Bounding the solutions of parameter dependent nonlinear ordinary differential equations

This paper presents two techniques for generating rigorous bounds on the solution of parameter dependent nonlinear ordinary differential equations. The first technique is an extension of differential inequalities that enables the construction of tight time varying state bounds by utilizing prior knowledge of the solution of the differential equations. The second technique provides a method for constructing pointwise in time convex and concave relaxations of the image of the solution of a system of parameter dependent differential equations on subsets of a Euclidean space. Two examples are presented to demonstrate the construction of the bounds. The examples include a brief discussion of a computer program written to automatically generate the bounds.