Is there anybody with an insightful note or explanation on how to establish the stability of the solution of ODE (bvp) using Poincare-Lyapunov Theorem?
Reduction of the bvp to initial values problem and generation of the Lyapunov function do not incorporate the boundary conditions. We believe these are not acceptable.
Do you know how to include the Neumann boundary condition into Jacobian matrix or how to generate Lyapunov function suitable to establish the stability of the solution of ODE bvp; see the attached problem.
The notion of stability for BVP is different than the notion of stability for IVP. Hence the standard linearisation theorem cannot be directly applied.
The main question that you want to address is: what does it mean for a solution of a BVP to be stable (with respect to what perturbations). So, you need a definition.
If the problem comes from a variational formulation based on the energy, then looking at the positive definiteness of the second variation is the standard way to define and prove stability. Intuitively, the question is to see if nearby solutions around a given solution have lower energy. You can see examples in my paper with Thomas Lessinnes that deals with Neuman conditions. In particular we give a full computation for a simple 2nd order system for which we establish stability. Maybe it would be helpful if this is what you have in mind. Otherwise, you will find references in there.
The reference is
T. Lessinnes and A. Goriely, 2017 Geometric conditions for the positive definiteness of the second variation in one-dimensional problems. Nonlinearity 30, 2023..
And I believe that it can be found on research gate (or ask for it).
The notion of stability for BVP is different than the notion of stability for IVP. Hence the standard linearisation theorem cannot be directly applied.
The main question that you want to address is: what does it mean for a solution of a BVP to be stable (with respect to what perturbations). So, you need a definition.
If the problem comes from a variational formulation based on the energy, then looking at the positive definiteness of the second variation is the standard way to define and prove stability. Intuitively, the question is to see if nearby solutions around a given solution have lower energy. You can see examples in my paper with Thomas Lessinnes that deals with Neuman conditions. In particular we give a full computation for a simple 2nd order system for which we establish stability. Maybe it would be helpful if this is what you have in mind. Otherwise, you will find references in there.
The reference is
T. Lessinnes and A. Goriely, 2017 Geometric conditions for the positive definiteness of the second variation in one-dimensional problems. Nonlinearity 30, 2023..
And I believe that it can be found on research gate (or ask for it).
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