation theorem, the control law factors in the posterior distribution of the state. We extend the well-known optimality of bang-bang evasion maneuvers in deterministic settings to the realm of realistic, stochastic evasion scenarios. First, we prove that an optimal evasion strategy always exists, and that the set of optimal solutions includes at least one bang-bang policy, rendering the resulting optimal control problem finite-dimensional. Second, leveraging this structure, we propose the closed-loop terminal-set-based evasion (TSE) strategy, and demonstrate its effectiveness in simulation against a proportional navigation pursuer. Monte Carlo simulations show that the TSE strategy outperforms traditional stochastic evasion strategies based on random telegraph, Singer, and weaving models.

Liraz Mudrik, Sean Kragelund, Isaac Kaminer

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.

Abram H. Clark, Liraz Mudrik, Colton Kawamura et al.

Designing autonomous drone swarms is hampered by a vast design space spanning platform, algorithmic, and numerical-strength choices. We perform large-scale agent-based simulations in three canonical scenarios: swarm-on-swarm battle, cooperative area search with attrition, and pursuit of scattering targets. We demonstrate how dimensional-analysis and data-scaling can be leveraged to collapse performance data onto scaling functions that are mathematically simple, yet counterintuitive and therefore difficult to predict a priori. These scaling laws reveal success-failure boundaries, including sharp break points which we show can be framed as an ``effective swarm size.'' Additionally, we show how this technique can be used to quantify trade-offs between agent count and platform parameters such as velocity, sensing or weapon range, and attrition rate. Furthermore, we show the benefits of embedding an optimal path planning loop within this framework, which can qualitatively improve the scaling laws that govern the outcome. The methods we demonstrate are highly flexible and would enable rapid, budget-aware sizing and algorithm selection for large autonomous swarms.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

This paper proposes the first fully distributed algorithm for finding the Generalized Nash Equilibrium (GNE) of a non-cooperative game with shared coupling constraints and general cost coupling at a user-prescribed finite time T. As a foundation, a centralized gradient-based prescribed-time convergence result is established for the GNE problem, extending the optimization Lyapunov function framework to gradient dynamics, the only known realization among existing alternatives that naturally decomposes into per-agent computations. Building on this, a fully distributed architecture is designed in which each agent concurrently runs three coupled dynamics: a prescribed-time distributed state observer, a gradient-based optimization law, and a dual consensus mechanism that enforces the shared-multiplier requirement of the variational GNE, thus guaranteeing convergence to the same solution as the centralized case. The simultaneous operation of these layers creates bidirectional perturbations between consensus and optimization, which are resolved through gain synchronization that matches the temporal singularities of the optimization and consensus layers, ensuring all error components vanish exactly at T. The Fischer-Burmeister reformulation renders the algorithm projection-free and guarantees constraint satisfaction at the deadline. Numerical simulations on a Nash-Cournot game and a time-critical sensor coverage problem validate the approach.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

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 and at a controllable rate remains an open challenge: gradient descent is blind to curvature, stochastic perturbation methods lack deterministic guarantees, and Newton-type approaches suffer from Hessian singularity. We present Curvature-Regularized Gradient Dynamics (CRGD), which augments the objective with a smooth penalty on the most negative Hessian eigenvalue, yielding an augmented cost that serves as an optimization Lyapunov function with user-selectable convergence rates to second-order stationary points. Numerical experiments on a nonconvex matrix factorization example confirm that CRGD escapes saddle points across all tested configurations, with escape time that decreases with the eigenvalue gap, in contrast to gradient descent, whose escape time grows inversely with the gap.

Liraz Mudrik, Isaac Kaminer, Sean Kragelund et al.

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-centric methods are standard: controllers are designed by repeatedly solving optimization problems, as in 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 naturally to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden the results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability. The framework provides systematic design tools for optimization algorithms in control and game-theoretic problems.

Liraz Mudrik, Yaakov Oshman

Using Bayesian decision theory, we modify the perfect-information, differential game-based guidance law (DGL1) to address the inevitable estimation error occurring when driving this guidance law with a separately-designed state estimator. This yields a stochastic guidance law complying with the generalized separation theorem, as opposed to the common approach, that implicitly, but unjustifiably, assumes the validity of the regular separation theorem. The required posterior probability density function of the game's state is derived from the available noisy measurements using an interacting multiple model particle filter. When the resulting optimal decision turns out to be nonunique, this feature is harnessed to appropriately shape the trajectory of the pursuer so as to enhance its estimator's performance. In addition, certain properties of the particle-based computation of the Bayesian cost are exploited to render the algorithm amenable to real-time implementation. The performance of the entire estimation-decision-guidance scheme is demonstrated using an extensive Monte Carlo simulation study.

Liraz Mudrik, Yaakov Oshman

In realistic pursuit-evasion scenarios, abrupt target maneuvers generate unavoidable periods of elevated uncertainty that result in estimation delays. Such delays can degrade interception performance to the point of causing a miss. Existing delayed-information guidance laws fail to provide a complete remedy, as they typically assume constant and known delays. Moreover, in practice they are fed by filtered estimates, contrary to these laws' foundational assumptions. We present an overarching strategy for tracking and interception that explicitly accounts for time-varying estimation delays. We first devise a guidance law that incorporates two time-varying delays, thereby generalizing prior deterministic formulations. This law is driven by a particle-based fixed-lag smoother that provides it with appropriately delayed state estimates. Furthermore, using semi-Markov modeling of the target's maneuvers, the delays are estimated in real-time, enabling adaptive adjustment of the guidance inputs during engagement. The resulting framework consistently conjoins estimation, delay modeling, and guidance. Its effectiveness and superior robustness over existing delayed-information guidance laws are demonstrated via an extensive Monte Carlo study.

Liraz Mudrik, Yaakov Oshman

Classical guidance laws aim at minimizing the miss distance, thus implicitly determining the minimum warhead lethality radius required against nominal targets. However, nonnominal targets or scenarios might render the designed warhead insufficient, causing a significant degradation in the single-shot kill probability (SSKP). We propose a guidance methodology that shifts the interceptor's objective from minimizing the miss distance to directly maximizing the SSKP, while taking into account the warhead's probabilistic lethality model. Complying with the generalized separation theorem, the new paradigm is based on modifying deterministic differential-game-based guidance laws using Bayesian decision theory. Extensive Monte Carlo simulations demonstrate consistent SSKP improvement over the standard and recently introduced estimation-aware guidance laws, when tested against nominal and nonnominal evasively maneuvering targets.