Backstepping Boundary Control of First-order Coupled Hyperbolic Partial Integro-differential Equations

We consider a feedback control problem of first-order coupled hyperbolic partial integro-differential equations with distributed and boundary inputs. As the distributed inputs to the system, the output feedback control is first applied under zero boundary inputs. Then, we apply a backstepping method to the design of the boundary inputs. Our main result shows that, for any initial data belonging to a linear space, the solution of the system becomes zero in finite time under both the output feedback control and the control law derived by the backstepping method. Further, the exact formula of feedback solutions is presented. Two applications to the parallel-flow heat exchange equations and the reactor equations of plug-flow type are given.