Solutions to Differential Algebraic Inequalities with Composite Bernstein Polynomials

Abstract: The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in mind, we introduce two collocation methods for solving general ordinary differential equations using composite Bernstein polynomials to preserve the basis properties while improving convergence. Of particular note is the integration based method which uses a minimal number of variables to describe the resulting composite polynomial, reducing computational effort. In addition, we exploit the convex hull property of the Bernstein polynomial basis in order to enforce inequality constraints in differential algebraic inequalities, highlighting the benefits of the basis in function approximation. Solutions to six numerical examples are provided as well as discussion of the advantages and disadvantages of the proposed solution methodologies.