Definition (Admissible control)

Let $t_{0},t_{1}\in J\subset \mathbb{R}$. A control input $u:[t_{0},t_{1}]\to \mathbb{R}^{m}$ is called an admissible control if it is piecewise Continuous on $[t_{0},t_{1}]$. We denote the class of admissible controls by $$C^{0}_{p}([t_{0},t_{0}],\mathbb{R}^{m})$$