A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

详细信息    查看全文

• 作者：Clara Carlota ; ^{ccarlota@uevora.pt" class="auth_mail" title="E-mail the corresponding author} ; Sí ; lvia Chá ; ^{silviaaccha@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; Antó ; nio Ornelas ^{antonioornelas@icloud.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：28B05 ; 34A60
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 July 2016
• 年：2016
• 卷：261
• 期：1
• 页码：296-318
• 全文大小：383 K

文摘

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p=1$p=1$, in any dimension d∈N$d\in \mathbb{N}$, by including a pointwise state-constraint.

More precisely, given a $\bar{x}(\cdot )\in W^{p,1}([a,b],{\mathbb{R}}^d)$ solving the convexified p  -th order differential inclusion ${\mathcal{L}}^p\bar{x}\left(t\right)\in \mathit{co}\{u^0\left(t\right),u^1\left(t\right),\dots ,u^m\left(t\right)\}$ a.e., consider the general problem consisting in finding bang-bang solutions (i.e. ${\mathcal{L}}^p\hat{x}\left(t\right)\in \{u^0\left(t\right),u^1\left(t\right),\dots ,u^m\left(t\right)\}$ a.e.) under the same boundary-data, ${\hat{x}}^{\left(k\right)}\left(a\right)={\bar{x}}^{\left(k\right)}\left(a\right)$ & ${\hat{x}}^{\left(k\right)}\left(b\right)={\bar{x}}^{\left(k\right)}\left(b\right)$ (k=0,1,…,p−1$k=0,1,\dots ,p-1$); but restricted, moreover, by a pointwise state constraint of the type $〈\hat{x}\left(t\right),\omega 〉\le 〈\bar{x}\left(t\right),\omega 〉\forall t\in [a,b]$ (e.g. ω=(1,0,…,0)$\omega =(1,0,\dots ,0)$ yielding ${\hat{x}}_1\left(t\right)\le {\bar{x}}_1\left(t\right)$).

Previous results in the scalar d=1$d=1$ case were the pioneering Amar & Cellina paper (dealing with a056ea60420">${\mathcal{L}}^{p}x(\cdot )=x^{\prime }(\cdot )$), followed by Cerf & Mariconda results, who solved the general case of linear differential operators L^p${\mathcal{L}}^p$ of order p≥2$p\ge 2$ with $C^0\left([a,b]\right)$-coefficients.

This paper is dedicated to: focus on the missing case p=1$p=1$, i.e. using ${\mathcal{L}}^{p}x(\cdot )=x^{\prime }(\cdot )+A(\cdot )x(\cdot )$; generalize the dimension of x(⋅)$x(\cdot )$, from the scalar case d=1$d=1$ to the vectorial d∈N$d\in \mathbb{N}$ case; weaken the coefficients, from continuous to integrable, so that A(⋅)$A(\cdot )$ now becomes a d×d$d\times d$-integrable matrix; and allow the directional vector ω   to become a moving AC function ω(⋅)$\omega (\cdot )$.

Previous vectorial results had constant ω  , no matrix (i.e. A(⋅)≡0$A(\cdot )\equiv 0$) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).