OPTIMAL CONTROL FOR SYSTEMS MODELED BY THE DIFFUSION-WAVE EQUATION

The article deals with an optimal control problem for the model system described by the one-dimensional inhomogeneous diffusion-wave equation that is a generalization of the wave equation to the case when the time derivative is replaced with the fractional Caputo derivative. In the general case, we consider both boundary and distributed controls which are Lebesgue p-summable functions, with p > 1 and p = 8. We state and study the two types of optimal control problems: The problem of finding a minimal norm control for a given control time and the performance problem of finding a control that brings the system to a given state in the minimal time for a given constraint on the control norm. The study bases on using an exact solution to the diffusion-wave equation, which allows us to reduce the optimal control problem to an infinite-dimensional l-moment problem. We also examine the similar finite-dimensional l-moment problem that uses an approximate solution to the diffusion-wave equation and analyze the well-posedness and solvability of this problem. Also, we exhibit some example of calculating the boundary control by using the finite-dimensional l-moment problem.