Optimization of the integro-differential hyperbolic model

We consider a model of processes in viscoelastic media. The force arising inside the body may depend not only on the current values but also on the entire time history of the motion (for example, polymers, emulsions, and suspensions exhibit memory properties). Such processes often lead to the necessity of studying hyperbolic integro-differential equations with Volterra integral term. The main goal is to prove the existence and uniqueness of generalized solutions to the corresponding initial-boundary value problem, as well as to investigate the existence of optimal control for systems described by these models. The main results are obtained using the method of a priori inequalities in negative norms. In particular, we justify inequalities in negative norms for the integro-differential operator in certain functional spaces. The work begins with a short description of relevant results with similar formulations and referencing works with physical justification of the model. Afterwards, we describe the problem statement, constraints on equation parameters, and functional spaces used for the investigation. Below are the main results, in particular definitions of generalized solutions and properties on the well-posed of the initial-boundary value problem as well as the existence of optimal control. Control is carried out through the right-hand side of the equation using a certain special operator. It is possible to examine specific examples of such control operators (illustrating various control mechanisms) and the corresponding control spaces. By utilizing proven inequalities and relying on general results of S.I. Lyashko from the theory of a priori inequalities in negative norms, we establish sufficient conditions for the existence of optimal control. In particular, we impose the restrictions on the admissible set of controls and the quality criterion.