The authors adapt results of a previous paper [C. Z. Xu, P. Ligarius, and J. P. Gauthier, Comput. Math. Appl. 29, No. 7, 13-21 (1995; Zbl 0829.93006)], dealing with an observer for the general infinite-dimensional dissipative bilinear distributed system
over a separable Hilbert space with scalar product to prove the convergence of the observer. The following candidate observer
for the system (1) is proposed. All the solutions of the systems considered throughout the paper are in the weak sense. Using the class of regularly persistent inputs which preserve some uniform observability, with respect to time, the convergence property of a very simple Luenberger-like observer is proved: Given a regularly persistent positive input the estimation error converges weakly to zero in when time goes to infinity. This result means, roughly speaking, that regularly persistent inputs are sufficiently rich to preserve an asymptotic “amount of observability” on some bounded intervals ensuring the convergence of the observer. The proposed observer is applied for a heat exchanger process. Some numerical simulations are presented to illustrate the convergence property of the designed observer.