In this paper we study asymptotic properties, primarily convergence to zero, of weak extremals of variational problems on unbounded intervals. The type of problems examined is δ∫∞Rg(r)F(r,u,u′)dr=0, where growth conditions on g(r), with convexity and regularity conditions on F, imply the convergence. The conditions listed in the paper are natural and are satisfied by large classes of functions. A typical growth condition on g which implies the desired convergence u′(r)→0 and u(r)→0 as r→∞ is the existence of 0<k<β and K such that rg′(r)g(r)−1≥β and ∫rRsk−1g′(s)−1ds≤Krk for all r∈[R,∞). The method of proof involves transformations on the Euler-Lagrange equation which determine the extremals, allowing Lyapunov-like arguments. Several results of this type are presented, and particular cases and examples are treated, indicating the extent and accuracy of the conditions.
Stability properties for solutions of general Euler-Lagrange systems
PUCCI, Patrizia
1992
Abstract
In this paper we study asymptotic properties, primarily convergence to zero, of weak extremals of variational problems on unbounded intervals. The type of problems examined is δ∫∞Rg(r)F(r,u,u′)dr=0, where growth conditions on g(r), with convexity and regularity conditions on F, imply the convergence. The conditions listed in the paper are natural and are satisfied by large classes of functions. A typical growth condition on g which implies the desired convergence u′(r)→0 and u(r)→0 as r→∞ is the existence of 0I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
