Optimal Control Problem Research Articles

Reducing M2 macrophage in lung fibrosis by controlling anti-M1 agent

Idiopathic pulmonary fibrosis (IPF) is a chronic lung disease characterized by excessive scarring and fibrosis due to the abnormal accumulation of extracellular matrix components, primarily collagen. This study aims to design and solve an optimal control problem to regulate M2 macrophage activity in IPF, thereby preventing fibrosis formation by controlling the anti-M1 agent. The research models the diffusion of M2 macrophages in inflamed tissue using a novel dynamical system with partial differential equation (PDE) constraints. The control problem is formulated to minimize fibrosis by regulating an anti-M1 agent. The study employs a two-step process of discretization followed by optimization, utilizing the Galerkin spectral method to transform the M2 diffusion PDE into an algebraic system of ordinary differential equations (ODEs). The optimal control problem is then solved using Pontryagin/s minimum principle, canonical Hamiltonian equations, and extended Riccati differential equations. The numerical simulations indicate that without control, M2 macrophage levels increase and stabilize, contributing to fibrosis. In contrast, the optimal control strategy effectively reduces M2 macrophages, preventing fibrosis formation within 120 days. The results highlight the potential of the proposed optimal control approach in modulating tissue repair processes and mitigating the progression of IPF. This study underscores the significance of targeting M2 macrophages and employing mathematical methods to develop innovative therapies for lung fibrosis.

Исследование задачи оптимального нагрева жидкого продукта жидким теплоносителем в емкости с рубашкой

Two problems of optimal control of heating of a liquid product in a heating tank with a liquid coolant have been posed and solved on the Pontryagin maximum principle. In the first task, the problem of the speed of heating the product to a given temperature is solved. In the second task, we are looking for a control in which the product is heated for a given time to a given temperature with minimal coolant consumption. The description of the control object with a flow diagram, assumptions simplifying the compilation of the model, a mathematical model of the control object, formulations of optimal control problems and the solutions obtained are given. In the problem of heating speed, analytical expressions are obtained to determine the temperature of the product, the coolant and the control time. An example of solving the problem of heating speed is given with an illustration of the dynamic characteristics of the object variables and the optimal control found. In the problem of minimizing coolant consumption, the solution is obtained numerically. The problem of finding optimal trajectories of variables and control is formulated as a boundary value problem with missing initial conditions, the search for which is realized using the difference method of solving the boundary value problem. Based on the found initial conditions, the Cauchy problem of an extended system of equations is solved, and the optimal trajectory of changes in product temperature and coolant flow is obtained. An example of solving the problem of minimizing coolant consumption is given with a demonstration of the dynamic characteristics of the variables of the object and the optimal control found.

Tube-MPC based trajectory tracking control for substation inspection robot

Abstract A Tube-MPC based trajectory tracking control method was developed to enhance the capabilities of substation inspection robots. First, the kinematic models of the inspection robots were established, and the general form of the optimization control problem for Tube-MPC was introduced. The nominal control law for the nominal system was derived using a linearized nominal model, which was employed to solve the Tube-MPC cost function, ensuring accurate tracking of the desired trajectory. The algorithm was then simulated and tested. Simulation results demonstrate that the Tube-MPC approach significantly outperforms conventional MPC algorithms in terms of trajectory tracking performance and robustness for substation inspection robots. Experimental results further confirm that the Tube-MPC based method effectively enhances both robustness and tracking accuracy during inspection robot operations.

Optimal control problem with nonregular mixed constraints via penalty functions

Optimal control problem with nonregular mixed constraints via penalty functions

Robust finite element solvers for distributed hyperbolic optimal control problems

Robust finite element solvers for distributed hyperbolic optimal control problems

HIV/AIDS and HBV co-infection with optimal control strategies and cost-effectiveness analyses using integer order model

