We consider a system of reaction–advection–diffusion partial differential equation (PDE) with a distributed input subject to an unknown and arbitrarily large time delay. Using Lyapunov technique, we derive a delay-adaptive predictor feedback controller to ensure the global stability of the closed-loop system in the $L^2$ sense. More precisely, we express the input delay as a 1-D transport PDE with a spatial argument leading to the transformation of the time delay into a spatially distributed shift. For the resulting mixed transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping design and certainty equivalence principle to derive the suitable adaptive control law that compensates for the effect of the unknown time delay. Our controller ensures the global stabilization in the $L^2$ sense. Our result is the first delay-adaptive predictor feedback controller with a PDE plant subject to a delayed distributed input. The feasibility of the proposed approach is illustrated by considering a mobile robot that spread a neutralizer over a polluted surface to achieve efficient decontamination with an unknown actuator delay arising from the noncollocation of the contaminant diffusive process and the moving neutralizer source. Consistent simulation results are presented to prove the effectiveness of the proposed approach.

With deep neural network approximations of PDE backstepping, for each new functional coefficient of the PDE plant, the gains are obtained through a function evaluation. In this paper we expand this framework to control of cascaded PDE systems from distinct classes: a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator for the control gain is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the DeepONet-approximated delay-compensated PDE backstepping controller, we guarantee exponential stability in the $L^{2}$ norm of the plant state and the $H^{1}$ norm of the input delay state.

Recently, the fixed-time consensus problem for multiple nonholonomic robot systems has gained much attention due to its practical value, high accuracy and strong robustness, and almost all of the existing studies have focused on kinematics control. However, the actual system is essentially dynamic, and the generalized force or torque driven by the design meets the actual requirements. Given this consideration, the fixed-time posture and velocity leader-following consensus problem is investigated for a class of nonholonomic robots, namely, the tractor-trailer vehicles with dynamics control. A nonlinear leader system capable of generating reference postures and velocities applicable to tractor-trailer vehicles is first formulated. Utilizing the newly proposed distributed observer capable of estimating the state and output matrix of the leader system in fixed time and the distributed internal model capable of reconstructing information in fixed time, all the tractor-trailer vehicles can acquire and construct the necessary reference information through the communication network in fixed time. Then, by combining the dynamic gain and state constraint methods, a fixed-time velocity ideal controller capable of overcoming nonholonomic constraints, exerting the desired control over postures, and with first-order derivability capable of being tracked by the actual velocity through the design of the control torque is devised at the kinematic level, which ensures that the fixed-time dynamics controller designed around it accomplishes the control objectives while sidestepping potential singularity issues that may arise from the use of existing non-smooth fixed-time control strategies.

In this paper we model the collective dynamics by two wave-like partial differential equations (PDEs), whose states in nature present the positions of the agents in 3-D space. The spatial arguments of PDEs can be treated as the continuum version of the identification (ID) number of the agents, so the agents communication topology is dependent on the domain of the spatial argument. Specifically, the domain is a cylindrical surface with constant radium in a 3-D cylindrical coordinates which helps to produce a rich family of 3-D deployment formations that are the PDEs' equilibria. In order to achieve the desired formation from any initial boundary conditions, we design the boundary controllers and observers by continuous backstepping. The PDEs are discretized by finite difference method to a distributed control law for the multi-agent system, in which only local information is needed to be share within the neighborhood. The leader agents on the boundary of the cylindrical surface can use its neighbors information to estimate every agent's position as needed in leaders' feedback. In addition, a variety of simulations demonstrate the effectiveness of the method.

Multi-frame depth fusion reduces only errors that vary across observations; a static active infrared stereo camera can retain a repeatable spatial residual field. We characterized a Gemini 336L over 1.5–3.5 m and performed a Gemini 335 check. Residuals after plane fitting comprised averageable temporal and fixed terms. After registration to checkerboard coordinates, sampled fixed fields admitted an exact finite-view mean-centered partition into view-invariant and view-varying constituents. Lateral displacement decorrelated the varying constituent at 1.5–2.5 m; fixed-pose optical substitution supported its speckle association. The invariant constituent persisted across same-range lateral views and showed fractional-disparity periodicity and axial phase-dependent correlation, supporting a disparity-locking association. At 3.5 m, eight lateral views retained 57% of fixed-field mean square, whereas four axial views reduced residual root-mean-square (RMS) from 13.87 to 7.48 mm (ratio 0.540). Median total and fixed-only single-frame disparity-equivalent residuals were 0.068 and 0.056 px. For static acquisition, total error is persistent bias squared plus temporal variance, not a conditional covariance bound. Results motivate combining lateral speckle and axial disparity-phase diversity. Axial fusion was verified for registered planar residual fields at one range endpoint, not across ranges or in full-scene three-dimensional (3D) reconstruction.