Canonical Duality Approach for Nonlinear Dynamical Systems

TL;DR

This paper introduces a canonical duality method to transform and solve a nonlinear population growth model, enabling the determination of global solutions efficiently through a concave maximization reformulation.

Contribution

The paper develops a novel canonical duality framework that converts a nonconvex nonlinear differential equation into a solvable concave maximization problem, providing a new approach for nonlinear dynamical systems.

Findings

Successfully reformulated the nonlinear population model as a concave maximization problem.

Demonstrated the effectiveness of the method through illustrative examples.

Achieved global optimal solutions for the nonlinear differential equation.

Abstract

This paper presents a canonical dual approach for solving a nonlinear population growth problem governed by the well-known logistic equation. Using the finite difference and least squares methods, the nonlinear differential equation is first formulated as a nonconvex optimization problem with unknown parameters. We then prove that by the canonical duality theory, this nonconvex problem is equivalent to a concave maximization problem over a convex feasible space, which can be solved easily to obtain global optimal solution to this challenging problem. Several illustrative examples are presented.