Formulation and multigrid solution of Cauchy-Riemann optimal control problems

Various variants of the formulation of optimal control problem of the reorientation of the axis of dynamic symmetry of the spacecraft (spacecraft), which is the axis of rotation, are considered. It is assumed that the solution to this problem should be found in the class of movements with one directional (flat) rotation, provided that before and after the reorientation of the axis of rotation, the angular velocity of the spacecraft is the same. In this case, the angular motion of the spacecraft is controlled according to the "rotary jet engine" scheme, when the control moment is limited by the ellipsoid of rotation. The corresponding mathematical model of spacecraft motion for the control problem under consideration is given. In addition to the formulation of the problem of the steepest reorientation of the axis of rotation of the spacecraft, the optimal control problem is also formulated, for which optimal control for the full control moment is found. In order to analyze the formulation of the problem under consideration, the results of its reduction to the boundary value problem and to the isoperimetric variational problem are also presented.

Dynamic optimization for air separation plants

This article addresses the problem of quantifying the uncertainty in planning aircraft ground movement operations using towbarless robotic tractors taking into account the inherent uncertainties of the problem, specifically, the uncertainties in the weight of the aircraft and in the rolling resistance of the wheels of the main landing gear. The tractor‐aircraft system is represented as a tractor‐trailer system with random parameters characterized by means of probability density functions. The quantification of the uncertainties is conducted within an optimal control framework using the formulation of a stochastic optimal control problem, which is solved through a stochastic collocation method based on generalized polynomial chaos. Specifically, the stochastic optimal control problem is converted into a set of deterministic optimal control problems, in which a reduced number of sample values of the random parameters are employed to solve particular instances of the problem. Using these sample values, it is possible to express the obtained optimal solutions as orthogonal polynomial expansions in terms of the uncertain parameters, which allows both statistical and global sensitivity analysis of the stochastic optimal solutions to be carried out in an efficient way. The objectives of this article are to understand the effects of the uncertainties in the model parameters of the tractor‐aircraft system on the solutions of the optimal control problem and to identify which uncertain parameters have more influence on the variability of these solutions. The ultimate goal is to determine the best approach for executing aircraft ground movement operations in the presence of such uncertainties.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

This study presents an algorithm for solving optimal control problems with the objective function of the Lagrange-type and multiple delays on both the state and control variables of the constraints, with bounds on the control variable. The full discretization of the objective functional and the multiple delay constraints is carried out by using the Simpson numerical scheme. The discrete recurrence relations generated from the discretization of both the objective functional and constraints are used to develop the matrix operators, which satisfy the basic spectral properties. The primal-dual residuals of the algorithm are derived in order to ascertain the rate of convergence of the algorithm, which performs faster when relaxed with an accelerator variant in the sense of Nesterov. The direct numerical approach for handling the multi-delay control problem is observed to obtain an accurate result at a faster rate of convergence when over-relaxed with an accelerator variant. This research problem is limited to linear constraints and objective functional of the Lagrange-type and can address real-life models with multiple delays as applicable to quadratic optimization of intensity modulated radiation theory planning. The novelty of this research paper lies in the method of discretization and its adaptation to handle linearly and proximal bound-constrained program formulated from the multiple delay optimal control problems.

Chapter 1 - Introduction and Motivation

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.

High order optimal feedback control of space trajectories with bounded control

Two-class freeway traffic regulation to reduce congestion and emissions via nonlinear optimal control

Brucellosis, a persistent zoonotic disease, presents significant challenges in understanding its transmission dynamics and in designing effective control strategies due to the complex interaction of biological and environmental factors. In this study, we develop a delayed differential equation model incorporating discrete time delays to represent both disease latency and the delayed effects of interventions, as well as environmental transmission mechanisms. These elements allow for a more accurate representation of the epidemiological processes that sustain brucellosis in both livestock and human populations. We derive threshold conditions for controlling disease prevalence and verify the existence of equilibrium points, analyzing their global asymptotic stability using Lyapunov functions. A key contribution of this work is the formulation and analysis of a mixed state-control delay optimal control problem, which explicitly considers the impact of delays on both state variables and control measures. This framework provides new insights into the timing of interventions and the optimal allocation of resources. Optimal strategies, including vaccination, culling, environmental disinfection and public awareness campaigns, are derived using the delay Pontryagin maximum principle. The study balances the direct costs of interventions with the indirect economic and societal impacts of the disease, offering actionable, cost-effective recommendations. Sensitivity analysis identifies key parameters influencing transmission and control effectiveness. The model is validated with epidemiological data from Inner Mongolia, demonstrating its applicability to endemic settings. Our findings highlight the critical role of mixed state-control delays in shaping optimal control strategies, showing that neglecting these factors can lead to ineffective or suboptimal results. This research advances both the theoretical and practical understanding of mixed delay optimal control problems, offering a valuable framework for managing brucellosis and other delay-sensitive infectious diseases.

Great efforts have been devoted to the intelligent control of autonomous systems. Yet, most of existing methods fail to effectively handle the uncertainty of their environment and models. Uncertain locations of dynamic obstacles pose a major challenge for their optimal control and safety, while their linearization or simplified system models reduce their actual performance. To address them, this paper presents a new model predictive control framework with finite samples and a Gaussian model, resulting in a chance-constrained program. Its nominal model is combined with a Gaussian process. Its residual model uncertainty is learned. The resulting method addresses an efficiently solvable approximate formulation of a stochastic optimal control problem by using approximations for efficient computation. There is no perfect distribution knowledge of a dynamic obstacle's location uncertainty. Only finite samples from sensors or past data are available for moment estimation. We use the uncertainty propagation of a system's state and obstacles' locations to derive a general collision avoidance condition under tight concentration bounds on the error of the estimated moments. Thus, this condition is suitable for different obstacles, e.g., bounding box and ellipsoid obstacles. We provide proved guarantees on the satisfaction of the chance-constraints corresponding to the nominal yet unknown moments. Simulation examples of a vehicle's control are used to show that the proposed method can well realize autonomous control and obstacle avoidance of a vehicle, when it operates in an uncertain environment with moving obstacles. It outperforms the existing moment methods in both performance and computational time.

Stochastic optimal control of mesostructure of supramolecular assemblies using dissipative particle dynamics and dynamic programming with experimental validation

In this contribution an efficient numerical method to solve multiobjective optimal control problems of fluid flow mixing governed by the advection equation is presented. To obtain well‐mixed behavior for non‐diffusive processes, the multi‐scale mixing of measure introduced in [1] is used in the formulation of the multiobjective optimal control problem. For the approximation of the Pareto set we combine reference point methods with an adjoint based optimization approach. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.