**Autors:** Perev, K. L.
**Title:** System Gramians Evaluation for Hyperbolic PDEs
**Keywords:** distributed parameter system, hyperbolic partial differential equation, time-space separation, method of Fourier, vibrating string, vibrating beam, controllability and observability gramians**Abstract:** The paper considers the problem of gramians computation for linear hyperbolic distributed parameter systems. Two different cases are considered: vibrating string and beam systems. The presented approach is based on directly deriving the equations solutions by using
time - space separation of variables and the Fourier series representation method. The initial problem framework is based on the state space formulation for infinite dimensional systems. This framework uses Riesz-spectral operators defined over Hilbert spaces and implements the concept of a $C_0$ – strongly continuous semigroup generated by bounded system operator. The solution of the hyperbolic partial differential equations is divided in two parts. The zero input part is due to the initial conditions and participates in obtaining the observability gramian of the system. The zero state part is a consequence of the input signal effect and is used to compute the controllability gramian.
**References**
**Issue**
| IFAC-PapersOnLine, vol. 55, issue 34, pp. 96 - 101, 2022, Canada, Elsevier |
Copyright IFAC-PapersOnLine Full text of the publication |