Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator (original) (raw)
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.
Quasi-Subdifferential Operators and Evolution Equations
Discrete and Continuous Dynamical Systems, 2013
ABSTRACT We introduce the concept of a quasi-subdifferential operator and that of a quasi-subdifferential evolution equation. We prove the existence of solutions to related problems and give applications to variational and quasi-variational inequalities.
Mathematics of Computation, 1982
Let 3Í> be the subdifferential of some lower semicontinuous convex function <t of a real Hubert space H, f E L2(0, T; H) and u" a continouous piecewise linear approximate solution of du/dt + 3i>(u) Bf, obtained by an implicit scheme. If u0 E Dom(), then du"/dt converges to du/dt in L2(0, T; H). Moreover, if u0 £EDom(33>), we construct a step function ij"(t) approximating / such that lim"^ + x /0r r\n \ dujdt-du/dt |2 dt = 0. When 0 is inf-compact and when the sequence of approximation of/is weakly convergent to/, then u" converges to u in C([0, T]; H) and t]"dun/dt is weakly convergent to tdu/dt.
Neumann boundary optimal control problems governed by parabolic variational equalities
2021
We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα. We find a explicit form for the optimal controls, we prove monotony properties and we obtain...
Evolution equations for nonlinear degenerate parabolic PDE
Nonlinear Analysis: Theory, Methods & Applications, 2006
We define a convex function on H −1 ( ) and characterize its subdifferential, by which we introduce an evolution equation as a weak formulation of a class of nonlinear, degenerate, singular and noncoercive parabolic PDEs associated with an arbitrary maximal monotone graph and with the Dirichlet boundary condition. ᭧
Nonlinear Analysis: Modelling and Control, 2022
The objective of our paper is to investigate the optimal control of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the optimal control results of the system are obtained using the C0-semigroup approach, fixed point theorem, and some other simple conditions on the nonlinear term as well as on operators involved in the model.