Gradient Flows, Second-Order Gradient Systems and Convexity
Copyright policy )Gradient Flows, Second-Order Gradient Systems and Convexity
We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $��$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla��$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two $\mathcal{C}^2$, convex and bounded from below functions $��_1$, $��_2$: if $||\nabla��_1||=||\nabla��_2||$, then $��_1=��_2 + k$, for some $k\in \mathbb{R}$.
convexity, Nonlinear differential equations in abstract spaces, dynamical systems, 540, Variational inequalities, Asymptotic properties of solutions to ordinary differential equations, Gradient flows, Second-order gradient equations, 510, Monotone systems involving ordinary differential equations, Convexity, Optimization and Control (math.OC), gradient flows, 26B25, 34C12, 34D05, 34G20, 37C10, 49J40, Dynamical systems, Dynamics induced by flows and semiflows, second-order gradient equations, FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Convexity of real functions of several variables, generalizations
convexity, Nonlinear differential equations in abstract spaces, dynamical systems, 540, Variational inequalities, Asymptotic properties of solutions to ordinary differential equations, Gradient flows, Second-order gradient equations, 510, Monotone systems involving ordinary differential equations, Convexity, Optimization and Control (math.OC), gradient flows, 26B25, 34C12, 34D05, 34G20, 37C10, 49J40, Dynamical systems, Dynamics induced by flows and semiflows, second-order gradient equations, FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Convexity of real functions of several variables, generalizations
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