Narrow Results

This paper deals with a novel method for Bat Algorithm (BA) based on optimal tuning of Fractional-Order Proportional Integral Derivative (FOPID) controller for governing the rotor speed of sensorless Brushless Direct Current (BLDC) motor. The BA is used for developing a novel optimization algorithm which can generate five degrees of freedom parameters namely $K_p$, $K_i$, $K_d$, $\lambda$ and $\mu$ of FOPID controller. The desired speed control and robust performance are achieved by using the FOPID closed loop speed controller with the help of BA for optimal tuning. The time domain specifications of a dynamic system for unit step input to FOPID controller for speed response such as peak time ($t_r$), Percentage of overshoot (PO), settling time ($t_s$), rise time ($t_r$) have been evaluated and the steady-state error ($e_{\mathrm{\text{ss}}}$) of sensorless speed control of BLDC motor has been measured. The simulation results are compared with Artificial Bee Colony (ABC) optimization method and Modified Genetic Algorithm (MGA) for evaluation of transient and steady state time domain characteristics. The proposed BA-based FOPID controller optimization technique is more efficient in improving the transient characteristic performance and reducing steady state error.

A co-infection model for human papillomavirus (HPV) and syphilis with cost-effectiveness optimal control analysis is developed and presented. The full co-infection model is shown to undergo the phenomenon of backward bifurcation when a certain condition is satisfied. The global asymptotic stability of the disease-free equilibrium of the full model is shown not to exist when the associated reproduction number is less than unity. The existence of endemic equilibrium of the syphilis-only sub-model is shown to exist and the global asymptotic stability of the disease-free and endemic equilibria of the syphilis-only sub-model was established, for a special case. Sensitivity analysis is also carried out on the parameters of the model. Using the syphilis associated reproduction number, ${{ℛ}}_{0s}$, as the response function, it is observed that the five-ranked parameters that drive the dynamics of the co-infection model are the demographic parameter $\mu$, the effective contact rate for syphilis transmission, ${\beta }_s$, the progression rate to late stage of syphilis ${\sigma }_2$, and syphilis treatment rates: ${\tau }_1$ and ${\tau }_2$ for co-infected individuals in compartments $H_i$ and $H_l$, respectively. Moreover, when the HPV associated reproduction number, ${{ℛ}}_{0h}$, is used as the response function, the five most dominant parameters that drive the dynamics of the model are the demographic parameter $\mu$, the effective contact rate for HPV transmission, ${\beta }_h$, the fraction of HPV infected who develop persistent HPV ${\rho }_1$, the fraction of individuals vaccinated against incident HPV infection $\varphi$ and the HPV vaccine efficacy ${\pi }_h$. Numerical simulations of the optimal control model showed that the optimal control strategy which implements syphilis treatment controls for singly infected individuals is the most cost-effective of all the control strategies in reducing the burden of HPV and syphilis co-infections.

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general Lévy noise.

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first-order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug. In the future, we believe that our optimal control framework may contribute to accelerate experimental research on chemotherapeutic drugs for advanced prostate cancer and ultimately provide a means to design and monitor its optimal delivery to patients.

This paper deals with control strategies for the Hegselmann–Krause opinion formation model with leadership. In this system, the control mechanism is included in the leader dynamics and the feedback control functions are determined via a stabilization procedure and with a model predictive optimal control process. Correspondingly, the issues of global stabilization, controllability, and tracking are investigated. The model predictive control scheme requires to solve a sequence of open-loop optimality systems discretized by an appropriate Runge–Kutta scheme and solved by a nonlinear conjugate gradient method. Results of numerical experiments demonstrate the validity of the proposed control strategies and their ability to drive the system to attain consensus.

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R₀) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E₀ when R₀ < 1. When R₀ > 1 endemic equilibrium E₁ exists and the system becomes locally asymptotically stable at E₁ under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

We report a mathematical model depicting gliomas and immune system interactions by considering the role of immunotherapeutic drug T11 target structure (T11TS). The mathematical model comprises a system of coupled nonlinear ordinary differential equations involving glioma cells, macrophages, activated cytotoxic T-lymphocytes (CTLs), immunosuppressive cytokine transforming growth factor-$\beta$ (TGF-$\beta$), immunostimulatory cytokine interferon-$\gamma$ (IFN-$\gamma$) and the concentrations of immunotherapeutic agent T11TS. For the better understanding of the circumstances under which the gliomas can be eradicated from a patient, we use optimal control strategy. We design the objective functional by considering the biomedical goal, which minimizes the glioma burden and maximizes the macrophages and activated CTLs. The existence and the characterization for the optimal control are established. The uniqueness of the quadratic optimal control problem is also analyzed. We demonstrate numerically that the optimal treatment strategies using T11TS reduce the glioma burden and increase the cell count of activated CTLs and macrophages.

