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Curtis and Beard, Successive collocation: An approximation to optimal nonlinear control, Proceedings of the American Control Conference 2001.

The problem is:

Minimize [tex]J(x)=\int_0^{10} x^Tx + u^Tu dt[/tex]

subject to

[tex]\dot{x}=Ax+Bu; \ \ x_0^T=(-12,20)[/tex]

where

[tex]A=\left(\begin{array}{cc}0&1\\-1&2\end{array}\right) [/tex]

[tex]B=\left(\begin{array}{cc}0\\1\end{array}\right) [/tex]

Answer for optimal cost is J*(x)=2221.

However I have try a few times but cannot reproduce this answer. I obtain 2346.5 instead using the methods of Pontryagin's Minimum Principle or Riccati equation. Probably I have misunderstood some concept here.

Using Pontryagin's Minimum Principle, I let the Hamiltonian

[tex]H=x_1^2 + x_2^2 + u^2 + \lambda_1x_2 + \lambda_2(-x_1+2x_2+u)[/tex]

From which I can obtain 5 equations.

[tex]\dot{x}=Ax+Bu [/tex]

[tex]\dot{\lambda}_1 = -\frac{\partial H}{\partial x_1}[/tex]

[tex]\dot{\lambda}_2 = -\frac{\partial H}{\partial x_2}[/tex]

[tex]\frac{\partial H}{\partial u}=0[/tex]

This linear system can be solve subject to the conditions

[itex]x_1(0)=-12, x_2(0)=20, \lambda_1(10)=0 , \lambda_2(10)=0.[/itex]

The solutions are plug into

[tex]J(x)=\int_0^{10} x^Tx + u^Tu dt [/tex].

Any clue where did I gone wrong? Or do anybody know a program that can compute the answer. I know there is a matlab command lqr but it only gives the feedback control not the value of the optimal cost.