Hepatitis B virus (HBV) and HIV/AIDS co-infection is a common infectious disease that has been spreading through different nations in the world. The main objective of the study is to minimize the number of HBV and HIV co-infected individuals in the community and the cost incurred to the effort applied towards protection and treatment control strategies. In this study, a novel HBV and HIV/AIDS co-infection model has been formulated and analyzed to investigate the effects of protection and treatment control strategies on the spreading dynamics of the HBV and HIV/AIDS co-infection in the community. In the qualitative analyses we have computed all the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points, we have proved the two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. In our understanding no one is formulated and analyzed the HBV and HIV/AIDS co-infection model with optimal control and cost-effective analyses so that we have formulated the associated optimal control problem and carried out the optimal control analysis on the HBV and HIV/AIDS co-infection model by implementing the Pontryagin’s minimum principle. Numerical simulations with various combinations of the control efforts implemented are then carried out to investigate the impacts of protection and treatments to tackle the HBV and HIV/AIDS co-infection diseases spreading in the community. Finally, we also carried out cost-effectiveness analysis for the implemented control strategies. From the findings of the numerical simulations we observed that implementing all the proposed controlling strategies simultaneously has fundamental impact to minimize and control the HBV and HIV/AIDS co-infection spreading in the community but cost-effectiveness analysis proved that implemented strategy 4 (implemented the HIV protection and HBV treatment controls simultaneously) is the most cost-effective strategy as compared with all other implemented strategies and we recommend for the health stake holders and policy makers to implement this strategy to tackle the HBV and HIV/AIDS co-infection spreading problem in the community.

Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints

Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints

Parabolic optimal control problems with combinatorial switching constraints, part III: branch-and-bound algorithm

Parabolic optimal control problems with combinatorial switching constraints, part III: branch-and-bound algorithm

External Optimal Control of the Semilinear Fractional Parabolic Coupled System

This paper deals with an external optimal control problem governed by coupled system of semilinear fractional parabolic equation and parabolic equation with Neumann exterior value condition. Under some assumptions on the nonlinearity, the well-posedness and boundedness of solutions for the system are proved. After showing the Lipschitz continuity and Fréchet differentiability of the control-to-state operator for the controlled system, we establish the existence theorem of solutions and the first-order necessary optimality conditions for an associated optimal control problem.

Fractional optimal control of systems using Bernoulli wavelets

In order to provide a new method for partial optimal control of systems, this work uses Bernoulli wavelets and uses Caputo's formula to solve partial optimal control problems (FOCPs) with inequality constraints. For the new optimization technique, the work uses partial order Bernoulli wave functions (F-BWFs) as basic functions. The answer using this method is expressed in terms of F-BWF where the coefficients have not yet been found. To simplify FOCPs in a system of nonlinear algebraic equations the procedure entails transforming inequality constraints into equality requirements by using operational matrices of fractional integration and F-BWFs and using the multipliers technique Lagrange. Numerical examples confirm the validity of the proposed strategy and demonstrate its correctness and efficiency when compared to the analytical or approximate answers provided by other methods.

Optimal Control problems of NS-$$\alpha $$ and NS-$$\omega $$ turbulence models: analysis and numerical tests

Optimal Control problems of NS-$$\alpha $$ and NS-$$\omega $$ turbulence models: analysis and numerical tests

Control Co-Design with Performance-Robustness Trade-Off using Tube-Based Stochastic Model Predictive Control

Abstract Control Co-Design (CCD) has been demonstrated to achieve superior solutions for closed-loop systems. However, limited work has addressed CCD problems under probabilistic disturbances. This paper addresses this gap by formulating a finite-horizon optimal control problem with chance constraints and proposing a novel CCD approach. This approach integrates tube-based Stochastic Model Predictive Control with constraint-tightening techniques to optimize performance and robustness while preventing instability and infeasibility. A nested CCD framework is introduced, along with a constrained multi-objective optimization algorithm that enables the performance-robustness trade-off. A method for quantifying the robustness of closed-loop systems under stochastic disturbances is presented. The proposed CCD approach is demonstrated on a numerical example and an engineering case of the satellite attitude control system. Results show that CCD can generate more well-spread Pareto fronts that cannot be reached by other design strategies. This helps designers explore more potential solutions with different dynamic characteristics. Selected non-dominated solution trajectories are visualized for qualitative comparisons. Future work will extend this to nonlinear applications.

Reinforcement learning-based near-optimal control for discrete time-delay singularly perturbed systems with unmeasurable states

This brief proposes a novel reinforcement learning-based composite near-optimal output feedback (OPFB) control method for discrete time-delay singularly perturbed systems (SPSs) with unknown dynamics and unmeasured system states. Firstly, an optimal control problem of the full-order time-delay SPS is formulated. The singular perturbation techniques are used to obtain the reduced-order subsystems and equivalent subproblems. Then, a state reconstruction method is presented to represent the state in terms of available input and output data. In addition, a reinforcement learning (RL) algorithm based on input and output data is developed. Theoretical analyses are discussed on the convergence of the proposed algorithm and the asymptotical stability of the system under the composite controller. Finally, numerical simulation results illustrate the effectiveness of the proposed approach.

