Optimal Control, Optimal Control

A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost functional. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

Oberle, H. J. and Grimm, W., "BNDSCO-A Program for the Numerical Solution of Optimal Control Problems," Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, 1989 The approach that has risen to prominence in numerical optimal control over the past two decades (i.e., from the 1980s to the present) is that of so called direct methods.

T., Practical Methods for Optimal Control Using Nonlinear Programming, SIAM Press, Philadelphia, Pennsylvania, 2001 ). In the latter case (i.e., a collocation method), the nonlinear optimization problem may be literally thousands to tens of thousands of variables and constraints. Given the size of many NLPs arising from a direct method, it may appear somewhat counter-intuitive that solving the nonlinear optimization problem is easier than solving the boundary-value problem. It is, however, the fact that the NLP is easier to solve than the boundary-value problem. The reason for the relative ease of computation, particularly of a direct collocation method, is that the NLP is sparse and many well-known software programs exist (e.g., SNOPT Gill, P. E., Murray, W. M., and Saunders, M. A., User's Manual for SNOPT Version 7: Software for Large-Scale Nonlinear Programming , University of California, San Diego Report, 24 April 2007 ) to solve large sparse NLPs. As a result, the range of problems that can be solved via direct methods (particularly direct collocation methods which are very popular these days) is significantly larger than the range of problems that can be solved via indirect methods. In fact, direct methods have become so popular these days that many people have written elaborate software programs that employ these methods. In particular, many such programs written in FORTRAN include DIRCOL , von Stryk, O., User's Guide for DIRCOL (version 2.1): A Direct Collocation Method for the Numerical Solution of Optimal Control Problems , Fachgebiet Simulation und Systemoptimierung (SIM), Technische Universitt Darmstadt (2000, Version of November 1999).

Usually the strategy is to solve for thresholds and regions that characterize the optimal control and use a numerical solver to isolate the actual choice values in time.

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