Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator (original) (raw)

Second order evolution equations with time-dependent subdifferentials

Journal of Evolution Equations, 2007

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.

Constrained Evolution for a Quasilinear Parabolic Equation

Journal of Optimization Theory and Applications, 2016

In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy-Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set of the space of square-integrable functions. Then, we consider convex sets of obstacle or double-obstacle type and prove rigorously the following property: if the factor in front of the feedback control is sufficiently large, then the solution reaches the convex set within a finite time and then moves inside it.

Minimal time sliding mode control for evolution equations in Hilbert spaces

ESAIM: Control, Optimisation and Calculus of Variations

This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

On an Optimal control Problem for Parabolic Equations

Consideration was given to the problem of optimal control of parabolic equations. The existence solution of the considering optimal control parabolic problem is proved. The gradient of the cost functional by the adjoint problem approach is obtained. Lipschitz continuity of the gradient is derived.

Optimal control of an evolution hemivariational inequality involving history-dependent operators

Communications in Nonlinear Science and Numerical Simulation, 2021

In this paper we consider a class of feedback control systems described by an evolution hemivariational inequality involving history-dependent operators. Under the mild conditions, first, we prove a priori estimates of the solutions to the feedback control system. Then, an existence theorem for the feedback control system is obtained by using the wellknown Bohnenblust-Karlin fixed point theorem. Moreover, we consider an optimal control problem driven by the feedback control system, and establish its solvability. Finally, a parabolic partial differential system with Clarke subgradient term is considered to illustrate the applicability of the theoretical results.

Control problems governed by a pseudo-parabolic partial differential equation

Transactions of the American Mathematical Society, 1979

We consider the solution^«) of the pseudo-parabolic initial-value problem (1 + M(x))y,(u) + L(x)y(u) = u in L2(Q), y(-,0;u) = 0 in L2(G), to be the state corresponding to the control u. Here M(x) and L{x) are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control un in a fixed ball in L\Q) such that (i) the endpoint of the corresponding statey(-, T; u0) lies in a given neighborhood of a target Z in L2(G) and (ii) u0 minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.

Convergence of optimal control problems governed by second kind parabolic variational inequalities

Journal of Control Theory and Applications, 2013

We consider a family of optimal control problems where the control variable is given by a boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem. This work solves the open problem left by the authors in IFIP TC7 CSMO2011.