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 of the bidomain system (II): uniqueness and regularity theorems for weak solutions

Motivated by the study of related optimal control problems, weak and strong solution concepts for the bidomain system together with two-variable ionic models are analyzed. A key ingredient for the analysis is the bidomain bilinear form. Global existence of weak and strong solutions as well as stability and uniqueness theorems is proven.     

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.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

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.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Hilbert space-valued forward-backward stochastic differential equations with Poisson jumps and applications

In this paper, we study a class of Hilbert space-valued forward-backward stochastic differential equations (FBSDEs) with bounded random terminal times; more precisely, the FBSDEs are driven by a cylindrical Brownian motion on a separable Hilbert space and a Poisson random measure. In the case where the coefficients are continuous but not Lipschitz continuous, we prove the existence and uniqueness of adapted solutions to such FBSDEs under assumptions of weak monotonicity and linear growth on the coefficients. Existence is shown by applying a finite-dimensional approximation technique and the weak convergence theory. We also use these results to solve some special types of optimal stochastic control problems.     

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.     

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.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

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。     

OPTIMAL CONTROL FOR LINEAR SYSTEM WITH QUADRATIC INDEFINITE CRITERION ON HILBERT SPACES

In this paper we consider optimal control problems for linear system on real separableHilbert spaces with quadratic criterion,in which the state weighted operators are indefinite.Wellposedness and solvability,existence and uniqueness of optimal control are discussed.Weprove that the closed-loop syntheses of optimal control are state linear feedback.Existence ofsolutions of related operator Riccati equations is investigated.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

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.     

Global attractor for the regularized Bénard problem

In this paper, we study the asymptotic behaviour of weak solutions for the regularized Bénard problem. We establish the global existence and uniqueness of weak solutions of this problem and give a proof for the existence of global attractor in the three-dimensional case for this system.     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non‐uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.     

Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.     

Externality and Hamilton-Jacobi equations

The relationship between optimal control problems and Hamilton-Jacobi-Bellman equations is well known [9]. In fact the value function, defined as the infimum of the cost functional, satisfies in the viscosity sense an appropriate Hamilton-Jacobi-Bellman equation. In this paper we consider several control problems such that the cost functional associated to each problem depends explicitly on the value functions of the other problems. This leads to a system of Hamilton-Jacobi-Bellman equations. This is known, in economic context [14] cap XI, as an externality problem. In these problems may occur a lack of uniqueness of the value functions. We give conditions to ensure existence, uniqueness of the value functions and an implicit integral representation formula. Moreover, under uniqueness assumption, we prove that the variational solutions of the associated Hamilton-Jacobi system converge asymptotically to the value functions. We prove also an uniqueness theorem in the case of viscosity solutions of Hamilton-Jacobi-Bellman system.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Weak solutions in nonlinear poroelasticity with incompressible constituents

We consider quasi-static nonlinear poroelastic systems with applications in biomechanics and, in particular, tissue perfusion. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. The system under consideration represents a nonlinear, implicit, degenerate evolution problem, which falls outside of the well-known implicit semigroup monotone theory. Previous literature related to proving existence of weak solutions for these systems is based on constructing solutions as limits of approximations, and energy estimates are obtained only for the constructed solutions. In comparison, in this treatment we provide for the first time a direct, fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solutions of the associated linear system (where the permeability is given as a function of space and time). The uniqueness proof for the associated linear problem is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions. The results of this work provide a foundation for addressing strong solutions, as well as uniqueness of weak solutions for nonlinear poroelastic systems.