Abstract
This paper presents a complete characterization of the local dynamics for optimal controlproblems in three‐dimensional systems of ordinary differential equations by using geometricalmethods. The particular structure of the Jacobian implies that the sixth‐order characteristicpolynomial is equivalent to a composition of two lower‐order polynomials, which are solvableby radicals. The classification problem for local dynamics is addressed by finding partitions,over an intermediate three‐dimensional space, which are homomorphic to the subspaces tangentto the complex, center and stable sub‐manifolds. The main results are: a local stability theoremand necessary conditions for the existence of fold, Hopf, double‐fold and fold‐Hopf bifurcations.
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Brito, P. Local dynamics for optimal control problemsof three‐dimensional ODE systems. Annals of Operations Research 89, 195–214 (1999). https://doi.org/10.1023/A:1018923623035
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DOI: https://doi.org/10.1023/A:1018923623035