I am looking at some papers dealing with the controllability of linear systems under positive controls.
M. Heymann, 1975, Controllability of Linear Systems with Positive Controls: Geometric Considerations.
R. F. Brammer, 1972, Controllability in Linear Autonomous Systems with Positive Controllers.
I was stuck in a problem while looking at the proof for conditions for the null-controllability of linear systems.
In the paper, the following linear system is considered. $$ \begin{align} \dot{x} &= Ax + Bu \\ \end{align} $$ where $x \in \mathbb{R}^{n}$ is the state, and $u \in \mathbb{R}^{m}$ is the control input. In the course of the proof, the condition $$\left<v, \int_{0}^{t}e^{A(t - s)}Bu(s)\,ds \right> \le 0 $$ is given for a vector $v \in \mathbb{R}^{n}$. Then, the paper says that it follows by continuity and a special choice of $u(\cdot)$ that $$<v, e^{At}Bu> \le 0$$ for all $t>0$ and $u \in \Omega$, where $<\cdot, \cdot>$ denotes the inner product operation and $\Omega$ denotes a constaraint set for control input $u$ of the linear system.
Here, I can't understand for which choice of $u(\cdot)$ the first condition leads to the second condition. Can anybody help me? I will really appreciate your help.
The standard choice would be
$$u(s) := B^T e^{-A^T s} \left[ \int_0^t e^{-As}BB^T e^{-A^T s} ds \right]^{-1} B u$$
assuming the system is controllable.