A class of boundary control systems with boundary observation is considered, for which the unbounded operators often lead to technical difficulties. An extended model for this class of systems is described and analyzed, which involves no unbounded operator except for the dynamics generator. A method for the resolution of the LQ-optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators, in the sense that, in closed-loop, the state trajectories converge to zero exponentially fast. The model consists of an extended abstract differential equation whose state components are the boundary input, the state (up to an affine transformation) and a Yosida-type approximation of the output of the nominal system. It is shown that, under suitable conditions, the model is well-posed and, in particular, that the dynamics operator is the generator of a C0-semigroup. Moreover, the model is shown to be observable and to carry controllability, stabilizability and detectability properties from the nominal system. A general method of resolution based on the problem of spectral factorization of a multi-dimensional operator-valued spectral density is described in order to solve a LQ-optimal control problem for this model. It is expected that this approach will lead hopefully to a good trade-off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems.
- Distributed-parameter systems
- LQR control method
- Spectral factorization