Existence and Convergence Results for Generalized Mixed Quasi-Variational Hemivariational Inequality Problem

The main purpose of this paper is threefold. One is to study the existence and convergence problem of solutions for a class of generalized mixed quasi-variational hemivariational inequalities. The second one is to study the existence of optimal control for such kind of generalized mixed quasi-variational hemivariational inequalities under given control u∈U. The third one is to study the relationship between the optimal control and the data for the underlying generalized mixed quasi-variational inequality problems and a class of minimization problem. As an application, we utilize our results to study the elastic frictional problem in a class of Hilbert spaces. The results presented in the paper extend and improve upon some recent results.

Application of the averaging method to the problems of optimal control of the impulse systems

The problem of optimal control at finite time interval for a system of differential equations with impulse action at fixed moments of time as well as the corresponding averaged system of ordinary differential equations are considered. It is proved the existence of optimal control of exact and averaged problems. Also, it is established that optimal control of averaged problem realize the approximate optimal synthesis of exact problem. The main result of the article is a theorem, where it is proved that optimal contol of an averaged problem is almost optimal for exact problem. Substantiation of proximity of solutions of exact and averaged problems is obtained.

Mathematical Analysis of a model for Chlamydia and Gonorrhea Codynamics with Optimal Control

A model for Chlamydia trachomatis (CT) and Gonorrhea codynamics, with optimal control analysis is studied and analyzed to assess the impact of targetted treatment for each of the diseases on their co-infections in a population. The model exhibits the dynamical feature of backward bifurcation when the associated reproduction number is less than unity. The global asymptotic stability of the disease-free equilibrium of the co-infection model is also proven not to exist, when the associated reproduction number is below unity. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established using the Pontryagin's Maximum Principle. Simulations of the optimal control model reveal that the intervention strategy which implements female Chlamydia trachomatis treatment and male gonorrhea treatment is the most effective in combating the co-infections of Chlamydia trachomatis and gonorrhea.

A co-infection model for Two-Strain Malaria and Cholera with Optimal Control

A mathematical model for two strains of Malaria and Cholera with optimal control is studied and analyzed to assess the impact of treatment controls in reducing the burden of the diseases in a population, in the presence of malaria drug resistance. The model is shown to exhibit the dynamical property of backward bifurcation when the associated reproduction number is less than unity. The global asymptotic stability of the disease-free equilibrium of the model is proven not to exist. The necessary conditions for the existence of optimal control and the optimality system for the model is established using the Pontryagin's Maximum Principle. Numerical simulations of the optimal control model reveal that malaria drug resistance can greatly influence the co-infection cases averted, even in the presence of treatment controls for co-infected individuals.

Existence of Optimal Control for a Nonlinear Partial Differential Equation of Hyperbolic Type

In this paper, we prove the existence of an optimal control for a nonlinear hyperbolic problem, examined in [3]. An estimation is used which makes it possible to extract from a minimizable sequence of controls and from the sequence of corresponding solutions weakly convergent sub sequences. To prove the passage to the limit in a true equality for every element of the minimizable sequence, Lebesgue’s theorem on the passage to the limit under the integral sign and the theorem of immersion have been used.

Stochastic Optimal Control Model of Dengue Disease

In this research article, an optimal control model of dengue disease with standard incidence rate is proposed. Maximum Principle was employed to derive the necessary conditions for the existence of optimal control. Stochastic version of the model is derived by introducing a random perturbation in the main parameters of the model equations. Numerical solution of the optimality was derived and computed to investigate the optimum control strategy that would be efficacious to be implemented in reducing the number of exposed and infected humans as well as illustrating the explicit differences in the dynamics of the models

Mathematical approaches for optimization of dengue fever dynamics

Background and Objective: For dengue outbreak prevention and vectors reduction, fundamental role of control parameters like vaccination against dengue virus in human population and insecticide in mosquito population have been addressed theoretically and numerically. For this purpose, an existing model was modified to optimize dengue fever. Methodology: Using Pontryagin’s maximum principle, the dynamics of infection for the optimal control problem was addressed, further, defined cost functional, established existence of optimal control, stated Hamiltonian for characterization of optimization. Results: Numerical simulations for optimal state variables and control variables were performed. Conclusion: Our findings demonstrate that with low cost of control variables, state variable such as recovered population increases gradually and decrease other state variables for host and vector population.

Stochastic Optimal Control Model of Haemorrhagic Conjunctivitis Disease

In this research article, a model for the transmission dynamics of haemorrhagic conjunctivitis disease is presented. The tool of dynamical system is employed in investigating the potency of the spreading of the epidemic. The analysis revealed the likelihood of the epidemic to spread when the basic reproduction number exceeds one. The model is reformulated as optimal control problem to assess the effectiveness of the proposed control strategy. Maximum Principle was employed to derive the necessary conditions for the existence of optimal control. Numerical solution of the optimality was derived and computed to investigate the optimum control strategy that would be efficacious to be implemented in reducing the number of exposed and infected individuals. Stochastic version of the model is deduced by introducing stochastic perturbations in the deterministic one. Numerical simulations are provided to illustrate the differences in the dynamics of the models and to understand the epidemic phenomenon.