The outbreak of COVID-19 resulted in high death tolls all over the world. The aim of this paper is to show how a simple SEIR model was used to make quick predictions for New Jersey in early March 2020 and call for action based on data from China and Italy. A more refined model, which accounts for social distancing, testing, contact tracing and quarantining, is then proposed to identify containment measures to minimize the economic cost of the pandemic. The latter is obtained taking into account all the involved costs including reduced economic activities due to lockdown and quarantining as well as the cost for hospitalization and deaths. The proposed model allows one to find optimal strategies as combinations of implementing various non-pharmaceutical interventions and study different scenarios and likely initial conditions.

Motivated by a mean field games stylized model for the choice of technologies (with externalities and economy of scale), we consider the associated optimization problem and prove an existence result. To complement the theoretical result, we introduce a monotonic algorithm to find the mean field equilibria. We close with some numerical results, including the multiplicity of equilibria describing the possibility of a technological transition.

The problem of corruption is of serious concern in all the nations, more so in the developing countries. This paper presents the formulation of a corruption control model and its analysis using the theory of differential equations. We found the equilibria of the model and stability of these equilibria are discussed in detail. The threshold quantity $R_0$ which has a similar implication here as in the epidemiological modeling is obtained for the present model. The corruption free equilibrium is found to be stable when $R_0$ is less than $1$ and unstable for $R_0>1$. The endemic equilibrium which signifies the presence of corrupted individuals in the society exists only when $R_0>1$. This equilibrium point is locally asymptotically stable whenever it exists. We perform extensive numerical simulations to support the analytical findings. Furthermore, we extend the model to include optimal control and the optimal control profile is obtained to get the maximum control within a stipulated period of time. Our presented results show that the level of corruption in the society can be reduced if corruption control efforts through media/punishments etc. are increased and put in place.

In this paper, we deal with the problem of optimal control of a deterministic model of hepatitis C virus (HCV). In the first part of our analysis, a mathematical modeling of HCV dynamics which can be controlled by antiretroviral therapy as fixed controls has been presented and analyzed which incorporates two mechanisms: infection by free virions and the direct cell-to-cell transmission. Basic reproduction number is calculated and the existence and stability of equilibria are investigated. In the second part, the optimal control problem representing drug treatment strategies of the model is explored considering control parameters as time-dependent in order to minimize not only the population of infected cells but also the associated costs. At the end of the paper, the impact of combination of the strategies in the control of HCV and their effectiveness are compared by numerical simulation.

Farming awareness is an important measure for pest controlling in agricultural practice. Time delay in controlling pest may affect the system. Time delay occurs in organizing awareness campaigns, also time delay may takes place in becoming aware of the control strategies or implementing suitable controlling methods informed through social media. Thus we have derived a mathematical model incorporating two time delays into the system and Holling type-II functional response. The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays. Stability changes occur through Hopf-bifurcation when time delays cross the critical values. Optimal control theory has been applied for cost-effectiveness of the delayed system. Numerical simulations are carried out to justify the analytical results. This study shows that optimal farming awareness through radio, TV etc. can control the delay induced bifurcation in a cost-effective way.

In this paper we present a new approach in the study of Aorto–Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor-like patch is found. A feedback procedure with Navier–Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI.

Immunotherapy has become a rapidly developing approach in the treatment of cancer. Cancer immunotherapy aims at promoting the immune system response to react against the tumor. In view of this, we develop a mathematical model for immune–tumor interplays with immunotherapeutic drug, and strategies for optimally administering treatment. The tumor–immune dynamics are given by a system of five coupled nonlinear ordinary differential equations which represent the interaction among tumor-specific CD4+T cells, tumor-specific CD8+T cells, tumor cells, dendritic cells and the immuno-stimulatory cytokine interleukin-2 (IL-2), extended through the addition of a control function describing the application of a dendritic cell vaccination. Dynamical behavior of the system is studied from the analytical as well as numerical points of view. The main aim is to investigate the treatment regimens which minimize the tumor cell burden and the toxicity of dendritic cell vaccination. Our numerical simulations demonstrate that the optimal treatment strategies using dendritic cell vaccination reduce the tumor cell burden and increase the cell count of CD4+T cells, CD8+T cells, dendritic cells and IL-2. The most influential parameters having significant impacts on the tumor cells are identified by employing the approach of global sensitivity analysis.

