I have to solve a hypersensitive optimal control problem (with ODEs) i.e. a problem in which the time interval of interest is long relative to the rates of expansion and contraction of the Hamiltonian dynamics in certain directions in a neighbourhood of the optimal solution.
The question is: how to solve efficiently this problem by means of indirect shooting methods?
A major drawback to solve a hypersensitive problem by indirect simple (or single) shooting method is that, this method requires a very good initial guess for the initial value of the costate variable (i.e. lambda(0) ) which has no physical interpretation. So, if one wants to solve this problem using a shooting method, I think multiple shooting method is the best choice. Generally, when the time interval in a boundary value problem is long, the simple shooting method is not recommended (Please refer to "Introduction to Numerical Analysis" by J. Stoer and R. Bulirsch Page 511).
Large time interval associated with singular perturbations. Such an optimal control problem are the stiffness. Iterative methods for solving such problems are often not working. Our experience suggests that the most reliable way - preforming initial approximation or its components. Even a rough qualitative approach often helps to get to the global optimum. Various asymptotics of singular perturbation problems, or even the parts of asymptotics to help build such approximation. For the shooting method it is possible to find the initial value of costate variable with the construction of some approximation of conditionally stable manifold in your problem.
You may refer to the book, Two-point Boundary Value problems: Shooting Methods, SM Roberts & JS Shipman, Elsevier. It deals stiff problems which are very sensitive. You may have to use the method of complementary functions with orthonormaliztion process.
You may also try some numerical softwares, for example, GPOPS, see http://www.gpops2.com/
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