Multi-objective Optimal Control Problem Research Articles

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

Health-conscious optimal control of Li-ion cell using simplified full homogenized macro-scale model

This article presents the optimal charging of a Li-ion cell based on a simplified full homogenized macro-scale (FHM) model. A solid electrolyte interface (SEI) layer model is included in the simplified FHM model to quantify cell degradation. With these models, a multi-objective optimal control problem subject to constraints from safety concerns is formulated to achieve health-conscious optimal charging. This constrained optimal control problem is converted to a nonlinear programming problem (NLP). A nonlinear model predictive control (NMPC) strategy is adopted by solving the NLP at each sampling time using the pseudo-spectral optimization method. The effect of the input current upper bound on the cell film resistance Rfilm and state of health (SoH) reveals that Rfilm and SoH are more sensitive to input current upper bound at lower values of input current upper bound. Simulation results show that the simplified model and pseudo-spectral method are crucial for reducing the computational load of the model. The proposed algorithm is more efficient in reducing health degradation than the conventional constant current constant voltage (CCCV) charging algorithm since it can explicitly handle the film resistance and capacity as health parameters. Multiple cycle charging simulations revealed that the health-conscious algorithm decreases health degradation and increases battery life.

Optimizing chemotherapy treatment outcomes using metaheuristic optimization algorithms: A case study.

This study explores the dynamics of a mathematical model, utilizing ordinary differential equations (ODE), to depict the interplay between cancer cells and effector cells under chemotherapy. The stability of the equilibrium points in the model is analysed using the Jacobian matrix and eigenvalues. Additionally, bifurcation analysis is conducted to determine the optimal values for the control parameters. To evaluate the performance of the model and control strategies, benchmarking simulations are performed using the PlatEMO platform. The Pure Multi-objective Optimal Control Problem (PMOCP) and the Hybrid Multi-objective Optimal Control Problem (HMOCP) are two different forms of optimal control problems that are solved using revolutionary metaheuristic optimisation algorithms. The utilization of the Hypervolume (HV) performance indicator allows for the comparison of various metaheuristic optimization algorithms in their efficacy for solving the PMOCP and HMOCP. Results indicate that the MOPSO algorithm excels in solving the HMOCP, with M-MOPSO outperforming for PMOCP in HV analysis. Despite not directly addressing immediate clinical concerns, these findings indicates that the stability shifts at critical thresholds may impact treatment efficacy.

Koopman-based surrogate models for multi-objective optimization of agent-based systems

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

Differential Stability in a Multi-Objective Optimal Control Problems with a Possibly Empty Solution Set

Differential Stability in a Multi-Objective Optimal Control Problems with a Possibly Empty Solution Set

Application of cell mapping to control optimization for an antenna servo system on a disturbed carrier

Abstract. The cell-mapping method, due to its global optimality, has been applied to solve multi-objective optimization problems (MOPs) and optimal control problems. However, the curse of dimensionality limits its application in high-dimensional systems. In this paper, the multi-parameter sensitivity analysis is investigated to reduce the parameter space dimension, which broadens the application of cell mapping to MOPs in high-dimensional parameter space. A post-processing algorithm for MOPs is introduced to help choose proper control parameters from the Pareto set. The proposed scheme is applied successfully in the control parameter optimization of an adaptive nonsingular terminal sliding-mode control for an antenna servo system on a disturbed carrier. Moreover, as the existing global optimal tracking control with an adjoining cell-mapping method may generate tracking-phase differences, an optimal-sliding-mode combined-control strategy is proposed. By using the combined-control strategy, the azimuth and pitch angles of the antenna system are controlled to catch up to a target trajectory with the minimum cost function and to keep high-precision tracking after that.

