Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate

In this article, we examine the dynamics of a Chikungunya virus (CHIKV) infection model with two routes of infection. The model uses four categories, namely, uninfected cells, infected cells, the CHIKV virus, and antibodies. The equilibrium points of the model, which consist of the free point for the CHIKV and CHIKV endemic point, are first analytically determined. Next, the local stability of the equilibrium points is studied, based on the basic reproduction number (R0) obtained by the next-generation matrix. From the analysis, it is found that the disease-free point is locally asymptotically stable if R0≤1, and the CHIKV endemic point is locally asymptotically stable if R0>1. Using the Lyapunov method, the global stability analysis of the steady-states confirms the local stability results. We then describe our design of an optimal recruitment strategy to minimize the number of infected cells, as well as a nonlinear optimal control problem. Some numerical simulations are provided to visualize the analytical results obtained.

Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions

In this paper we develop a new approach for optimal control problems with general jointly varying state endpoints (also called coupled endpoints). We present a new transformation of a nonlinear optimal control problem with jointly varying state endpoints and pointwise equality control constraints into an equivalent optimal control problem of the same type but with separately varying state endpoints in double dimension. Our new transformation preserves among other properties the controllability (normality) of the considered optimal control problems. At the same time it is well suited even for the calculus of variations problems with joint state endpoints, as well as for optimal control problems with free initial and/or final time. This work is motivated by the results on the second order Sturm-Liouville eigenvalue problems with joint endpoints by Dwyer and Zettl (1994) and by the sensitivity result for nonlinear optimal control problems with separated state endpoints by the authors (2018). p, li { white-space: pre-wrap;

A fast-moving horizon estimation method based on the symplectic pseudospectral algorithm

In this paper, a fast-moving horizon state estimation algorithm for nonlinear continuous systems with measurement noises and model disturbances is developed. The optimization problem required to be solved at each sampling instant is formulated into a backward nonlinear optimal control problem over the finite past. Once prior knowledge of the observed system is available, constraints can be further imposed. The highly efficient and accurate symplectic pseudospectral algorithm is taken as the core solver, which leads to the symplectic pseudospectral moving horizon estimation (SP-MHE) method. The developed SP-MHE is first evaluated by numerical simulations for a hovercraft. Then the developed method is extended to parameter estimation and applied to a chaotic system with an unknown parameter. Simulation results show that the SP-MHE can generate accurate estimations even under large sampling periods or large noise where regular filters fail. In addition, the SP-MHE exhibits excellent online efficiency, suggesting it can be used for scenarios where the sampling period is relatively small.

Helicopter Safe Landing Trajectory after Main Rotor Actuator Failures

Main rotor actuator failure leads to catastrophic accidents for single main rotor helicopters. This paper focuses on safe landing trajectories after an actuator is locked in place by the remaining actuators, without introducing other control inputs. A general swashplate geometry is described, and new reconfiguration solutions for the control mixer are presented. The safe landing trajectories are obtained by formulating a nonlinear optimal control problem based on a nonlinear helicopter dynamic model and geometry constraints due to actuator failure. Safe landing trajectory results are shown with various initial forward velocities of all actuator failure cases. The safe initial speed boundaries are also explored by employing speed sweeps.

Relaxation methods for optimal control problems

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.

Nonlinear optimal tracking control of spacecraft formation flying with collision avoidance

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.