An optimal class of non-degenerate potentials for second-order ordinary differential equations

作者：Shen Kaiming; Zhang Meirong^*

来源：Boundary Value Problems, 2015, 2015(1): 189.

DOI：10.1186/s13661-015-0451-0

• non-degenerate potential
• eigenvalue
• optimal control
• Pontryagin maximum principle
• boundary value problem

By considering the Dirichlet boundary condition x(0) = x(1) = 0, we say that q is an element of L-1[0, 1] is a non-degenerate potential if the ordinary differential equation x" + q(t) x = 0 has only the trivial solution x(t) = 0 which verifies the boundary condition. Starting with a non-degenerate positive constant potential B, in this paper, we will apply the Pontryagin maximum principle (PMP) in optimal control theory to find the optimal bound r = r(A, B) for any A is an element of[-infinity, B) such that any potential q is an element of L-1[0, 1] satisfying A <= q <= B and integral([0,1]) q(t) dt > r(A, B) is necessarily non-degenerate. Such a non-degeneracy problem can be considered as the dual problem in a series of papers by Li et al.

• 出版日期2015-10-16
• 单位清华大学