Optimizing breast cancer treatment using hyperthermia: A single and multi-objective optimal control approach

In recent decades, the study and development of numerical strategies for cancer treatment have become increasingly feasible, thanks to the advancement of numerical techniques and the increased availability of higher-performance computers. This has enabled researchers to tackle more complex and realistic problems associated with cancer treatment. Among the deadliest types of carcinomas, breast cancer is one of the most extensively studied, primarily due to its high annual incidence and mortality rate. From a medical standpoint, hyperthermia has emerged as one of the most promising treatment modalities. In general, this procedure involves applying heat to the tumor using an applicator over a period of time in order to destroy pathological cells. The applicator provides a source of energy that is used to raise the temperature inside the tumor. Traditionally, a constant heat source has been utilized during the treatment period. This article proposes single and multi-objective optimal control problems for breast cancer treatment by hyperthermia, and solutions for these problems are provided. In this instance, the heat source is treated as a control variable considering two types of strategies: i) increasing monotone control and ii) non-monotone control. For this purpose, the extent of tissue damage is considered a constraint in each application to ensure the degree of tissue damage. As expected, the findings demonstrate that the best result was achieved when employing a strategy that considers only the lateral limits, without any additional constraints. In addition, for every case study, a more appropriate optimal strategy can be employed for cancer treatment. Moreover, for the multi-objective optimal control problem, a satisfactory balance between the objectives can be selected such that a specific optimal strategy can be established for each patient.

Robust Constrained Multi-Objective Guidance of Supersonic Transport Landing Using Evolutionary Algorithm and Polynomial Chaos

Landing of supersonic transport (SST) suffers from a large uncertainty due to its highly sensitive aerodynamic properties in the subsonic domain, as well as the wind gusts around runways. At the vehicle design stage, a landing trajectory optimization under wind uncertainty in a multi-objective solution space is desired to explore the possible trade-off in its key flight performance metrics. The proposed algorithm solves this robust constrained multi-objective optimal control problem by integrating non-intrusive polynomial chaos expansion into a constrained evolutionary algorithm. The computationally tractable optimization is made possible through the conversion of a probabilistic problem into an equivalent deterministic representation while maintaining a form of the multi-objective problem. The generated guidance trajectories achieve a significant reduction of the uncertainty in their terminal states with a marginal modification in the control history of the deterministic solutions, validating the importance of the consideration of robustness in trajectory optimization.

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Adaptive authority dynamic game for human-machine cooperative control

An adaptive dynamic allocation strategy for driving authority based on the non-cooperative game was designed to solve the decision conflict problem between the driver and automatic system in human-machine driving. The coupling dynamics model of human-machine interaction was established according to the coupling mechanism between the driver and the automatic system. Considering the driver’s driving status and the driving risk of vehicles, the driving authority dynamic allocation strategy was designed. On this basis, the human-machine cooperative multi-objective optimal control problem is constructed, and the dynamic allocation control system was designed based on the non-cooperative game theory. The optimal control method was used to solve the Nash equilibrium solution and achieve the optimal authority allocation. The effectiveness of the proposed adaptive authority dynamic allocation strategy was verified under normal and emergency conditions. Results showed that the proposed strategy can effectively reduce human-machine decision conflicts and achieve harmonious human–machine cooperative driving.

The Clarke Coderivative of the Frontier Map in a Multi-objective Optimal Control Problem

The Clarke Coderivative of the Frontier Map in a Multi-objective Optimal Control Problem

Multi-objective optimal control of tungiasis diseases with terminal demands

In this paper, we aim to minimize the epidemic size of tungiasis disease and economic costs simultaneously, with terminal demands for infected humans. A human–jigger parasite control system with four control schemes for humans and jiggers is established. We propose a multi-objective optimal control problem with terminal constraints, in which the accumulated number of infected humans and control costs are involved. By applying the modified normal boundary intersection algorithm and the interior point scheme, numerical simulations for different combinations of control schemes are carried out, and actual data in Madagascar are used. Effective combination schemes are indicated from the perspectives of disease eradication, cost saving and time saving. Once these effective combinations are properly performed, the disease can be controlled. When only minimizing the epidemic size, the combination of the optimal treatments and adulticiding efforts is the best choice in the rainy season; the combination of the optimal personal protections and treatments is the preferential option in the dry season. When only minimizing the economical cost, the combination of the optimal adulticide and larvicide is the better selection in the rainy season; the combination of the optimal personal protections, treatments and adulticiding efforts is the prior choose in the dry season. Thus, there is a trade-off between the two objectives for all the effective combinations, decision-makers may choose an appropriate one to control the disease.

