The Bryson-Denham optimal control problem is a benchmark test problem for optimal control algorithms.

The parameter *u* (acceleration) is adjusted over the time horizon from a starting time of zero to a final time of one. The variable *x* is the position and *v* is the velocity.

$$\min J = \frac{1}{2} \; \int_0^1 u^2(t) dt$$

$$\mathrm{subject\;to}$$

$$\frac{dx(t)}{dt} = v(t) $$

$$\frac{dv(t)}{dt} = u(t) $$

$$x(0) \; = \; x(1) \; = \; 0$$

$$v(0) \; = \; -v(1) \; = \; 1$$

$$x(t) \le \ell, \; \ell=\frac{1}{9}$$

Attach:bryson_denham_solution.png Δ

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