Rafael Vazquez

Rafael Vazquez, Miroslav Krstic

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.

Rafael Vazquez, Miroslav Krstic

This paper develops an extension of infinite-dimensional backstepping method for parabolic and hyperbolic systems in one spatial dimension with two actuators. Typically, PDE backstepping is applied in 1-D domains with an actuator at one end. Here, we consider the use of two actuators, one at each end of the domain, which we refer to as bilateral control (as opposed to unilateral control). Bilateral control laws are derived for linear reaction-diffusion, wave and 2X2 hyperbolic 1-D systems (with same speed of transport in both directions). The extension is nontrivial but straightforward if the backstepping transformation is adequately posed. The resulting bilateral controllers are compared with their unilateral counterparts in the reaction-diffusion case for constant coefficients, by making use of explicit solutions, showing a reduction in control effort as a tradeoff for the presence of two actuators when the system coefficients are large. These results open the door for more sophisticated designs such as bilateral sensor/actuator output feedback and fault-tolerant designs.

Rafael Vazquez, Miroslav Krstic

An explicit output-feedback boundary feedback law is introduced that stabilizes an unstable linear constant-coefficient reaction-diffusion equation on an $n$-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere) using only measurements from the boundary. The backstepping method is used to design both the control law and a boundary observer. To apply backstepping the system is reduced to an infinite sequence of 1-D systems using spherical harmonics. Well-posedness and stability are proved in the $H^1$ space. The resulting control and output injection gain kernels are the product of the backstepping kernel used in control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the $n$-ball.

Rafael Vazquez, Francisco Gavilan, Eduardo F. Camacho

This work presents a model predictive controller (MPC) that is able to handle linear time-varying (LTV) plants with PWM control. The MPC is based on a planner that employs a PAM or impulsive approximation as a hot-start and then uses explicit linearization around successive PWM solutions for rapidly improving the solution by means of linear programming. As an example, the problem of rendezvous of spacecraft for eccentric target orbits is considered. The problem is modeled by the LTV Tschauner-Hempel equations, whose transition matrix is explicit; this is exploited by the algorithm for rapid convergence. The efficacy of the method is shown in a simulation study.

Qian Yin, Jiaxing Li, Jiaqi Cheng et al.

Earth observation satellite imaging scheduling is a challenging NP-hard combinatorial optimisation problem central to space mission operations. While next-generation agile Earth observation satellites (EOS) increase operational flexibility, they also significantly raise scheduling complexity. The lack of a unified, open-source benchmark makes it difficult to compare algorithms across studies. This paper introduces EOS-Bench, a comprehensive framework for systematic and reproducible evaluation of scheduling methods. By integrating high-fidelity orbital dynamics and platform constraints, EOS-Bench generates 1,390 scenarios and 13,900 benchmark instances, spanning from small-scale validation cases to large coordination problems with up to 1,000 satellites and 10,000 requests. We further propose a scenario characterisation scheme to quantify structural difficulty based on factors such as opportunity density, task flexibility, conflict intensity, and satellite congestion. A multidimensional evaluation protocol is introduced, assessing performance across five metrics: task profit, completion rate, workload balance, timeliness, and runtime. The framework is evaluated using mixed-integer programming, heuristics, meta-heuristics, and deep reinforcement learning across both agile and non-agile settings. Results show that EOS-Bench effectively distinguishes solver performance across scales and conditions, revealing trade-offs between solution quality and computational efficiency, and providing deeper insight into scenario complexity. EOS-Bench offers a unified and extensible open testbed for advancing research in Earth observation satellite scheduling. The code and data are available at https://github.com/Ethan19YQ/EOS-Bench.

Jose Antonio Rebollo, Rafael Vazquez, Claudio Bombardelli

Non-Gaussian tails dominate collision probability estimates in conjunction assessment, yet capturing them without Monte Carlo sampling is challenging, especially when process noise is included. We present a closed-form, grid-free solution to the Fokker-Planck equation by proving that an exponential-of-quadratic-form ansatz is structurally preserved under advection and diffusion. The probability density function propagates via a compact ODE system, significantly cheaper than Monte Carlo and without spatial discretization. As an application, the method performs orbit uncertainty propagation under stochastic forcing representative of atmospheric drag. Results demonstrate the method faithfully captures non-Gaussian features, asymmetric tails, and stochastic broadening, matching a Monte Carlo benchmark.

Maxence Lamarque, Luke Bhan, Rafael Vazquez et al.

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

John A. Evans, Michael A. Scott, Kendrick Shepherd et al.

In this paper, we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute, and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.