Review of some new results in the control theory of classical distributed systems and systems with integral memory

The report will address the issues of controllability for some classical systems of mechanics (membranes, plates) and systems with integral memory. The control can be applied to the boundary of the domain occupied by the system, to a part or to the entire domain. In some cases, an absolute value restriction is additionally imposed on the control function. In all cases, the purpose of control is to drive the mechanical system to rest in a finite time. If this goal is achievable for classical systems, then for systems with memory, as a rule, uncontrollability is observed. Moreover, it is possible to describe a whole class of mechanics models for which, for example, the property of boundary controllability is missing. There are examples of integral memory equations that are controllable for the entire domain occupied by the system. There are also opposite examples when there is no controllability even for the entire domain. The latter property is associated with special kernels in the integral term of the equation, which are Abelian-type kernels or series of decreasing exponential functions with slowly increasing exponents.