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On the stable equilibrium points of gradient systems. (English) Zbl 1129.34320

Summary: This paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f. In particular, it is shown that, under the assumption that f is real analytic, local minimality is necessary and sufficient for stability. Under the weaker assumption that f is indefinitely continuously differentiable, local minimality is neither necessary nor sufficient for stability.

MSC:

34D20 Stability of solutions to ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
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