On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

Previously, boundary control problems for the second order parabolic type equation in the bounded domain were studied. In this paper, a boundary control problem associated with a fourth-order parabolic equation in a bounded two-dimensional domain was considered. On the part of the considered domain’s boundary, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. By the method of separation of variables the given problem is reduced to a Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method and an estimate was found for the minimal time at which the given average temperature in the domain is reached

Global Carleman estimates for the fourth order parabolic equations and application to null controllability

The main objective of this paper is to establish the null controllability for the fourth order semilinear parabolic equations with the nonlinearities involving the state and its gradient up to second order. First of all, based on optimal control theory of partial differential equations and global Carleman estimates obtained in \cite{gs} for fourth order parabolic equation with $L^2(Q)$-external force, we establish the global Carleman estimates for the $L^2(Q)$ weak solutions of the same system with a low regularity external term and some linear terms including the derivatives of the state up to second order. Then we prove the null controllability of the fourth order semilinear parabolic equations by such global Carleman estimates and the Leray-Schauder's fixed points theorem.Comment: arXiv admin note: text overlap with arXiv:2211.0042

Insensitizing controls for a fourth order semi-linear parabolic equations

This paper is concerned with the existence of insensitizing controls for a fourth order semilinear parabolic equation. Here, the initial data is partially unknown, we would like to find controls such that a specific functional is insensitive for small perturbations of the initial data. In general, this kind of problems can be recast as a null controllability problem for a nonlinear cascade system. We will first prove a null controllability result for a linear problem by global Carleman estimates and dual arguments. Then, by virtue of Leray-Schauder's fixed points theorem, we conclude the null controllability for the cascade system in the semi-linear case.Comment: arXiv admin note: text overlap with arXiv:2211.00428, arXiv:2211.0064

Hierarchic control for the coupled fourth order parabolic equations

In this paper, we obtain a null controllability result for a coupled fourth order parabolic system based on the Stackelberg-Nash strategies. For this purpose, we first prove the existence and uniqueness of Nash equilibrium pair of the original system and its explicit expression is provided. Next, we investigate the null controllability of Nash equilibrium to the corresponding optimal system. By duality theory, we establish an observability estimate for the coupled fourth order parabolic system. Such an estimate is obtained by a new global Carleman estimate we derived