In this paper, a cholera disease transmission mathematical model has been developed. According to the transmission mechanism of cholera disease, total human population has been classified into four subpopulations such as (i) Susceptible human, (ii) Exposed human, (iii) Infected human and (iv) Recovered human. Also, the total bacterial population has been classified into two subpopulations such as (i) Vibrio Cholerae that grows in the infected human intestine and (ii) Vibrio Cholerae in the environment. It is assumed that the cholera disease can be transmitted in a human population through the consumption of contaminated food and water by Vibrio Cholerae bacterium present in the environment. Also, it is assumed that Vibrio Cholerae bacterium is spread in the environment through the vomiting and feces of infected humans. Positivity and boundedness of solutions of our proposed system have been investigated. Equilibrium points and the basic reproduction number $(R_0)$ are evaluated. Local stability conditions of disease-free and endemic equilibrium points have been discussed. A sensitivity analysis has been carried out on the basic reproduction number $(R_0)$. To eradicate cholera disease from the human population, an optimal control problem has been formulated and solved with the help of Pontryagin’s maximum principle. Here treatment, vaccination and awareness programs have been considered as control parameters to reduce the number of infected humans from cholera disease. Finally, the optimal control and the cost-effectiveness analysis of our proposed model have been performed numerically.

In Japan, the first case of Coronavirus disease 2019 (COVID-19) was reported on 15th January 2020. In India, on 30th January 2020, the first case of COVID-19 in India was reported in Kerala and the number of reported cases has increased rapidly. The main purpose of this work is to study numerically the epidemic peak for COVID-19 disease along with transmission dynamics of COVID-19 in Japan and India 2020. Taking into account the uncertainty due to the incomplete information about the coronavirus (COVID-19), we have taken the Susceptible-Asymptomatic-Infectious-Recovered (SAIR) compartmental model under fractional order framework for our study. We have also studied the effects of fractional order along with other parameters in transfer dynamics and epidemic peak control for both the countries. An optimal control problem has been studied by controlling social distancing parameter.

Chikungunya fever, caused by Chikungunya virus (CHIKV) and transmitted to humans by infected Aedes mosquitoes, has posed a global threat in several countries. In this paper, we investigated a modified within-host Chikungunya virus (CHIKV) infection model with antibodies where two routes of infection are considered. In a first step, the basic reproduction number ${{ℛ}}_0$ was calculated and the local and global stability analysis of the steady states is carried out using the local linearization and the Lyapunov method. It is proven that the CHIKV-free steady-state $E_0$ is globally asymptotically stable when ${{ℛ}}_0\le 1$, and the infected steady-state $E_1$ is globally asymptotically stable when ${{ℛ}}_0>1$. In a second step, we applied an optimal strategy in order to optimize the infected compartment and to maximize the uninfected one. For this, we formulated a nonlinear optimal control problem. Existence of the optimal solution was discussed and characterized using some adjoint variables. Thus, an algorithm based on competitive Gauss–Seidel-like implicit difference method was applied in order to resolve the optimality system. The theoretical results are confirmed by some numerical simulations.

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease (HFMD) infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number ($R_0$) of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls (two treatment controls with quarantine control). We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows the reliability of our model from the practical point of view.

In this paper, we propose and analyze a nonlinear deterministic malaria disease model for the impact of temperature variability on malaria epidemics. Firstly, we analyzed the invariant region and the positivity solution of the model. The basic reproduction number with respect to disease free-equilibrium is calculated by the next-generation matrix method. The local stability and global stability of the equilibrium points are shown using the Routh–Hurwitz criterion and the Lyapunov function, respectively. A disease-free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one and endemic equilibrium exists otherwise. Moreover, we have shown the sensitivity analysis of the basic reproduction number and the model exhibits forward and backward bifurcation. Secondly, we apply the optimal control theory to describe the model with incorporates three controls, namely using treated bed nets, treatment of infected with anti-malaria drugs and for vector killing using insecticide spray strategy. Pontraygin’s maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the simulation result of optimal control problem and analysis of cost-effectiveness show that a combination of using treated bed nets and treatment is the most effective and least-cost strategy to prevent the malaria disease.

This paper briefly reviews the dynamics and the control architectures of unmanned vehicles; reinforcement learning (RL) in optimal control theory; and RL-based applications in unmanned vehicles. Nonlinearities and uncertainties in the dynamics of unmanned vehicles (e.g. aerial, underwater, and tailsitter vehicles) pose critical challenges to their control systems. Solving Hamilton–Jacobi–Bellman (HJB) equations to find optimal controllers becomes difficult in the presence of nonlinearities, uncertainties, and actuator faults. Therefore, RL-based approaches are widely used in unmanned vehicle systems to solve the HJB equations. To this end, they learn the optimal solutions by using online data measured along the system trajectories. This approach is very practical in partially or completely model-free optimal control design and optimal fault-tolerant control design for unmanned vehicle systems.