Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

We consider the optimal control of solutions of first order Hamilton–Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.     

Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

ABSTRACT

In this paper, we study a distributed optimal control problem of a coupled nonlinear system of reaction–diffusion equations. The system consists of three partial differential equations to represent cancer cell density, matrix-degrading enzymes concentration and oxygen concentration, and an ordinary differential equation to describe the extracellular matrix concentration. Our aim is to minimize the growth of cancer cells by controlling the production of matrix-degrading enzymes. First, we prove the existence and uniqueness of solutions of the direct problem. Then, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and prove the existence of weak solutions of the adjoint problem.     

Dynamics for Controlled 2D Generalized MHD Systems with Distributed Controls

We study the dynamics of a piecewise (in time) distributed optimal control problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal distributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of Generalized MHD equations are derived. Next, the existence of a solution of optimal control problemis proved also optimality system is derived. Finally, the long-time decay properties for the optimal solutions is established.     

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.     

一类弱耦合动力系统的参数识别问题

研究一类弱耦合反应－扩散动力系统的参数识别问题。通过构造上下解，证明了反应－扩散方程组解的存在惟一性；给出了求解参数识别问题的最优化系，从而可以选取适当的梯度法或者共轭梯度法，实现对系统参数的识别。     

Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls

Summary. An optimal control problem for impressed cathodic systems in electrochemistry is studied. The control in this problem is the current density on the anode. A matching objective functional is considered. We first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition. We then prove the existence of an optimal solution. Next, we derive a necessary condition of optimality and establish an optimality system of equations. Finally, we define a finite element algorithm and derive optimal error estimates. Received March 10, 1993 / Revised version received July 4, 1994     

具有年龄结构的种群线性动力系统的最优收获控制问题

§ 1　Introduction and setting of the problemThe optimal control of age-dependent population dynamics has been intensivelystudied in the last two decades and there is now a vast stock of literature on the topic ofoptimal control problems ofage-structured population dynamics.(see [1 -9] ) .To the bestof our knowledge,the works of Brokate[3,4] are the firstto deal with this topic.Since then,many authors devote to the optimal harvesting problem.In this aspect,we refere to thefundamental papers o…     

一类海冰-海水耦合热力学系统参数辨识问题的最优性

针对一维的海冰-海水耦合热力学系统,以该系统中的物理参数为辨识量,以温度偏差为目标函数,建立了一个参数辨识模型,并证明了该问题最优解的存在性,从而为这类海冰-海水耦合热力学系统参数辨识问题的数值计算提供数学理论依据.     

一类由抛物-双曲耦合组支配的边界最优控制问题

本文讨论了一类由抛物-双曲耦合组支配的第三边值最优控制问题.证明了该问题最优控制的存在性及其bang-bang性质.     

Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons

We formulate an S-I-R (Susceptible, Infected, Immune) spatiotemporal epidemic model as a system of coupled parabolic partial differential equations with no-flux boundary conditions. Immunity is gained through vaccination with the vaccine distribution considered a control variable. The objective is to characterize an optimal control, a vaccine program which minimizes the number of infected individuals and the costs associated with vaccination over a finite space and time domain. We prove existence of solutions to the state system and existence of an optimal control, as well as derive corresponding sensitivity and adjoint equations. Techniques of optimal control theory are then employed to obtain the optimal control characterization in terms of state and adjoint functions. To illustrate solutions, parameter values are chosen to model the spread of rabies in raccoons. Optimal distributions of oral rabies vaccine baits for homogeneous and heterogeneous spatial domains are compared. Numerical results reveal that natural land features affecting raccoon movement and the relocation of raccoons by humans can considerably alter the design of a cost-effective vaccination regime. We show that the use of optimal control theory in mathematical models can yield immediate insight as to when, where, and what degree control measures should be implemented.     

Optimal control of parabolic variational inequalities

Optimal control of parabolic variational inequalities is studied in the case where the spatial domain is not necessarily bounded. First, strong and weak solutions concepts for the variational inequality are proposed and existence results are obtained by a monotone and a finite difference technique. An optimal control problem with the control appearing in the coefficient of the leading term is investigated and a first order optimality system in a Lagrangian framework is derived.     

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.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

Solvability for a Coupled System of Fractional p-Laplacian Differential Equations at Resonance

In this paper, by using the coincidence degree theory, the existence of solutions for a coupled system of fractional p-Laplacian differential equations at reso-nance is studied. The result obtained in this paper extends some known results. An example is given to illustrate our result.     

Optimal control of solutions of the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation

An optimal control problem for a second-order Sobolev type equation with a relatively polynomially bounded operator pencil is considered. We prove the existence and uniqueness of a strong solution of the Showalter-Sidorov problem for this equation. Necessary and sufficient conditions for the existence and uniqueness of an optimal control of such solutions are obtained. We study the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Existence of strong solutions and global attractors for the coupled suspension bridge equations

In this paper, we show the existence of the strong solutions for the coupled suspension bridge equations. Furthermore, existence of the strong global attractors is investigated using a new semigroup scheme. Since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.     

Global stabilization in nonlinear discrete systems with time-delay

A class of scalar nonlinear difference equations with delay is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are given. Applications in economics and other fields lead to consideration of associated optimal control problems. An optimal control problem of maximizing a consumption functional is stated. The existence of optimal solutions is established and their stability (the turnpike property) is proved.     

Interval optimal control problem in a Hilbert space

An optimal control problem for a system involving an interval parameter is considered. The concepts of a universal optimal state and a universal optimal control are introduced. The existence and uniqueness of a universal solution to the interval optimal control problem is proved, and an algorithm for its determination is presented. The interval optimal control problem for a system described by the boundary value problem for a second-order ordinary differential equation is solved as an example.     

Solvability for a coupled system of fractional differential equations at resonance

By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions for a coupled system of fractional differential equations at resonance. An example is given to illustrate our result.