STABILIZATION IN A GRADIENT SYSTEM WITH A CONSERVATION LAW

Suppose SIGMA-mu(j) = 1 and F:R --> R is C1 with F' piecewise C1. For the finite system of ordinary differential equations u(i) = F'(u(i)) [ GRAPHICS] mu(j)F'(u(j)) = 0, I prove that every bounded solution stabi lizes to some equilibrium as t --> infinity. For this system, SIGMA-mu (j)u(j) is conserved and the quantity SIGMA-mu(j)F(u(j)) is nonincreas ing and serves as a Lyapunov function, but the set of equilibria can b e connected and degenerate. Essential use is made of a result related to one of Hale and Massat that an omega-limit set that lies in a C1 hy perbolic manifold of equilibria must be a singleton.