The time-optimal design of linear dynamic systems

The following general time-optimal design problem is studied: determine a real constant square matrix, , subject to specified constraints, to minimize the transit time between specified endpoints while satisfying the vector differential equation . Two specific kinds of constraints on are considered: 1) where the individual elements of are free but the matrix as a whole must satisfy where is a specified homogeneous function of the elements of and is a given upper bound and 2) where the elements of are individually bounded. Theoretical results show that both problems can be solved by first solving a related minimum cost fixed time problem. The latter problem is solved iteratively by using the generalized Newton-Raphson method for two point boundary value problems to provide a set of linear equations at each iteration. The cost function is then minimized subject to these equations using appropriate optimization techniques.