Erratum to “Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems”

Erratum to “Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems”

Steady Navier–Stokes Equations with Regularized Directional Do-Nothing Boundary Condition: Optimal Boundary Control for a Velocity Tracking Problem

We consider the steady Navier–Stokes equations with mixed boundary conditions, where a regularized directional do-nothing (RDDN) condition is defined on the Neumann boundary portion. An auxiliary Stokes reference flow, which also works as a lifting of the inhomogeneous Dirichlet boundary values, is used to define the RDDN condition. Our aim is to study the minimization of a velocity tracking cost functional with controls localized on a part of the boundary. We prove the existence of a solution for this optimal control problem and derive the corresponding first order necessary optimality conditions in terms of dual variables. All results are obtained under appropriate assumptions on the size of the data and the controls, which, however, are less restrictive when compared with the case of the classical do-nothing outflow condition. This is further confirmed by the numerical examples presented, which include scenarios where only noisy data is available.

Exponential turnpike property for particle systems and mean-field limit

Abstract This work is concerned with the exponential turnpike property for optimal control problems of particle systems and their mean-field limit. Under the assumption of the strict dissipativity of the cost function, exponential estimates for both optimal states and optimal control are proven. Moreover, we show that all the results for particle systems can be preserved under the limit in the case of infinitely many particles.

An optimal control problem with respect to variable thicknesses for a vibrating elastic plate in a contact with a rigid obstacle

Abstract We deal with an optimal control problem governed by a hyperbolic variational inequality describing perpendicular vibrations of a simply supported anisotropic elastic plate against a rigid obstacle. A variable thickness of a plate plays the role of a control variable. The state problem is not uniquely solved. In order to overcome the problem of a priori estimates of states we restrict the admissible set to solutions obtained through a penalization method. We verify the existence of an optimal thickness function as a limit of the sequence of thicknesses solving the case of the penalized state problem.

Optimal control of an influenza model incorporating pharmacological and non-pharmacological interventions

The burden of influenza virus infection poses a challenge due to its significant negative impact on public health. The implementation of intervention measures including vaccination, treatment, and isolation can help alleviate this influence. In this paper, we consider an optimal control problem with both pharmacological and non-pharmacological interventions, which involves widely concerned issues such as the decline of vaccine-based immunity and the emergence of drug resistance. We prove the existence of the optimal control, solve the optimal control problem through applying the Pontryagin’s maximum principle, and conduct some numerical experiments to seek out effective prevention and control strategies. We arrive at a conclusion that the best strategy to control the outbreak of influenza is to isolate infected individuals as soon as possible when medical resources are abundant, such as staying at home, avoiding crowded places and so on. Epidemiologically, we find that reducing the waning rate of vaccine-based immunity is also an effective strategy when there is energy available, and may be better than increasing treatment rates.

A variational principle of an electrohydrodynamic fluid

Electrohydrodynamic fluids possess extensive applications, spanning from the formation of magnetic fields in planets and stars to oil recovery. In this paper, a variational principle for an electrohydrodynamic fluid is presented, and formulated through the semi-inverse method. The Ritz method is utilized to derive an approximate solution. Furthermore, the potential application of this principle to optimal control problems constrained by an electrohydrodynamic fluid is explored.

Model predictive control for risk-based inspection and maintenance planning of offshore wind turbines

ABSTRACT Offshore wind turbines (OWTs) are highly susceptible to fatigue deterioration under extreme environmental conditions, making optimal inspection and maintenance (I&M) planning critical for their safe and sustainable operation. The dynamic nature of weather and complex deterioration processes challenge the long-term effectiveness of maintenance strategies. Moreover, I&M policies often become outdated when deterioration models are updated. Unlike traditional approaches that rely on a fixed life-span horizon, effective I&M planning must adapt to evolving conditions and varying planning horizons. This paper introduces a model predictive control (MPC) method for adaptive, risk-based I&M planning of OWTs. The closed-loop MPC approach provides flexibility by dynamically adjusting inspection and maintenance strategies based on evolving fatigue risks. We propose an augmented state-space model that captures the impact of probabilistic I&M actions on fatigue failure risks over time. A finite-horizon optimal control problem is then formulated to derive the MPC controller, which balances fatigue risk and I&M costs over the planning horizon. The proposed method is demonstrated using a fatigue-prone OWT component. Results show that the MPC controller effectively manages the trade-off between fatigue risk and maintenance costs while remaining adaptable to different planning horizons and decision-making preferences.