A novel engine and battery coupled thermal management strategy for connected HEVs based on switched model predictive control under low temperature

Under a low-temperature environment, electric vehicles face serious environmental adaptability problems, and efficient vehicle thermal management strategies are urgently needed. This paper presents a novel engine–battery coupled thermal management strategy for connected hybrid electric vehicles (HEVs). An improved system structure for an engine–battery coupled thermal management system (engine–battery CTMS) is designed to avoid unnecessary heat loss. The control requirements of the engine–battery CTMS include minimum engine fuel consumption, minimum power battery aging damage and minimum system energy consumption, which constitutes a multi-objective optimal control problem in a finite time domain. Based on model predictive control (MPC) theory, a switched nonlinear MPC (NMPC) control strategy is proposed to solve the optimal control problem of the complex coupled multi-input multi-output system. To verify the effectiveness of the proposed strategy, three comparative experiments of the centralized NMPC-based and rule-based methods combined with the improved system structure and the unimproved system structure are designed. The results of the cosimulation experiment between MATLAB/Simulink and AMEsim under various driving cycles and different ambient temperatures show that the improved structure and switched control strategy confer great advantages in reducing the controller computation burden, engine fuel consumption, and power battery aging damage.

Extending Life of Lithium-Ion Battery Systems by Embracing Heterogeneities via an Optimal Control-Based Active Balancing Strategy

This article formulates and solves a multiobjective fast charging-minimum degradation optimal control problem (OCP) for a lithium-ion battery module made of series-connected cells equipped with an active balancing circuitry. The cells in the module are subject to heterogeneity induced by manufacturing defects and nonuniform operating conditions. Each cell is expressed via a coupled nonlinear electrochemical, thermal, and aging model and the direct collocation approach is employed to transcribe the OCP into a nonlinear programming problem (NLP). The proposed OCP is formulated under two different schemes of charging operation: 1) same charging time (OCP-SCT) and 2) different charging time (OCP-DCT). The former assumes simultaneous charging of all cells irrespective of their initial conditions, whereas the latter allows for different charging times of the cells to account for heterogeneous initial conditions. The problem is solved for a module with two series-connected cells with intrinsic heterogeneity among them in terms of state of charge and state of health. Results show that the OCP-DCT scheme provides more flexibility to deal with heterogeneity, boasting of lower temperature increase, charging current amplitudes, and degradation. Finally, a comparison with the common practice of constant current (CC) charging over a long-term cycling operation shows that promising savings, in terms of retained capacity, are attainable under both control (OCP-SCT and OCP-DCT) schemes.

Identification of Motor Control Objectives in Human Locomotion via Multi-Objective Inverse Optimal Control

Abstract Predictive simulations of human motion are a precious resource for a deeper understanding of the motor control policies encoded by the central nervous system. They also have profound implications for the design and control of assistive and rehabilitation devices, for ergonomics, as well as for surgical planning. However, the potential of state-of-the-art predictive approaches is not fully realized yet, making it difficult to draw convincing conclusions about the actual optimality principles underlying human walking. In the present study, we propose a novel formulation of a bilevel, inverse optimal control strategy based on a full-body three-dimensional neuromusculoskeletal model. In the lower level, prediction of walking is formulated as a principled multi-objective optimal control problem based on a weighted Chebyshev metric, whereas the contributions of candidate control objectives are systematically and efficiently identified in the upper level. Our framework has proved to be effective in determining the contributions of the selected objectives and in reproducing salient features of human locomotion. Nonetheless, some deviations from the experimental kinematic and kinetic trajectories have emerged, suggesting directions for future research. The proposed framework can serve as an inverse optimal control platform for testing multiple optimality criteria, with the ultimate goal of learning the control objectives that best explain observed human motion.2

Local stability of solutions to a parametric multi-objective optimal control problem

This paper studies local stability of solutions to a parametric multi-objective optimal control problem with constraints. By combining the scalarization method and non-scalarization methods, we show that if the unperturbed problem has a locally strong Pareto solution, then the Pareto solution set is locally Hölder continuous at a reference parameter.

