Controllability conditions for evolutionary systems of linear partial differential equations - Makarov, Alexander; Nikolenko, Iryna (Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 2025)
Many works are devoted to control theory but most of them are related to ordinary differential equations. Of the partial differential equations, mathematical physics equations are most often considered, for example, wave equations. In the article by Makarov O.A. "Controllability of an evolutionary system of partial differential equations. Visnyk of V. N. Karazin Kharkiv National University, Series "Mathematics, Applied Mathematics and Mechanics", 2016, Vol. 83. pp. 47-56" a system of partial differential equations has previously been considered. The complete controllability of such systems in the L. Schwartz space under certain control conditions has been investigated in this work. In particular, it was proved that if the eigenvalues of the system matrix are real, then there is a time-independent control. In addition, the case when the eigenvalues are imaginary was investigated. The purpose of this article is to study the controllability of a system of linear partial differential equations for an arbitrary matrix of the system under the constraints on the search for control in the form u(x,t)=u(x)·(-α t), where the vector of the function u(x) belongs to the L. Schwartz space. Necessary and sufficient conditions for the complete controllability of this system are obtained and examples of both controllable and uncontrollable systems are given. As a consequence of this theorem, sufficient conditions for the complete controllability of the system are obtained, the real parts of the eigenvalues of the matrix P(s) are bounded from above or below. In addition, it is proved that if the spatial variable belongs to the space R, then such a system is completely controllable. Examples are given for each case. A partial differential equation of the second order in time is also considered. It has been proven that if the roots of the characteristic equation satisfy the conditions of the criterion, then this equation is completely controllable. Thus, the Helmholtz equation are completely controllable.