To find the optimum control law u = u(x) for the process x˙ = f(x, u), the Hamiltonian H = pf 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)); $p˙=−fx′p$ 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 ‖ξ‖ 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˙ = MHXP + HPXM + MHPPM + HXX, 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.

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