Relaxed control

Remarks

Every ordinary admissible control $u \colon [0, T] \to U$ gives the relaxed control $\mu_t = \delta_{u(t)}$, the Dirac measure at $u(t)$. Its relaxed equation is $x' = f(t, x, u(t))$, the original one. Example: $U = \set{-1, 1}$ and $\mu_t = \frac12 \delta_{-1} + \frac12 \delta_1$ for all $t$. It is a relaxed control, since $\int_U h \, d\mu_t = \frac12 (h(-1) + h(1))$ is constant in $t$. For $f(t, x, u) = u$ its trajectory has $x' = 0$, a velocity that no ordinary control with values in $\set{-1, 1}$ gives. Non-example: $\mu_t = \delta_{-1} + \delta_1$ is not a relaxed control, since $\mu_t(U) = 2$, so $\mu_t$ is not a probability measure. The velocities of relaxed controls fill the convex hull of $f(t, x, U)$, so the convexity hypothesis of Filippov's existence theorem holds for the relaxed problem. A relaxed control is a Young measure on $[0, T]$ with values in $U$.