{'fraction_non_alphanumeric': 0.022686025408348458, 'fraction_numerical': 0.0, 'mean_word_length': 5.231638418079096, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, '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}}
Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ℐ with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ℐ in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ℐ in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ℐ. Finally, we define partial quadratic functionals associated with ℐ and a (strong) regularity of ℐ, which we relate to the positivity and nonnegativity of ℐ.