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The aim of this work is to reconstruct a spacewise dependent heat source in a degenerate heat equation using the final measurement temperature. Based on the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established, and the global uniqueness and stability of the minimizer are also obtained from the necessary condition. Finally, a numerical algorithm on the basis of the conjugate gradient method is designed to treat the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm designed in the paper is quite efficient and stable, and the unknown heat source is recovered very well. [ABSTRACT FROM AUTHOR]

A complex mode-locking (entrainment) topology underlying the continuous stirred tank reactor reaction model subjected to impulsive perturbations is identified. Employing high-resolution stability diagrams, we exhibit the global structure of mode-locking oscillations and describe how they are interconnected and how their complexity unfolds with control parameters varying. The scenarios shown in the bi-parametric planes revealed that the skeleton of Arnold's tongues is organized according to the symmetric Stern-Brocot sum tree. Moreover, the mode-locking organization is controlled by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations. Interestingly, the mode-locking order is unfolded in an elusive way, that is, in perfect agreement with the reciprocal of the Stern-Brocot sum tree. The findings reported here contribute to providing a description and classification of mode-locking oscillations for the impulsive system.