A grid‐based searching algorithm for observer‐based multiobjective control of T–S fuzzy stochastic jump‐diffusion systems

Most studies on a multiobjective optimal control problem (MOCP) with nonlinear stochastic jump-diffusion system (NSJDS) constraints assume the state vector is available, but in practice, this is not necessarily the case. The observer-based MOCP is worthy of further research. Furthermore, the hybrid multiobjective differential evolution algorithm (HMODEA) is usually employed to help solve MOCPs with dynamical system constraints, and such problems often have a higher computational burden. To address these two issues, a grid-based front-squeezing searching algorithm (GBFSA) is proposed to solve the observer-based MOCP with NSJDS. Takagi-Sugeno (T-S) fuzzy model is used to approximate the NSJDS and convert the problem into an MOCP with linear matrix inequality (LMI) constraints. Then, the GBFSA efficiently searches for the Pareto front by merging the LMIs and the squeezing theorem. To automatically select a preferred Pareto controller, the minimum Manhattan distance (MMD) approch is applied. Mathematical proofs are given to show that the obtained Pareto optimal controller can concurrently stabilize the associated NSJDS and achieve the desired performance indices. In addition, computational complexity and convergence analysis are also provided.

Optimal Lane-Free Crossing of CAVs Through Intersections

Connected and autonomous vehicles (CAVs), unlike conventional cars, will utilise the whole space of intersections and cross in a lane-free order. This article formulates such a lane-free crossing of intersections as a multi-objective optimal control problem (OCP) that minimises the overall crossing time, as well as the energy consumption due to the acceleration of CAVs. The constraints that avoid collision of vehicles with each other and with road boundaries are smoothed by applying the dual problem theory of convex optimisation. The developed algorithm is capable of finding the lower boundary of the crossing time of a junction which can be used as a benchmark for comparing other intersection crossing algorithms. Simulation results show that the lane-free crossing time is better by an average of 40% as compared to the state-of-the-art reservation-based method, whilst consuming the same amount of energy. Furthermore, it is shown that the lane-free crossing time through intersections is fixed to a constant value regardless of the number of CAVs.

Risk-neutral multiobjective optimal control of random Volterra integral equations

We analyze risk-neutral multiobjective optimal control problems, governed by Volterra integral equations with random inputs, and subject to expectation-type final/pure state constraints and mixed pointwise control-state constraints. Moreover, controls are unbounded, and inclusion-form pure state as well as control-state constraints are given by measurable set-valued mappings, whose images are nonempty closed subsets of infinite dimensional spaces. With such constraints, the multiplier corresponding to pure state constraints belongs to the dual space of the Banach space-valued continuous functions. We set up a decomposition result for this dual, and develop a Lagrange multiplier theorem for an abstract multi-criteria optimization model, in which a Robinson-type constraint qualification is applied. Then, making use of the obtained outcomes, exploiting higher integrability and regularity of the random field control/state variables, and employing Banach space-valued integration techniques, we derive Fritz-John necessary optimality conditions, with integrable functions and (countably additive) measures singular with respect to the Lebesgue measure as multipliers, for local weak Pareto solutions of studied problems.

Optimization of Chemotherapy Using Hybrid Optimal Control and Swarm Intelligence

This study aimed to minimize the tumor cell population by using minimal medicine for chemotherapy treatment while maintaining the effector-immune cell population at a healthy threshold. Therefore, a mathematical model was developed in the form of ordinary differential equations (ODE), and the solution to Multi-Objective Optimal Control Problem (MOOCP) was obtained using Multi-Objective Optimization algorithms. In this study, the interaction of the tumor-effector cell population with chemotherapy was investigated using Pure MOOCP and Hybrid MOOCP methods. The handling of constraints and the Pontryagin Maximum Principle (PMP) differ among these methods. Swarm Intelligence (SI) and Evolutionary Algorithms (EA) were used to process the results of these methods. The numerical outcomes of SI and EA are displayed via Pareto Optimal Front. In addition, the solutions from these algorithms were further analyzed using the Hypervolume Indicator. The findings of this study demonstrate that the hybrid method outperforms Pure MOOCP via Multi-Objective Differential Evolution (MODE). The MODE produces a point on the Pareto Optimal Front with minimal distance to the origin, where the distance represents the best solution.