General formulation for the numerical solution of optimal control problems

A new improved computational method for a class of optimal control problems is presented. The state and the costate (adjoint) variables are approximated using a set of basis functions. A method, similar to a variational virtual work approach with weighing coefficients, is used to transform the canonical equations into a set of algebraic equations. The method allows approximating functions that need not satisfy the initial conditions a priori. A Lagrange multiplier technique is used to enforce the terminal conditions. This enlarges the space from which the approximating functions can be chosen. Orthogonal polynomials are used to obtain a set of simultaneous equations with fewer non-zero entries. Such a sparse system results in substantial computational economy. Two examples, a time-invariant system and a time-varying system with quadratic performance index, are solved using three different sets of orthogonal polynomials and the power series to demonstrate the feasibility and efficiency of this method.