The Dynamics of Biological Models with Optimal Harvesting

This paper aims to introduce a concept of an equilibrium point of a dynamical system which will call it almost global asymptotically stable. We also propose and analyze a prey-predator model with a suggested function growth in prey species. Firstly the existence and local stability of all its equilibria are studied. After that the model is extended to an optimal control problem to obtain an optimal harvesting strategy. The discrete time version of Pontryagin's maximum principle is applied to solve the optimality problem. The characterization of the optimal harvesting variable and the adjoint variables are derived. Finally these theoretical results are demonstrated with numerical simulations.

A priori error estimates for a linearized fracture control problem

A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived.

Pontryagin’s Maximum Principle for Optimal Control of Stochastic SEIR Models

In this paper, we study the necessary conditions as well as sufficient conditions for optimality of stochastic SEIR model. The most distinguishing feature, compared with the well-studied SEIR model, is that the model system follows stochastic differential equations (SDEs) driven by Brownian motions. Hamiltonian function is introduced to derive the necessary conditions. Using the explicit formulation of adjoint variables, desired necessary conditions for optimal control results are obtained. We also establish a sufficient condition which is called verification theorem for the stochastic SEIR model.

Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

Consider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.

A Fast Computational Method for the Minimum Duration Transfer Trajectories of Space-to-Ground Vehicles

On the conditions that the spacecraft engine is in finite thrust mode and the maneuver time is given, it takes a long time to compute the minimum duration transfer trajectories of space-to-ground vehicles, which is mainly because the initial values of the adjoint variables involved in the optimization model have no definite physical meanings and the model is sensitive to them. In order to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment for the decision makers, we propose a fast method for computing the minimum duration transfer trajectories of space-to-ground vehicles with the given position of the landing point and the arbitrary maneuver point. First, the optimization model based on the hybrid method is established to compute the minimum duration transfer trajectory. Then, the region composed of maneuverable points is gridded and the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectories at all gridded points are computed and saved to a database. Finally, the predicted values of the initial values of the adjoint variables and the values of partial state variables at any maneuver point within the region composed of maneuverable points are computed by using a binary cubic interpolation method. Finally, the minimum duration transfer trajectory is obtained by the hybrid method which takes the neighborhood of the predicted values as the search ranges of the initial values of the adjoint variables and the values of partial state variables. Simulation results demonstrate that the proposed method, which requires only 2.93% of the computational time of the hybrid method, can improve substantially the computational time of the minimum duration transfer trajectory of a space-to-ground vehicle under the guarantee of ensuring accuracy. The methodology of converting the time domain into the space domain is well applied in this paper.

A continuous adjoint-based aeroacoustic shape optimization for multi-mode duct acoustics

A multi-mode adjoint-based optimization method is proposed for the noise reduction optimization in multi-mode duct acoustics problems. The objective is to minimize the amplitude of sound from an inlet duct on the wall and integral line while maintaining the aerodynamic performance. The complete detailed derivation of the adjoint equations and their corresponding adjoint boundary conditions are presented firstly based on the multi-mode linear Euler equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. The sensitivity derivative computed by the continuous adjoint method is validated by comparing with that obtained using finite difference method. Up to 50 design variables are involved in the adjoint optimization to ensurely provide an adequate design space. And a quasi-Newton Broyden–Fletcher–Goldfarb–Shanno algorithm is utilized to determine an improved intake duct geometry based on the objective function gradient provided by the adjoint solution. Finally, two multi-mode optimization of a typical inlet duct confirms the flexibility of the multi-mode adjoint-based framework and the efficiency of the multi-mode adjoint-based technique.

Dynamics Response Control of a Mindlin-Type Beam

Optimal boundary control for damping the vibrations in a Mindlin-type beam is considered. Wellposedness and controllability of the system are investigated. A maximum principle is introduced and optimal control function is obtained by means of maximum principle. Also, by using maximum principle, control problem is reduced to solving a system of partial differential equations including state, adjoint variables, which are subject to initial, boundary and terminal conditions. The solution of the system is obtained by using MATLAB. Numerical results are presented in table and graphical forms.