A trajectory is obtained for discrete points, what is the procedure for measuring the smoothness of this trajectory. The answer to this question will help me get a clear picture about the convergence rate of Legendre Pseudospectral method, where the rate of convergence is defined as 1/( N^2m/3−1 ). Here m is defined as the smoothness of the optimal trajectory and N is the number of nodes or points. This rate of convergence formula and further discussions can be found in the paper titled " Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems" written by Wei Kang .
Hi I think the paper is talking about the smoothness of the dynamics and not of the trajectory no?
There're some papers that present mesh refinement techniques for nonlinear optimal control problems. One typical method is hp mesh refinement scheme, which originates from the finite element method in computational mechanics. Once the state variables are considered not smooth enough in a sub-interval, one should choose to either divide this sub-interval into more small sub-intervals (corresponding to h method) or increase the degree of approximation polynomial (corresponding to p method).
For example, you can refer to one of my papers titled "An hp symplectic pseudospectral method for nonlinear optimal control" , as attached below.
Respected Dr. Richard Epenoy, this paper talks about the rate of convergence for the optimal cost computed using Legendre PS method. The paper also discusses in detail about how the value of the cost function converges to the optimal cost as the number of nodes (N) is increased and also depends on the smoothness of the optimal trajectory (m). The formula by which this convergence rate is determined is given by: 1/( N^2m/3−1 ). So, the paper talks about convergence of the method and not particularly about the convergence of the dynamics, even though I believe the dynamics will always play a part in the ultimate converge rate of the Legendre PS method.
Respected Dr.Xinwei Wang, first of all thank you for suggesting me the paper. It has cleared certain doubts of mine but my initial question about what is meant by smoothness of trajectory generated by joining discrete points still remains unanswered. Is there a way by which the smoothness of the trajectory generated by joining the values of state or control at the Legendre Gauss Lobatto(lgl) points can be quantified? More specifically is there a measure of the same?
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