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.

  • $\begingroup$ For your information, there are other papers on the topic that you may find interesting. For instance, the papers by Saperstone, "Global controllability of linear systems with positive controls", Frias, Verduzco, Leyva, and Carrillo, "On controllability of linear systems with positive control", Joseph, "Controllability of a Linear System with Nonnegative Sparse Controls", and Loheac, Trelat, and Zuazua, "Nonnegative control of finite-dimensional linear systems" $\endgroup$
    – KBS
    Mar 22, 2022 at 12:30
  • $\begingroup$ @RBH I understand that the inequality can be written as $\int_{0}^{t} \left<v, e^{A(t-s)}Bu(s) \right> ds \le 0$, but can $v$ and $e^{-As}$ be commuted? $\endgroup$
    – minii93
    Mar 22, 2022 at 16:12
  • $\begingroup$ @KBS Thank you for your recommendations. I will look at those papers too :) $\endgroup$
    – minii93
    Mar 22, 2022 at 16:14
  • $\begingroup$ @minii93 I am sorry, that was a mistake. It seems that one can conclude from the inequality in your comment. $\endgroup$
    – ahdahmani
    Mar 22, 2022 at 19:58
  • $\begingroup$ @RBH I'm not sure if there is a way to go any further from the inequality. But thank you for your trial! $\endgroup$
    – minii93
    Mar 22, 2022 at 23:59

1 Answer 1


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.

  • $\begingroup$ Thank you for your answer :) The choice in your answer clearly leads to the given inequality. However, I still have trouble because the proof in the paper seems to require that $u(s)$ be an admissible control, which means that $u(s)$ also should be in the constraint set $\Omega$ for all $t>0$. (I am sorry that I forgot to mention this requirement. I realized this later after looking at the paper several times.) Is $u(s)$ in your answer satisfies $u(s) \in \Omega$ given that $u \in \Omega$? Or is there any other choice to enforce $u(s) \in \Omega$? $\endgroup$
    – minii93
    Mar 31, 2022 at 1:47
  • $\begingroup$ All the related papers refer to the result of R. F. Brammer, but I am really doubtful whether the proof in the paper is solid, because there is no explanation on how to choose $u(s)$, and no other literature that reproduces the result is found. $\endgroup$
    – minii93
    Mar 31, 2022 at 1:55

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