{'fraction_non_alphanumeric': 0.054474708171206226, 'fraction_numerical': 0.0038910505836575876, 'mean_word_length': 4.718518518518518, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stability analysis of Euler‐Bernoulli beam with input delay in the boundary control

In this paper, we investigate an Euler‐Bernoulli system with input delay in the boundary control. Suppose that there is no delay in observation, y(t), of the system, and a partial input delay in the boundary control. The collocated boundary feedback control law u(t) = αy(t) + βy(t − τ) is applied to obtain the closed loop system. By spectral analysis and Lyapunov method, we show that: when α>|β|, the closed loop system is exponentially stable for any τ>0; when α<|β|, the system is unstable for any τ>0; when α = |β|, the system is asymptotically stable for almost all τ>0. Finally, we provide numerical simulations to show the spectral distribution for different values of α and β.