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Consider the function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ which is continuously differentiable and strongly convex. Let $x^*$ to be the unique global minimizer of $f$.

Assume that $L_1\|x\| \leq \nabla f(x) \leq L_2 \|x\|$. Consider the dynamic system $\dot{z} = A z + B u$ with $(A,B)$ being stabilizable. Design a controller $u$ depending on $x,z$ for the following dynamics such that $x(t)$ and $z(t)$ both converge to $x^*$ from any initial state $x(0), z(0)$: \begin{align*}\dot{x} = -\nabla f(z),\quad \dot{z} = A z + B u.\end{align*}

user147687
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  • You need more conditions. If $A$ is Hurwitz and $B=0$, then $(A,B)$ is trivially stabilizable, but $z\to 0$ in all cases, and we can always choose $x(0)$ so that $x \not\to x^*$. – copper.hat May 04 '14 at 15:52
  • There is one more condition: There exists $u^ $ such that $Ax^ + Bu^ * = 0$. – user147687 May 20 '14 at 20:53

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Consider the variable given by $e=z-x^\ast$. Taking its derivative, you get \begin{equation*} \dot{e}=\dot{z}=A(e+x^\ast) + Bu. \end{equation*} Now, assuming that $B$ is invertible, with the coordinate change $u=v-B^{-1}Ax^\ast$ the last equation yields \begin{equation*} \dot{e}=Ae+Bv. \end{equation*} Since $(A,B)$ are stabilizable, there exists a feedback law $\phi=-Ke$ such that the origin is globally asymptotically stable for \begin{equation*} \dot{e}=(A-KB)e. \end{equation*} This implies $e(t)\to0$. Consequently, $z=x^\ast$ and $x=x^\ast$.

ℂybernetician
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