Optimal Control of an Infinite Order Hyperbolic System with Multiple Time-Varying Lags

Distributed parameter systems with delays can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elastic-plastic material, fluid dynamics, diffusion-reaction processes, transmission of the signals at the certain distance by using electric long lines, etc., all lie within this area. The object that we are studying (temperature, displacement, concentration, velocity, etc.) is usually referred to as the state. We are interested in the case where the state satisfies proper differential equations that are derived from basic physical laws, such as Newton’s law, Fourier’s law etc. The space in which the state exists is called the state space, and the equation that the state satisfies is called the state equation. In particular, we are interested in the cases where the state equations are one of the following types: partial differential equations, integro-differential equations, or abstract evolution equations. Equations with deviating arguments appeared in the Euler’s works. However , systematic research of such equations began only in the 20th century, as a result of the development of applied sciences and particularly automatic control theory. Consequently, equations with deviating arguments are a well-known mathematical tool for representing many physical problems . Historically, they have achieved great popularity among mathematicians, physists and engineers. During the recent twenty years, equations with deviating arguments have been applied not only in applied mathematics, physics and automatic control, but also in some problems of economy and biology. Currently, the theory of the equations with deviating arguments, constitutes a very important subfield of mathematical control theory. Consequently, the equations with deviating arguments are widely applied in the optimal control problems of distributed parameter systems with time delays.