Output tracking problem for a class of 3-D hyperbolic PDEs
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Abstract
The output tracking problem (OTP) refers to the procedure of determining an input control that ensures the system output follows a specified trajectory. In this paper, we investigate the OTP for controlled systems governed by a class of 3-D hyperbolic partial differential equations (PDEs), aiming for an exact alignment between the system output and the desired trajectory. Utilizing backstepping techniques, semigroup theory, the Hahn-Banach theorem, Laplace transforms, and Green's functions, our approach begins by establishing the OTP solution as a bounded feedback law, with Green's functions allowing the solution of a Cauchy-Euler equation. This is followed by deriving a detailed formulation for the complete state of the corresponding control problem. Finally, the approach provides an explicit formula for the OTP solution, using the desired trajectory and the solutions of specific kernel PDEs.