## Abstract

To find the optimum control law u = u(x) for the process x = f(x, u), the Hamiltonian H = p′f is formed. The optimum control law can be expressed as u = u^{*} = σ(p, x), where u^{*} maximizes H. The transformation from the state x to the “costate” p entails the analytic solution of the nonlinear system: x = f(x, σ(p, x)); p = - f^{′}_{x} p with boundary conditions at two points. Since such a solution generally can not be found, we seek a quasi-optimum control law of the form u = σ(P + Mξ, x), where x = X + ξ with small, and X, P are the solutions of a simplified problem, obtained by setting ξ = 0 in the above two-point boundary-value problem. We assume that P(X) is known. It is shown that the matrix M satisfies a Riccati equation, - M = MH_{XP} + H_{PX} M + MH_{PP}M + H_{XX} and can. be computed by solving a linear system of equations. A simple example illustrates the application of the technique to a problem with a bounded control variable.

Original language | English (US) |
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Pages (from-to) | 437-443 |

Number of pages | 7 |

Journal | Journal of Fluids Engineering, Transactions of the ASME |

Volume | 88 |

Issue number | 2 |

DOIs | |

State | Published - 1966 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mechanical Engineering