Noriaki Yamazaki

Local solvability of a constrained gradient system of total variation

A 1-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in R^N is formulated by use of subd-ifferentials of a singular energy - the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result a local-in-time solution of 1-harmonic map flow equation is constructed as a limit of the solutions of p-harmonic (p > 1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.

Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.

A Boundary Control Problem for the Equation and Dynamic Boundary Condition of Cahn–Hilliard Type

A dynamic boundary condition is a type of partial differential equation that describes the dynamics of a system on the boundary. Combining with the heat equation in a smooth-bounded domain, the characteristic structure of “total mass conservation” appears, namely, the volume in the bulk plus the volume on the boundary is conserved. Based on this interesting structure, an equation and dynamic boundary condition of Cahn–Hilliard type was introduced by Goldstein–Miranville–Schimperna. In this paper, based on the previous work of Colli–Gilardi–Sprekels, a boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type is considered. The optimal boundary control that realizes the minimal cost under a control constraint is determined, and a necessary optimality condition is obtained.

Optimal Control Problem of Positive Solutions to Second Order Impulsive Differential Equations

In this paper, we consider optimal control problem of second order impulsive differential equations. We show the existence and uniqueness of positive solutions to our problem for each given control functions. Also, we consider the control problem of positive solutions to our equations. Then, we prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover we give an example of the main results.

WELL-POSEDNESS AND PERIODIC STABILITY FOR QUASILINEAR PARABOLIC VARIATIONAL INEQUALITIES WITH TIME-DEPENDENT CONSTRAINTS

We study variational inequalities with time-dependent constraints for quasilinear parabolic PDE of divergence from. Introducing a general condition on the constraints, we prove existence. uniqueness and order property of solution. Some applications are given.

Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation

We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data.

Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

Attractors for nonautonomous multivalued evolution systems generated by time-dependent subdifferentials

In a real separable Hilbert space, we consider nonautonomous evolution equations including time-dependent subdifferentials and their nonmonotone multivalued perturbations. In this paper, we treat the multivalued dynamical systems associated with time-dependent subdifferentials, in which the solution is not unique for a given initial state. In particular, we discuss the asymptotic behaviour of our multivalued semiflows from the viewpoint of attractors. In fact, assuming that the time-dependent subdifferential converges asymptotically to a time-independent one (in a sense) as time goes to infinity, we construct global attractors for nonautonomous multivalued dynamical systems and its limiting autonomous multivalued dynamical system. Moreover, we discuss the relationship between them.

New Class of Doubly Nonlinear Evolution Equations Governed by Time-Dependent Subdifferentials

We discuss a new class of doubly nonlinear evolution equations governed by time-dependent subdifferentials in uniformly convex Banach spaces, and establish an abstract existence result of solutions. Also, we show non-uniqueness of solution, giving some examples. Moreover, we treat a quasi-variational doubly nonlinear evolution equation by applying this result extensively, and give some applications to nonlinear PDEs with gradient constraint for time-derivatives.

Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem.