Rate of Convergence to Equilibrium and Łojasiewicz-Type Estimates

A well known result states that the Łojasiewicz gradient inequality implies some estimates of the rate of convergence to equilibrium for solutions of gradient systems. We generalize this result to gradient-like systems satisfying certain angle condition and Kurdyka–Łojasiewicz inequality and to even more general situation. We apply the results to a broad class of second order equations with damping.

[1]  J. Bolte,et al.  Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity , 2009 .

[2]  Alain Haraux,et al.  APPLICATIONS OF THE ŁOJASIEWICZ–SIMON, GRADIENT INEQUALITY TO GRADIENT-LIKE EVOLUTION EQUATIONS , 2009 .

[3]  C. Lageman.  Pointwise convergence of gradient‐like systems , 2007 .

[4]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[5]  K. Kurdyka.  On gradients of functions definable in o-minimal structures , 1998 .

[6]  R. Chill,et al.  Every ordinary differential equation with a strict Lyapunov function is a gradient system , 2012 .

[7]  Ralph Chill,et al.  On the Łojasiewicz–Simon gradient inequality , 2003 .

[8]  A. Haraux,et al.  Convergence of Solutions of Second-Order Gradient-Like Systems with Analytic Nonlinearities , 1998 .

[9]  Alain Haraux,et al.  Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity , 2001 .

[10]  Alberto Fiorenza,et al.  Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations , 2006 .

[11]  Convergence of Global and Bounded Solutions of a Second Order Gradient like System with Nonlinear Dissipation and Analytic Nonlinearity , 2008 .

[12]  E. Fašangová,et al.  Convergence to equilibrium for solutions of an abstract wave equation with general damping function , 2016 .

[13]  Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy , 2011 .

[14]  J. Bolte,et al.  On damped second-order gradient systems , 2014, 1411.8005.