Similarity between two distributed parameter systems

The relationship between two distributed parameter systems can be linked by a homeomorphic mapping, and the core is to study the minimizer of the functional to measure the degree of their similarity. We prove the existence and the necessary conditions (a maximum principle) for the minimizer. The similarity degree between two distributed parameter systems is thus defined by the functional, which extends the conjugacy in dynamical systems. As applications, we consider parabolic systems that satisfy different similarities. We prove a Hartman-Grobman theorem for general parabolic systems. We also demonstrate asymptotic similarity for the general quasilinear parabolic systems, indicating the Clausius statement of the second law of thermodynamics. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.