Weak Solutions of a Reaction Diffusion System with Superdiffusion and Its Optimal Control

The existence and uniqueness of weak solutions to the 2-dimensional reaction diffusion system with superdiffusion and the optimal control of such model are investigated in this paper. Fractional function spaces, Galerkin ap- proximation method and Gronwall inequality are used to obtain the existence and uniqueness of weak solutions. On this basis, an optimal control prob- lem of such superdiffusive system is further considered by using the minimal sequence.     

An optimal birth control problem for a dynamical population model with size-structure

This work is concerned with an optimal control problem for a size-structured population model, which takes fertility as the control variable. The existence and uniqueness of solutions to the basic state system and the dual system are proven via the Banach fixed point theorem. Necessary optimality conditions of first order are established in the form of an Euler-Lagrange system by the use of tangent-normal cone technique. The existence of a unique optimal controller is established by means of Ekeland’s variational principle. An example and some comments are presented.     

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

A turbidostat is an apparatus with feedback control system used to continuously culturing microorganisms. The dilution rate of the turbidostat can be regulated by the control system when the concentration of microorganism, detected by photoelectricity system or other devices, reaches a preset value. Based on the design ideas of the turbidostat, a differential equation with impulsive state feedback control is proposed for a kind of turbidostat system in this paper. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of periodic solution of order one are obtained according to the preset value and the types of the positive equilibrium of the corresponding system without impulsive control. Furthermore, it is pointed out that the system either tends to a stable state or has a periodic solution. Finally, the theoretical results are verified by numerical simulations.     

Synthesis of nonlinear control systems invariant under perturbations

For nonlinear control systems with perturbations, we consider the problem of synthesis of perturbation-invariant characteristics (invariant functions) with the use of feedbacks. The existence of invariant functions is related to a decomposition of the control system for which the quotient system is independent of perturbations. We present conditions for the existence of such quotient systems, which are certain systems of partial differential equations. The synthesizing controls are found from these equations.     

Mild Solution and Fractional Optimal Control of Semilinear System with Fixed Delay

This paper considers fractional optimal control of a semilinear system with fixed delay in a reflexive Banach space. The existence and uniqueness of mild solution are obtained using the Weissinger’s fixed point theorem. The existence of optimal control for the system governed by fractional-order semilinear equation with fixed delay in state is presented. To show the effectiveness of the developed theory, an example is given.     

Robust State-Feedback Controllability of Linear Systems to a Hyperplane in a Class of Bounded Controls

The paper deals with a linear time-dependent dynamic system with scalar control and input uncertainty (disturbance). Two admissible classes of input uncertainty realizations are considered: the class of measurable bounded functions and the class of measurable quadratically integrable functions. The problem to be studied is the existence of a state feedback control with measurable bounded time realizations transferring the system to a given hyperplane (a target set) from any initial position in a prescribed time for any admissible input uncertainty realization. Necessary and sufficient conditions for the existence of such a control are derived, based on the explicit construction of this control by using an auxilary zero-sum linear-quadratic differential game with a cheap control for the minimizing player. Examples illustrting the theoritical results are presented.     

Dynamic analysis of a turbidostat model with the feedback control

In this paper, a mathematical model with impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of positive order one periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentration of microorganism and substrate. By investigating the periodic solution, the period and the initial point of the periodic solution are given. The results show that turbidostat with impulsive state feedback control tends to an order one periodic solution.     

An optimal control problem for a pest, predator, and plant system

The paper is devoted to an optimal control problem for a system of three nonlinear parabolic equations from population dynamics. The equations model a trophic chain consisting of a predator, a pest and a plant species. The existence and uniqueness of the positive solution for the system are proved. The control variable is connected with the action of a pesticide. Our goal is to minimize the density of the pest and to maximize the plant density. The existence of the optimal solution is proved. The first and second order optimality conditions are established.     

非线性经济增长模型的最优控制问题

研究了带有中性技术进步生产函数边界条件的非线性经济增长模型的最优控制问题.利用Banach空间不动点原理,得到了系统解的存在唯一性,利用Gronwall不等式得到了系统解关于控制序列的连续依赖性,借助于法锥和共轭系统,得到了控制问题最优解存在的必要条件.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.     

具终端观测且与年龄相关的非线性种群扩散系统的最优控制

讨论了一类具终端观测且与年龄相关的非线性时变种群扩散系统的最优分布控制问题利用偏微控制理论和先验估计,证明了系统最优分布控制的存在性,得到了控制为最优的一阶必要条件,并进而讨论了系统的最优反馈控制问题.     

Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control

In this study the optimal control of fed-batch glycerol fermentation is investigated based on an impulsive dynamical system. Considering the sudden increase of the glycerol and alkali in fed-batch culture of biodissimilation of glycerol to 1,3-propanediol, this paper proposes a nonlinear impulsive system of fed-batch culture. The existence, uniqueness and regularity properties of piecewise solution for the system are proved. In view of the controllability of volumes of glycerol added to the reactor instantaneously, the paper constructs an optimal control model based on the nonlinear impulsive system and the existence of the optimal control is obtained. The control variables here are the moments and the sizes of jumps in the states at the discrete instants and the objective is to maximize the productivity of 1,3-propanediol over one cycle.     

Nonlinear optimal control problems of degenerate parabolic equations with logistic time-varying delays of convolution type

In this article, we consider a bioeconomic model for optimal control problems which are governed by degenerate parabolic equations governing diffusive biological species with logistic growth terms and multiple time-varying delays. The time-varying delays are given in a convolution form. The existence, uniqueness and regularity results to the state equations with homogeneous Dirichlet and Neumann boundary conditions are established. The vanishing viscosity method is used to obtain the existence result. Afterwards, we formulate the optimal control problem in two cases. Firstly, we suppose that this biological species causes damage to environment (e.g. forest, agriculture): the optimal control is the trapping rate and the cost functional is a combination of damage and trapping costs. Secondly, an optimal harvesting control of a biological species is considered: the optimal control is a distribution of harvesting effort on the biological species and the cost functional measure the difference between economic revenue and cost. The existence and the condition of uniqueness of the optimal solution are obtained. A nonlinear optimality system is derived, characterizing the optimal control.     

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Conditions for the Existence of Two Limit Cycles in a System with Hysteresis Nonlinearity

This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.     

Quadratic control problems

The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.     

The existence results for optimal control problems governed by a variational inequality

In this paper, we establish the existence of the optimal control for an optimal control problem where the state of the system is defined by a variational inequality problem with monotone type mappings. Moreover, as an application, we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle.     

有两种可修复方法的复杂可修复系统的最优控制

研究了具有两种可修复方法的复杂可修复系统的最优控制问题,首先将此类系统方程转化为对应的Volterra积分方程的形式,然后利用算子半群理论证明了系统解的存在唯一性,再利用范数指标函数作为衡量控制变量的标准,研究有此类系统的最优控制问题,证明了对应的最优控制问题的解的存在唯一性.     

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.     

Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory *

For the d–dimensional reflecting stochastic differential equations (1) with non-smooth boundary and unbounded domain the existence of a strong solution, (weak solution) is obtained under the conditions that the coefficients are less than linear growth and they are non-Lipschitz, (and the diffusion coefficient is non-degenerate, the drift coefficient is bounded and measurable only). Moreover, the Girsanov theorem and the martingale representation theorem with respect to system (1) are also derived. Then by using the Ekeland lemma and the martingale method the existence, necessary and sufficient conditions for an optimal control and an optimal control are obtained. The results are then applied to solve an optimal control problem for a stochastic population model