Observers for hyperbolic systems with multiple delays in the nonlocal boundary condition and its application to secure image encryption

In this article, we propose a step-by-step observer design for a first-order linear hyperbolic system with multiple delays in the nonlocal boundary condition. In the proposed approach inspired by the backstepping methodology, the observer gains are obtained following seven successive steps. Next, we present the application of this observer design in the context of image encryption. We consider a system where a local nonlinear term causing a chaotic vibration is added to the nonlocal boundary condition with time lags. A synchronization system for this chaotic system can be constructed using the same observer design method. The chaotic states generated by both systems can be used to encrypt and decrypt image data. The time lags, and weight functions included in the nonlocal boundary condition play the role of common encryption keys. By considering a system that includes arbitrarily many delays, this approach allows for a significant increase in the number of common encryption keys compared to conventional methods, resulting in greatly improved communication security.