The control problem for a system of linear loaded differential equations with nonseparated multi-point intermediate conditions

The possibilities of modern computational and measurement techniques allow using the most adequate mathematical models of control of the considered dynamic processes depending on their adequate practical purpose. A mathematical description of various dynamic processes of control in which the future flow of processes depends not only on the present but also is significanlty determined by the history of the process, is performed by using ordinary differential equations with memory of various types, also called equations with aftereffect or loaded differential equations. In this work the problem of control and optimal control of a system of linear loaded differential equations is considered, for which, along with classical boundary (initial and terminal) conditions, nonseparated multipoint intermediate conditions are given. It is assumed that at the loading points the phase-state function of the system has left-side limits and some nonseparated multipoint conditions are satisfied. Similar problems arise, for example, when, during the observation of a dynamic process, phase states are measured at some moments of time and the information is continuously transmitted by feedback. These problems have important practical and theoretical value, hence the need for their investigation in various formulations arises naturally. In this work necessary and sufficient conditions for complete controllability for the considered system of linear loaded differential equations is formulated. A constructive approach for the solution of control problem is given and conditions for the existence of program control and motion are formulated. An explicit form of the control action for the control problem is constructed and a method for solving the optimal control problem is proposed. An analitical form of the control action for the control problem is constructed, as well as an approach for solving the optimal control problem is proposed.