Convergence rate for a Radau hp collocation method applied to constrained optimal control

Abstract

For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.

Modified Radau Collocation method for Solving Optimal Control Problems with Nonsmooth Solutions Part I: Lavrentiev Phenomenon and the Search Space

“…It has been shown in Refs. [23,24,25,26,27] that under conditions of smoothness and coercivity, a Gaussian quadrature direct LGR collocation will converge to a local minimizer of the continuous optimal control problem. A locally minimizing solution may not, however, be obtained when the problem does not satisfy such coercivity conditions (for example, a problem with a nonsmooth optimal control).…”

“…In addition, a convergence theory has recently been developed using Gaussian quadrature collocation. Research on this theory had demonstrated that, under certain assumptions of the smoothness and coercivity, an hp Gaussian quadrature method that employs either LG or LGR collocation points converges to a local minimizer of the optimal control problem [23,24,25,26,27] While Gaussian quadrature orthogonal collocation methods are well suited to solving optimal control problems whose solutions are smooth, it is often the case that the solution of an optimal control problem has a nonsmooth optimal control [28]. The difficulty in solving problems with nonsmooth control lies in determining when the nonsmoothness occurs.…”

“…For optimal control problems where the solution is nonsmooth the convergence theory developed in Refs. [23,24,25,26,27] is not applicable. Consequently, when the solution of an optimal control problem is nonsmooth, an hp method may not converge to a local minimizer of the optimal control problem.…”

Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation

“…In a p method, the number of intervals is fixed, and convergence is achieved by increasing the degree of the approximation in each interval. To achieve maximum effectiveness, p methods have been developed using orthogonal collocation at Gaussian quadrature points [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. For problems whose solutions are smooth and well-behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [20,21,22,23,24].…”

“…To achieve maximum effectiveness, p methods have been developed using orthogonal collocation at Gaussian quadrature points [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. For problems whose solutions are smooth and well-behaved, a Gaussian quadrature orthogonal collocation method converges at an exponential rate [20,21,22,23,24]. Gauss quadrature collocation methods use either Legendre-Gauss (LG) points [8,12,13,20,21,22,23], Legendre-Gauss-Radau (LGR) points [11,12,13,14,20,23,24], or Legendre-Gauss-Lobatto (LGL) points [7].…”